Update

Alexander Shvydky (see his work here) pointed out that many of the results in this tutorial can also be obtained using spherical Fourier transformations. For details, see GraefS2 and Gauß-type Quadratures on S2. Some t-designs still remain especially interesting, including N=132, l_max=19 and N=186, l_max=23, because they need only 3 and 4 distinct weights. That is attractive for technical applications such as laser fusion reactor design.

Mars Mission: The Lost Harmony

2064, Low Mars Orbit, Mission Commander: You

Prologue

The red planet hung below us, lit by the faint glow of Phobos and Deimos. Our fleet of 42 spacecraft waited in orbit. Each ship was one node in a large celestial array, ready for your command.

The mission had been planned for years. The 43rd ship, Horizon Prime, carried a quantum computing array that was supposed to compute the formation in real time. Two weeks ago, a rogue asteroid destroyed it. We reached Mars with the fleet intact, but without the machine meant to guide it.

Now we rely on an aging mainframe. It comes from a time before real-time gravitational corrections and neural orbital optimization. Jobs that should take nanoseconds now move slowly while we enter equations by hand.

Mission Objective: The Lost Signal

Our task is unchanged. We must arrange the fleet as a spherical t-design. This formation minimizes interference from the Martian surface and maximizes the spatial resolution of our interferometric imaging array.

The Challenge

Every ship needs a carefully chosen orbit. Together, the fleet must sample the celestial sphere in a way that lets us resolve subsurface structure on Mars, detect deep crustal anomalies, and perhaps even hint at past microbial life buried beneath the dust.

The Problem

Without Horizon Prime, reaching the highest possible l_max is much harder.

Bonus Task

The admiral can still send more ships if needed. So we also want to know how many additional ships are required to increase l_max by one or two.

Understanding l_max

In simple terms, l_max determines our maximum imaging resolution. The higher the value, the finer the details we can extract from our interferometric measurements.

In a perfect world, with N = 43 ships and the quantum processor, we might have reached an ideal spherical t-design.
With N = 42 ships and only our sluggish classical computer, our goal is to push l_max as high as possible.

The Battle Against Time

We sit aboard the Aurora Ascendant while the mainframe grinds through the equations. The fleet must remain stable, but every orbital correction burns fuel. Time is short. We have one chance to get this right before we are forced to spend resources just to hold position.

The fate of the mission rests on your decision. What is your command?

Problem Analysis

Our task is to find a spherical t-design. That means placing points on the unit sphere so that all spherical harmonics cancel up to and including a given l_max.

Fortunately, the old mainframe still carries an archived internet snapshot from 2025. It contains several useful references:

sloanes archive

The closest match is 3.42.8, a 42-point spherical 8-design, so l_max=8. The smallest l_max=9 solution needs 48 ships. For l_max=10 it needs 60 ships.

https://web.maths.unsw.edu.au/~rsw/Sphere/EffSphDes/sf.html

This source lists ship counts 42, 50, and 62 for l_max = 8, 9, and 10.

https://web.maths.unsw.edu.au/~rsw/Sphere/EffSphDes/ss.html

Here the ship counts for l_max = 7, 9, and 11 are 32, 48, and 70.

The admiral is not impressed by those numbers, so we ask the ancient AI on the mainframe. It suggests weighted spherical designs such as Lebedev rules. In principle, weights add freedom and may allow a higher l_max:

"For most computational integration tasks, having a weighted spherical design is not a problem—it’s entirely standard. From a purely numerical integration perspective, weights are absolutely normal and pose no fundamental barrier to computing accurate results."

Unfortunately, the Lebedev approach often relies on negative or highly uneven weights. That does not work with the antennas on our ships. So we keep searching. Deep in the archive, we find an old tutorial covering exactly this problem:

The old Tutorial from 2025

This Tutorial

Shows how to construct perfect weighted t-designs for l_max = 10, 11, and 12 with as few points as possible.
Shows how to design the fitness function and use evolutionary optimization to find exact solutions.
Shows how to use all CPU cores by parallelizing the optimisation process.

This example also shows why fcmaes fits this kind of problem well. The search space is continuous, constrained, and highly non-linear. A restart-based global optimizer can work through it without requiring gradients or a carefully crafted initial design.

Tutorial Code

The complete code for this tutorial is at tdesign.py

Fitness Function

First we define t_design_error. It computes the error of a weighted t-design from a list of N spherical points, given by their theta and phi angles, and a list of weights.

t_design_error checks whether the spherical harmonics cancel up to and including a given l_max, and measures the remaining error.
It computes real spherical harmonics from the associated Legendre function in scipy.special.lpmv . We could also use scipy.special.sph_harm_y and convert its complex result, but experiments showed that this is slower. Because numpy/scipy evaluates all points at once, an array-based implementation is already fast enough. There is no need for numba.
The original version of the symmetry function was contributed by Stephen Pollaine, a specialist in fusion-reactor laser design. The Laboratory for Laser Energetics (LLE) at the University of Rochester used a spherical 9-design with 60 lasers placed on a modified soccer ball, achieving symmetry for all t ⇐ l_max = 9. Weighted designs would be much harder to realize there because the irregular point distribution would require each laser to run at a different intensity.

def real_spherical_harmonics(l_max, theta, phi):
    theta = np.asarray(theta)
    phi   = np.asarray(phi)
    # Compute cos(theta) for the Legendre functions.
    cos_theta = np.cos(theta)
    Y = {}  # dictionary to store the spherical harmonics
    for l in range(l_max+1):
        for m in range(l+1):
            # Compute the normalization factor
            norm = np.sqrt((2*l+1)/(4*np.pi) * math.factorial(l-m)/math.factorial(l+m))
            # Compute the associated Legendre function for order m and degree l.
            P_lm = lpmv(m, l, cos_theta)
            if m == 0:
                # For m = 0, no extra trigonometric factor is needed.
                Y[(l, 0)] = norm * P_lm
            else:
                # For m > 0, compute both the cosine and sine components.
                Y[(l, m)]  = np.sqrt(2) * norm * P_lm * np.cos(m * phi)
                Y[(l, -m)] = np.sqrt(2) * norm * P_lm * np.sin(m * phi)
    return Y

def weighted_spherical_harmonics(l_max, theta, phi, weights):
    weights = normalize_weights_to_average_one(weights)
    # 1) Compute the SH values at all points (unweighted).
    Y = real_spherical_harmonics(l_max, theta, phi)
    # 2) Multiply each Y_{l,m} by the corresponding weight and sum up.
    W = {}
    for l in range(l_max+1):
        for m in range(-l, l+1):
            # element-wise multiply by w_i and sum
            W[(l,m)] = np.sum(Y[(l,m)] * weights)
    return W

def symmetry_error(Y, N, l_max):
    """
    For each degree l from 0 to l_max, compute the sum over m of the square of the
    (pointwise) sums of the spherical harmonic values. Then multiply by 4*pi/(N^2).

    Parameters:
      Y : dict mapping (l, m) -> array of shape (N,)
      N : int, number of points
      l_max : maximum degree

    Returns:
      s : numpy array of shape (l_max+1,)
    """
    s = np.zeros(l_max+1)
    # For l = 0 (only m=0 exists)
    s[0] = np.abs(np.sum(Y[(0, 0)]))**2
    for l in range(1, l_max+1):
        for m in range(-l, l+1):
            s[l] += np.abs(np.sum(Y[(l, m)]))**2
    s[np.abs(s) < 1.e-20] = 0.
    return s * 4*np.pi / (N**2)

def symmetry(pts, l_max, weights=None):
    """
    Compute a symmetry measure for a set of points.
    pts has 2 columns, it is assumed to be [theta, phi];

    Returns:
      An array of length l_max+1.
    """
    pts = np.array(pts)
    # Assume pts[:,0]=theta, pts[:,1]=phi
    if weights is None:
        Y = real_spherical_harmonics(l_max, pts[:, 0], pts[:, 1])
    else:
        Y = weighted_spherical_harmonics(l_max, pts[:, 0], pts[:, 1], weights)
    N = pts.shape[0]
    error = symmetry_error(Y, len(pts), l_max)
    # Create a multiplier: 1 / (2*l + 1) for l=0,...,l_max
    mult = 1. / (2*np.arange(0, l_max+1) + 1)
    return error * mult

def t_design_error(points, l_max, weights=None):
    syms = symmetry(points, l_max, weights)
    return sum(syms[1:l_max+1])

Utilities

Next we need a few helper functions:

cartesian_to_spherical converts Cartesian 3D points to spherical (theta,phi) coordinates.
x_to_points extracts the spherical (theta,phi) points from the argument vector.
normalize_weights_to_average_one rescales the weights so their sum is N, matching the unweighted case where every weight is 1.
fibonacci_sphere is a fairly rough approximation suggested by the AI, but it works surprisingly well as an initial guess. I tried "better" alternatives, but the gains were small.

def cartesian_to_spherical(points):
    spherical_coords = np.empty((len(points),2))
    for i, (x, y, z) in enumerate(points):
        theta = np.arccos(z)  # Polar angle
        phi = np.arctan2(y, x)  # Azimuthal angle
        if phi < 0:
            phi += 2*np.pi
        spherical_coords[i] = (theta, phi)
    return spherical_coords

def x_to_points(x): # stack theta, phi into an array of shape (N,2)
    N = len(x)//2
    return np.stack([x[:N], x[N:]], axis=1)

def normalize_weights_to_average_one(weights):
    weights = np.array(weights)
    N = len(weights)
    sum = np.sum(weights)
    if sum == 0:
        return np.ones(N)
    alpha = N / sum
    return alpha * weights

def fibonacci_sphere(N):
    points = []
    phi = np.pi * (3. - np.sqrt(5.))  # Golden angle
    for i in range(N):
        y = 1 - (i / float(N - 1)) * 2  # y goes from 1 to -1
        radius = np.sqrt(1 - y * y)          # radius at y
        theta = phi * i                      # golden angle increment
        x = np.cos(theta) * radius
        z = np.sin(theta) * radius
        points.append([x, y, z])
    return np.array(points)

Parallel Optimisation using the BiteOpt algorithm.

Now we can run the optimization. We use the fcmaes parallel retry mechanism together with the BiteOpt algorithm.

BiteOpt has one major drawback:

Parallelization is only available at the restart level. It cannot parallelize population evaluation inside a single run.

fcmaes also provides algorithms such as Differential Evolution, CMA-ES, and CRFM-NES that support parallel population evaluation. In this case, however, BiteOpt’s dynamic adaptation makes up for that limitation.

def optimize_weights(N, l_max, workers=20, max_evals=1000000, max_iters=1):

    def fit(x):
        points = x_to_points(x[:2*N])
        weights = x[2*N:]
        return t_design_error(points, l_max, weights)

    x0 = np.array(list(cartesian_to_spherical(fibonacci_sphere(N)).flatten()) + [1]*N)

    dim = N*3 # we encode the input by concatenating the theta, phi and weight vectors
    # apply BiteOpt using parallel restart
    for i in range(max_iters):
        result = retry.minimize(wrapper(fit),
                                bounds=Bounds([0]*dim,[np.pi]*N + [2*np.pi]*N + [2]*N),
                                num_retries=workers, workers=workers,
                                stop_fitness = 0,
                                optimizer=Bite_cpp(max_evals, guess=x0, stop_fitness=0))
        x0 = result.x
    points = x_to_points(result.x[:2*N])
    weights = normalize_weights_to_average_one(result.x[2*N:])

Results

The results below matter for two reasons. First, they improve on the unweighted point counts from the references above. Second, they illustrate that fcmaes can find exact constructions in a difficult continuous optimization problem with many variables and constraints.

N=42, l_max = 10

We found a perfect solution for l_max=10 using only 42 points.

42 points (theta,phi) 42 different weights:
theta: [0.912090306934545, 2.746761190754236, 2.2986410362170338, 1.745862929099355, 0.6087172979572448, 0.8750179958146024, 2.003199964794502, 1.914019919169261, 1.3719018453624228, 2.0783535824454447, 3.0416516361988566, 1.9030536604204784, 2.5555780724806496, 0.6356259587455823, 1.0585497682737501, 1.4679930768047569, 0.38392536310433395, 2.42688314878376, 1.9802333048552152, 2.123718951551832, 1.5987738995397742, 1.377057139594396, 2.061148509982718, 1.4517070680861281, 1.4353585379167855, 1.1643720284459766, 0.8907621209089062, 1.795193579750673, 1.4049928360005761, 0.9936439341556538, 1.5103957757421282, 1.1852333762660772, 0.27975686369694425, 2.447217284242543, 1.4814488883229116, 0.4161660936306942, 2.53426390482221, 2.5343394636596, 0.9022157881722463, 2.049939403516931, 0.8567644078993933, 1.8001113473988923]
phi: [1.6189186869012622, 4.417890964353101, 3.925717844888819, 4.165114459914451, 1.024244687866028, 3.571479923677028, 5.6761304055135895, 6.205336584239582, 3.8328097495962488, 0.4743672484760749, 0.986147115054497, 3.5546001474686792, 2.0704647095249578, 4.915642480324595, 2.1949792038737597, 4.610023292590779, 3.8262559858367093, 5.258320077680217, 2.977183013149219, 4.598896945880547, 2.0519720817232328, 6.162612488037494, 2.370264024699624, 5.604941266160562, 1.5141547998093383, 5.098644649280042, 2.8593015359409786, 5.096594585219709, 3.238975312951073, 4.308797093290831, 0.4550427435469594, 0.9465296848185534, 6.155149089475483, 1.0303914048801714, 2.6508963308118045, 2.2568415894699476, 3.1129991266539556, 6.209840414334106, 0.2816873934360353, 1.6296533929828394, 5.769891533320035, 1.03323260646458]
weights: [0.8702450096407891, 0.891068953969204, 0.8955518026815953, 0.8990632826491868, 0.9112149039905258, 0.919575366373204, 0.9247050688209141, 0.9249481421631185, 0.9374736698059064, 0.9441339800127497, 0.9460695594614713, 0.9624139527428935, 0.9626781153524502, 0.9694810363793928, 0.9706018847174712, 0.9731112520873746, 0.9777700998281489, 0.9796868643955328, 0.9803515510110513, 0.9886895472366434, 0.9945191426107445, 1.0026057961079573, 1.006543582504591, 1.006936853157852, 1.02102322906593, 1.0237625319134556, 1.0366919458662687, 1.0463681488559085, 1.0481872408048207, 1.0484653593209248, 1.0569115307744548, 1.0613689554089596, 1.0660550338853683, 1.0686923989463608, 1.0696719846263136, 1.0700103666558665, 1.07433486681468, 1.0891862263724188, 1.0909117206860226, 1.091665763486926, 1.0964660660903478, 1.1007872127242062]

symmetries =  [1.         0.         0.         0.         0.         0.
 0.         0.         0.         0.         0.         0.1241871
 0.06575862 0.02474867]
symmetry error = 0.0

There are only minor errors at l=11, l=12, and l=13.

N=48, l_max = 11

We found a perfect solution for l_max=11 using only 48 points.

48 points (theta,phi) 2 different weights:
theta: [1.2336997948453468, 1.9078928587444466, 1.9078928587444466, 1.2336997948453468, 1.4598542464697268, 1.6817384071200663, 1.6817384071200663, 1.4598542464697268, 2.7853163157342573, 0.3562763378555358, 0.3562763378555358, 2.7853163157342573, 2.7853163157330116, 0.35627633785678164, 0.35627633785678164, 2.7853163157330116, 1.45985424646999, 1.6817384071198034, 1.6817384071198034, 1.45985424646999, 1.907892858745844, 1.2336997948439494, 1.2336997948439494, 1.907892858745844, 0.8824843417477756, 2.2591083118420174, 2.2591083118420174, 0.8824843417477756, 1.8616874665689487, 1.2799051870208444, 1.2799051870208444, 1.8616874665689487, 2.259108311842379, 0.8824843417474144, 0.8824843417474144, 2.259108311842379, 1.2799051870219895, 1.8616874665678038, 1.8616874665678038, 1.2799051870219895, 2.370415958253471, 0.7711766953363222, 0.7711766953363222, 2.370415958253471, 0.771176695336047, 2.370415958253746, 2.370415958253746, 0.771176695336047]
phi: [4.8299771574945405, 1.453208149685046, 4.594800803274839, 1.6883845039047471, 5.943919993516647, 0.33926531366293894, 3.480857967252732, 2.8023273399268542, 3.4646084238352346, 2.8185768833443516, 5.960169536934145, 0.3230157702454418, 5.035404750628221, 1.2477805565513658, 4.389373210141159, 1.8938120970384273, 4.373123666720354, 1.9100616404592323, 5.051654294049025, 1.231531013130561, 6.165597130069957, 0.11758817710962952, 3.2591808306994223, 3.024004476480164, 4.331919117226351, 1.9512661899532349, 5.092858843543028, 1.1903264636365583, 2.416651002899945, 3.866534304279641, 0.7249416506898482, 5.558243656489738, 3.5220625167467206, 2.7611227904328657, 5.902715444022658, 0.3804698631569275, 5.437330631074623, 0.8458546761049632, 3.9874473296947563, 2.29573797748483, 2.7174950230862995, 3.5656902840932867, 0.42409763050349364, 5.859087676676093, 5.1364866108865845, 1.146698696293002, 4.288291349882795, 1.9948939572967912]
weights: [0.9495675564588315, 0.9495675564588315, 0.9495675564588315, 0.9495675564588315, 0.9495675564588315, 0.9495675564588315, 0.9495675564588315, 0.9495675564588315, 0.9495675564588315, 0.9495675564588315, 0.9495675564588315, 0.9495675564588315, 0.9495675564588315, 0.9495675564588315, 0.9495675564588315, 0.9495675564588315, 0.9495675564588315, 0.9495675564588315, 0.9495675564588315, 0.9495675564588315, 0.9495675564588315, 0.9495675564588315, 0.9495675564588315, 0.9495675564588315, 1.0504324435411683, 1.0504324435411683, 1.0504324435411683, 1.0504324435411683, 1.0504324435411683, 1.0504324435411683, 1.0504324435411683, 1.0504324435411683, 1.0504324435411683, 1.0504324435411683, 1.0504324435411683, 1.0504324435411683, 1.0504324435411683, 1.0504324435411683, 1.0504324435411683, 1.0504324435411683, 1.0504324435411683, 1.0504324435411683, 1.0504324435411683, 1.0504324435411683, 1.0504324435411683, 1.0504324435411683, 1.0504324435411683, 1.0504324435411683]

symmetries =  [1.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
 0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
 0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
 1.14621038e-01 9.45072435e-02 7.03566689e-05]
symmetry error = 0.0

There are only minor errors at l=12, l=13, and l=14.

N=58, l_max = 12

We found a perfect solution for l_max=12 using only 58 points.

58 points (theta,phi) 58 different weights:
theta: [1.4625732945102392, 1.5124900499321432, 0.749948790760706, 1.8055944536325494, 2.061310567834104, 2.23226434495121, 1.6456869322125696, 1.564732590741013, 1.4957407024337483, 0.8041339533280961, 0.9749327633824536, 2.2530865495040144, 2.4196140732903504, 1.1234157722963025, 2.9120625356827894, 0.538112422091656, 2.579071228464289, 1.1673808518901907, 1.1958027838302598, 1.6991198381568426, 2.6515073694563775, 2.198845072848418, 0.3573606091636152, 0.5486085347557721, 0.8883386793240915, 0.6789416862870981, 1.6670247565036371, 1.9595090731976386, 1.0824457996934733, 1.7217704594909287, 2.895787806997049, 1.117677748388637, 1.8673844360870282, 1.2191549430750221, 1.927285154465101, 0.7154107707199917, 1.3440884565485327, 2.0172009133418474, 1.3940910225834287, 2.1886125856021015, 1.554395366080808, 0.9025077721183872, 2.4776210135817545, 2.681104400095342, 1.0223476998510461, 1.7910367280457309, 2.4510517150186244, 1.657081274078465, 0.36673989020787107, 1.2852150762492365, 2.0607184906845792, 1.2815538491964542, 2.15571242979144, 0.19737192872794945, 1.9462168306606145, 1.4615780340408329, 2.414340849387238, 0.7441936799638402]
phi: [6.093046430729653, 2.497984534320207, 4.661672382384337, 6.062793086597214, 0.5029541743695377, 0.9998465280485642, 4.763853560130005, 2.0417176855383357, 1.6291640801891727, 0.4138427514745532, 3.645646402536016, 3.5181797115811313, 1.5350347118450354, 2.17878262234884, 4.66427341850738, 1.5386371524678506, 0.52761351708754, 0.06308351359734513, 4.730861138756998, 5.594023873585617, 5.872128820146829, 3.383892846989375e-08, 0.4911347637466857, 3.993892843564511, 1.0227092186157694, 2.328329180993931, 2.913667422738485, 2.397175382603447, 4.18228214464703, 0.8458413657665363, 1.517021562991158, 2.749032932035534, 1.3415982723199287, 5.744060604676913, 5.128239532310928, 3.1093108232963274, 1.1678137651302691, 1.8546938350783773, 5.215498358161127, 5.64678437287598, 4.295364236137228, 5.279284759766376, 4.140574036565011, 3.1297111908066637, 1.6539868090786796, 3.4021765877688606, 2.3041965870967176, 0.27416250207874604, 5.263709270011244, 3.2452205366176208, 4.518646321073305, 0.6095694593202984, 2.9201672205986777, 2.905130104600923, 3.966649614743391, 3.7743774240100443, 4.997897634883493, 5.966385688898415]
weights: [0.5950876792889014, 0.8563462352677825, 0.8737840492391372, 0.8763365519399436, 0.8765735986256921, 0.8871945100879534, 0.8901570382398424, 0.9032691525738665, 0.9125744573276605, 0.9148515573746437, 0.9367675355371524, 0.9496026529334867, 0.9510053307449832, 0.9553472437314717, 0.963340169956153, 0.967971978400785, 0.9821983006942473, 0.9854613463006211, 0.9873931636061192, 0.9915489714651599, 0.9918567334238754, 0.9932899851366229, 0.9963303152777864, 1.0046071663622314, 1.007529744120311, 1.0095982144815758, 1.0144689262853812, 1.0161286682563666, 1.0265946992516373, 1.0303076649347143, 1.031102598230703, 1.033585586061885, 1.03594510604714, 1.0370671188671112, 1.039242114356098, 1.0426135657875877, 1.044301868705987, 1.0464884771017846, 1.0495777954528496, 1.049865102651683, 1.0534229170116134, 1.0545234695792765, 1.055638455125709, 1.0568405650969255, 1.0568586778257856, 1.060560981064404, 1.0632468849300942, 1.0633170371517284, 1.063811351416183, 1.0691801255766094, 1.0698326759861898, 1.075775796958918, 1.0760014680058143, 1.0762435812041282, 1.0840576179023762, 1.085668298658389, 1.088599071254965, 1.089108051121954]

symmetries =  [1.         0.         0.         0.         0.         0.
 0.         0.         0.         0.         0.         0.
 0.         0.0934051  0.06649197 0.01707206]
symmetry error = 0.0

There are only minor errors at l=13, l=14, and l=15.

Summary

BiteOpt parallel restart is well suited to searching for weighted spherical t-designs.
We found N=42, l_max=10, N=48, l_max=11, and N=58, l_max=12 spherical t-designs. These go well beyond what is possible without weights.
Maybe some future space mission commander will read this when planning a mission.
Note that we found a significant performance boost of about 15% using Python 12 compared to Python 10 on our 16 core AMD 9950x CPU using Linux Mint 22.
Note that we found The Answer to the Ultimate Question of Life, the Universe, and Everything , namely the question: "how many points on a unit sphere are required, so that all the spherical harmonics cancel out to zero up to and including a given l_max = 10" . Exercise: Can you find the correct answer using any existing AI applied to this prompt?

Laser Fusion Reactor Design

As mentioned earlier, the Laboratory for Laser Energetics (LLE) at the University of Rochester used a spherical 9-design, arranging 60 lasers to converge at the center of the sphere (see Shvydky). Weighted designs are harder to realize in this setup because the irregular point distribution would require a different intensity for each laser.

We can partially address this by adjusting the optimization process. The construction of uniform t-designs uses techniques we can adapt here (see Improved Snub Cube). In particular, permutation and mirroring let us derive 12 symmetric points from a single reference point.

def generate_orbits(N, x): # generate 12 symmetries for each reference point

    def generate_orbit(A, B, C): # apply 12 symmetries
        cart = []
        for perm in [(A, B, C), (B, C, A), (C, A, B)]: # permute
            for signs in [(1, 1, 1), (1, -1, -1), (-1, 1, -1), (-1, -1, 1)]:
                x, y, z = [sign * coord for sign, coord in zip(signs, perm)]
                cart.append((x, y, z))
        return cartesian_to_spherical(cart)

    ref_points = x_to_points(x)
    theta = ref_points[:,0]
    phi = ref_points[:,1]
    x, y, z = spherical_to_cartesian(theta, phi)
    points = []
    for A, B, C in zip(x,y,z):
        points.extend(generate_orbit(A, B, C))
    return np.array(points[:N])

This method greatly reduces the number of decision variables. We only optimize the reference points and generate the 12 symmetries automatically. The number of weights stays the same, but the search space becomes smaller, which speeds up the optimization.

    def fit(x):
        points = generate_orbits(N, x[:2*n])
        weights = x[2*n:]
        return t_design_error(points, l_max, weights)

Experiments revealed another benefit. The number of distinct weights often drops sharply, so only a few different intensity levels are needed. For N = 72, for example, we only need two laser types with different intensities. Below are the corresponding solutions for 48, 60, 72, 132, and 192 lasers:

48 beams: 2 intensities, l_max = 11
60 beams: 5 intensities, l_max = 12
72 beams: 2 intensities, l_max = 14
132 beams: 3 intensities, l_max = 19
192 beams: 4 intensities, l_max = 23

We also show intensity distribution plots. Compare them to Shvydky for uniform designs. In particular, the beam designs with at least 72 beams show excellent uniformity. These plots are computed using Equation 1 from Shvydky.

Note that in the plot the intensities are normalized. So 1.0 means full intensity and 0.0 means minimal relative intensity.
Note that Gauss-Quadratures shows similar results, although the focus here was not to minimize the number of weights / different beam strengths.

The minimal relative intensities for 48, 60, 72 and 132 lasers are 0.982725, 0.984542, 0.995359 and 0.999907. The uniform N=60 design proposed by Shvydky has a minimal relative intensity of 0.985446, almost equal to the weighted N=60 design.

N=48, l_max = 11, 2 different weights

48 points (theta,phi) 2 different weights:
theta: [1.2336997948453468, 1.9078928587444466, 1.9078928587444466, 1.2336997948453468, 1.4598542464697268, 1.6817384071200663, 1.6817384071200663, 1.4598542464697268, 2.7853163157342573, 0.3562763378555358, 0.3562763378555358, 2.7853163157342573, 2.7853163157330116, 0.35627633785678164, 0.35627633785678164, 2.7853163157330116, 1.45985424646999, 1.6817384071198034, 1.6817384071198034, 1.45985424646999, 1.907892858745844, 1.2336997948439494, 1.2336997948439494, 1.907892858745844, 0.8824843417477756, 2.2591083118420174, 2.2591083118420174, 0.8824843417477756, 1.8616874665689487, 1.2799051870208444, 1.2799051870208444, 1.8616874665689487, 2.259108311842379, 0.8824843417474144, 0.8824843417474144, 2.259108311842379, 1.2799051870219895, 1.8616874665678038, 1.8616874665678038, 1.2799051870219895, 2.370415958253471, 0.7711766953363222, 0.7711766953363222, 2.370415958253471, 0.771176695336047, 2.370415958253746, 2.370415958253746, 0.771176695336047]
phi: [4.8299771574945405, 1.453208149685046, 4.594800803274839, 1.6883845039047471, 5.943919993516647, 0.33926531366293894, 3.480857967252732, 2.8023273399268542, 3.4646084238352346, 2.8185768833443516, 5.960169536934145, 0.3230157702454418, 5.035404750628221, 1.2477805565513658, 4.389373210141159, 1.8938120970384273, 4.373123666720354, 1.9100616404592323, 5.051654294049025, 1.231531013130561, 6.165597130069957, 0.11758817710962952, 3.2591808306994223, 3.024004476480164, 4.331919117226351, 1.9512661899532349, 5.092858843543028, 1.1903264636365583, 2.416651002899945, 3.866534304279641, 0.7249416506898482, 5.558243656489738, 3.5220625167467206, 2.7611227904328657, 5.902715444022658, 0.3804698631569275, 5.437330631074623, 0.8458546761049632, 3.9874473296947563, 2.29573797748483, 2.7174950230862995, 3.5656902840932867, 0.42409763050349364, 5.859087676676093, 5.1364866108865845, 1.146698696293002, 4.288291349882795, 1.9948939572967912]
weights: [0.9495675564588315, 0.9495675564588315, 0.9495675564588315, 0.9495675564588315, 0.9495675564588315, 0.9495675564588315, 0.9495675564588315, 0.9495675564588315, 0.9495675564588315, 0.9495675564588315, 0.9495675564588315, 0.9495675564588315, 0.9495675564588315, 0.9495675564588315, 0.9495675564588315, 0.9495675564588315, 0.9495675564588315, 0.9495675564588315, 0.9495675564588315, 0.9495675564588315, 0.9495675564588315, 0.9495675564588315, 0.9495675564588315, 0.9495675564588315, 1.0504324435411683, 1.0504324435411683, 1.0504324435411683, 1.0504324435411683, 1.0504324435411683, 1.0504324435411683, 1.0504324435411683, 1.0504324435411683, 1.0504324435411683, 1.0504324435411683, 1.0504324435411683, 1.0504324435411683, 1.0504324435411683, 1.0504324435411683, 1.0504324435411683, 1.0504324435411683, 1.0504324435411683, 1.0504324435411683, 1.0504324435411683, 1.0504324435411683, 1.0504324435411683, 1.0504324435411683, 1.0504324435411683, 1.0504324435411683]

symmetries =  [1.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
 0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
 0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
 1.14621038e-01 9.45072435e-02 7.03566689e-05]
symmetry error = 0.0
minimal relative intensity = 0.982725 rms = 0.00432292

N=60, l_max = 12, 5 different weights

60 points (theta,phi) 5 different weights:
theta: [1.6274940267722833, 1.51409862681751, 1.51409862681751, 1.6274940267722833, 0.8638557178204358, 2.2777369357693575, 2.2777369357693575, 0.8638557178204358, 0.7101900904428751, 2.4314025631469183, 2.4314025631469183, 0.7101900904428751, 2.4226326692470517, 0.7189599843427416, 0.7189599843427416, 2.4226326692470517, 1.9389988215692207, 1.2025938320205727, 1.2025938320205727, 1.9389988215692207, 0.9865818998776599, 2.1550107537121335, 2.1550107537121335, 0.9865818998776599, 1.0615792547795029, 2.0800133988102907, 2.0800133988102907, 1.0615792547795029, 0.674609763729876, 2.466982889859917, 2.466982889859917, 0.674609763729876, 1.9719368622435383, 1.1696557913462549, 1.1696557913462549, 1.9719368622435383, 2.8875499597023726, 0.2540426938874208, 0.2540426938874208, 2.8875499597023726, 1.816324864404675, 1.3252677891851181, 1.3252677891851181, 1.816324864404675, 1.6347041340303299, 1.5068885195594635, 1.5068885195594635, 1.6347041340303299, 0.4325408030670978, 2.7090518505226955, 2.7090518505226955, 0.4325408030670978, 1.4551037073027548, 1.6864889462870385, 1.6864889462870385, 1.4551037073027548, 1.1560352805339638, 1.9855573730558294, 1.9855573730558294, 1.1560352805339638]
phi: [0.8624800626257317, 5.420705244553854, 2.2791125909640613, 4.004072716215525, 3.0669957854499774, 3.216189521729609, 0.0745968681398157, 6.2085884390397705, 4.799414810465431, 1.4837704967141547, 4.625363150303947, 1.6578221568756386, 2.14899866351929, 4.134186643660296, 0.9925939900705033, 5.290591317109083, 5.344893613415652, 0.9382916937639341, 4.079884347353727, 2.2033009598258593, 0.4461617235020494, 5.837023583677537, 2.6954309300877437, 3.5877543770918425, 2.034436987188264, 4.248748319991322, 1.1071556664015294, 5.176029640778057, 2.466259676091016, 3.81692563108857, 0.6753329774987773, 5.607852329680809, 0.5580462589752347, 5.725139048204351, 2.5835463946145585, 3.6996389125650277, 4.455454775662978, 1.8277305315166086, 4.9693231851064015, 1.3138621220731848, 3.2074792013766307, 3.0757061058029556, 6.217298759392748, 0.06588654778683765, 4.466347818334601, 1.816837488844985, 4.9584301424347785, 1.3247551647448081, 3.4205806052235346, 2.8626047019560517, 6.004197355545845, 0.2789879516337414, 0.4177261924096632, 5.865459114769923, 2.72386646118013, 3.559318845999456, 4.585923306932992, 1.6972620002465937, 4.838854653836387, 1.4443306533431994]
weights: [0.8935966646215763, 0.8935966646215763, 0.8935966646215763, 0.8935966646215763, 0.8935966646215763, 0.8935966646215763, 0.8935966646215763, 0.8935966646215763, 0.8935966646215763, 0.8935966646215763, 0.8935966646215763, 0.8935966646215763, 0.9930272225438518, 0.9930272225438518, 0.9930272225438518, 0.9930272225438518, 0.9930272225438518, 0.9930272225438518, 0.9930272225438518, 0.9930272225438518, 0.9930272225438518, 0.9930272225438518, 0.9930272225438518, 0.9930272225438518, 1.0249403615485184, 1.0249403615485184, 1.0249403615485184, 1.0249403615485184, 1.0249403615485184, 1.0249403615485184, 1.0249403615485184, 1.0249403615485184, 1.0249403615485184, 1.0249403615485184, 1.0249403615485184, 1.0249403615485184, 1.0346957277725266, 1.0346957277725266, 1.0346957277725266, 1.0346957277725266, 1.0346957277725266, 1.0346957277725266, 1.0346957277725266, 1.0346957277725266, 1.0346957277725266, 1.0346957277725266, 1.0346957277725266, 1.0346957277725266, 1.053740023513527, 1.053740023513527, 1.053740023513527, 1.053740023513527, 1.053740023513527, 1.053740023513527, 1.053740023513527, 1.053740023513527, 1.053740023513527, 1.053740023513527, 1.053740023513527, 1.053740023513527]

symmetries =  [1.         0.         0.         0.         0.         0.
 0.         0.         0.         0.         0.         0.
 0.         0.05187415 0.08413808 0.05052016]
symmetry error = 0.0
minimal relative intensity = 0.985146 rms = 0.00255375

N=72, l_max = 14, 2 different weights

The Lebedev13 quadrature design also utilizes 74 points, but it achieves only l_max = 13. A crucial difference is that the Lebedev13 scheme includes negative weights, which pose a practical challenge in applications where the quadrature weights must be realized physically—such as when adjusting laser beam strengths. Negative weights are not physically meaningful in this context because they imply a subtraction of intensity, which isn’t feasible with standard laser modulation techniques.

72 points (theta,phi) 2 different weights:
theta: [1.570796326794894, 1.5707963267948994, 1.5707963267948994, 1.570796326794894, 2.1243706856920457, 1.0172219678977474, 1.0172219678977474, 2.1243706856920457, 0.553574358897149, 2.5880182946926444, 2.5880182946926444, 0.553574358897149, 1.1491880742954592, 1.992404579294334, 1.992404579294334, 1.1491880742954592, 1.2171645701120242, 1.924428083477769, 1.924428083477769, 1.2171645701120242, 1.045409483694764, 2.096183169895029, 2.096183169895029, 1.045409483694764, 0.7040578522196116, 2.4375348013701816, 2.4375348013701816, 0.7040578522196116, 0.8494277341123436, 2.2921649194774494, 2.2921649194774494, 0.8494277341123436, 2.7221843035747035, 0.41940835001508975, 0.41940835001508975, 2.7221843035747035, 1.310828575047656, 1.830764078542137, 1.830764078542137, 1.310828575047656, 2.3000023719275777, 0.8415902816622157, 0.8415902816622157, 2.3000023719275777, 1.2494562217538965, 1.8921364318358966, 1.8921364318358966, 1.2494562217538965, 1.7266673281712817, 1.4149253254185115, 1.4149253254185115, 1.7266673281712817, 1.4191070012698126, 1.7224856523199807, 1.7224856523199807, 1.4191070012698126, 2.9232201418418287, 0.21837251174796468, 0.21837251174796468, 2.9232201418418287, 0.6139373949535742, 2.5276552586362193, 2.5276552586362193, 0.6139373949535742, 2.1736833987774284, 0.9679092548123646, 0.9679092548123646, 2.1736833987774284, 1.6727818147410136, 1.4688108388487795, 1.4688108388487795, 1.6727818147410136]
phi: [2.1243706856920457, 4.15881462148754, 1.0172219678977474, 5.265963339281839, 3.3134750731082e-15, 6.283185307179583, 3.14159265358979, 3.1415926535897962, 4.712388980384695, 1.5707963267948912, 4.712388980384684, 1.5707963267949019, 0.9888124407285062, 5.29437286645108, 2.152780212861287, 4.130405094318299, 2.3606150997793494, 3.922570207400237, 0.7809775538104439, 5.5022077533691425, 0.49272141503149497, 5.790463892148091, 2.648871238558298, 3.634314068621288, 0.8864164746259138, 5.396768832553672, 2.2551761789638793, 4.028009128215707, 5.191722370928203, 1.0914629362513832, 4.233055589841176, 2.05012971733841, 3.824730121200286, 2.4584551859793002, 5.600047839569093, 0.6831374676104929, 5.950248391530775, 0.3329369156488108, 3.474529569238604, 2.8086557379409824, 3.624553208138237, 2.6586320990413492, 5.800224752631142, 0.48296055454844417, 4.9867343042770065, 1.2964510029025795, 4.438043656492372, 1.8451416506872138, 6.129619756331894, 0.15356555084769227, 3.2951582044374854, 2.988027102742101, 1.413099631286041, 4.870085675893545, 1.728493022303752, 4.554692284875834, 2.3427063559058823, 3.940478951273704, 0.7988862976839112, 5.484299009495675, 3.3192493760002035, 2.9639359311793827, 6.105528584769176, 0.17765672241041075, 1.6947122593271187, 4.588473047852467, 1.4468803942626745, 4.836304912916912, 5.676698233760886, 0.6064870734187001, 3.748079727008493, 2.535105580171093]
weights: [0.8928571428599884, 0.8928571428599884, 0.8928571428599884, 0.8928571428599884, 0.8928571428599884, 0.8928571428599884, 0.8928571428599884, 0.8928571428599884, 0.8928571428599884, 0.8928571428599884, 0.8928571428599884, 0.8928571428599884, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025]

symmetries =  [1.         0.         0.         0.         0.         0.
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 0.         0.         0.         0.10261877 0.06483876 0.        ]
symmetry error = 0.0
minimal relative intensity = 0.995359 rms = 0.00105520

N = 132, l_max = 19, 3 different weights

132 points (theta,phi) 3 different weights:
theta: [1.5707963267950238, 1.5707963267947695, 1.5707963267947695, 1.5707963267950238, 0.553574358897899, 2.5880182946918944, 2.5880182946918944, 0.553574358897899, 1.0172219678969978, 2.1243706856927957, 2.1243706856927957, 1.0172219678969978, 1.4976050337520537, 1.6439876198377397, 1.6439876198377397, 1.4976050337520537, 1.3134187045583376, 1.8281739490314557, 1.8281739490314557, 1.3134187045583376, 0.26803908522608905, 2.8735535683637043, 2.8735535683637043, 0.26803908522608905, 0.6054877901357145, 2.536104863454079, 2.536104863454079, 0.6054877901357145, 1.8742434700066268, 1.2673491835831665, 1.2673491835831665, 1.8742434700066268, 1.8027405044420433, 1.3388521491477499, 1.3388521491477499, 1.8027405044420433, 2.2981654067764015, 0.8434272468133919, 0.8434272468133919, 2.2981654067764015, 2.0764912507690347, 1.0651014028207584, 1.0651014028207584, 2.0764912507690347, 2.361253227213127, 0.7803394263766662, 0.7803394263766662, 2.361253227213127, 1.9456057712320352, 1.1959868823577582, 1.1959868823577582, 1.9456057712320352, 0.7406460241206059, 2.4009466294691872, 2.4009466294691872, 0.7406460241206059, 0.8426590105438804, 2.298933643045913, 2.298933643045913, 0.8426590105438804, 1.4590139017260522, 1.682578751863741, 1.682578751863741, 1.4590139017260522, 1.8272143206921447, 1.3143783328976484, 1.3143783328976484, 1.8272143206921447, 2.6800139676761177, 0.4615786859136756, 0.4615786859136756, 2.6800139676761177, 0.9418439606502862, 2.199748692939507, 2.199748692939507, 0.9418439606502862, 2.0021748791777574, 1.1394177744120357, 1.1394177744120357, 2.0021748791777574, 1.3800390499810915, 1.7615536036087018, 1.7615536036087018, 1.3800390499810915, 0.7598973341099453, 2.381695319479848, 2.381695319479848, 0.7598973341099453, 1.0802394852462442, 2.061353168343549, 2.061353168343549, 1.0802394852462442, 2.335266709934967, 0.8063259436548262, 0.8063259436548262, 2.335266709934967, 2.097344824550846, 1.0442478290389476, 1.0442478290389476, 2.097344824550846, 0.45356129582117977, 2.6880313577686135, 2.6880313577686135, 0.45356129582117977, 1.7078336512804537, 1.4337590023093396, 1.4337590023093396, 1.7078336512804537, 2.072327710740324, 1.0692649428494694, 1.0692649428494694, 2.072327710740324, 2.9586974725889315, 0.1828951810008618, 0.1828951810008618, 2.9586974725889315, 2.618224044560076, 0.5233686090297168, 0.5233686090297168, 2.618224044560076, 1.395920744396228, 1.745671909193565, 1.745671909193565, 1.395920744396228, 1.1647030812045267, 1.9768895723852666, 1.9768895723852666, 1.1647030812045267, 1.5177778194937113, 1.6238148340960818, 1.6238148340960818, 1.5177778194937113]
phi: [0.5535743588978989, 5.729610948281687, 2.5880182946918944, 3.695167012487692, 3.1415926535895515, 3.1415926535900347, 2.418923349795825e-13, 6.283185307179345, 4.712388980384839, 1.5707963267947471, 4.7123889803845405, 1.5707963267950462, 0.2580842833579041, 6.025101023821682, 2.883508370231889, 3.399676936947697, 1.4951073632213168, 4.788077943958269, 1.6464852903684766, 4.63670001681111, 0.27974715780034876, 6.003438149379237, 2.8618454957894444, 3.421339811390142, 3.6943097012082466, 2.5888756059713396, 5.730468259561133, 0.5527170476184536, 5.750780231626697, 0.5324050755528893, 3.673997729142682, 2.609187578036904, 3.8937313778775176, 2.3894539293020687, 5.531046582891862, 0.7521387242877248, 4.399556695570758, 1.8836286116088283, 5.0252212651986214, 1.257964041980965, 1.9193749064347827, 4.3638104007448035, 1.2222177471550106, 5.060967560024576, 0.33285478436922783, 5.950330522810359, 2.8087378692205656, 3.4744474379590207, 4.4363595306593, 1.8468257765202865, 4.988418430110079, 1.2947668770695067, 4.5463100377260695, 1.7368752694535163, 4.878467923043309, 1.404717384136277, 0.1500093124051802, 6.1331759947744064, 2.991583341184613, 3.2916019659949733, 3.9786382254207737, 2.3045470817588125, 5.446139735348606, 0.8370455718309808, 0.3881421332626837, 5.8950431739169025, 2.7534505203271094, 3.529734786852477, 0.6058525335553483, 5.677332773624238, 2.535740120034445, 3.747445187145141, 3.685013702523128, 2.598171604656458, 5.739764258246251, 0.5434210489333353, 5.416854777248093, 0.8663305299314932, 4.007923183521286, 2.2752621236583, 4.298338249585249, 1.9848470575943375, 5.126439711184131, 1.1567455959954558, 2.3239203913376283, 3.959264915841958, 0.817672262252165, 5.465513044927421, 5.318580389105739, 0.9646049180738474, 4.106197571663641, 2.176987735515946, 2.188690196328243, 4.094495110851343, 0.9529024572615502, 5.330282849918036, 3.717897812019414, 2.565287495160172, 5.706880148749965, 0.5763051584296215, 1.1232942687449117, 5.159891038434674, 2.018298384844882, 4.264886922334704, 4.2056613215936265, 2.0775239855859597, 5.219116639175753, 1.0640686680038332, 3.2980253280517142, 2.985159979127872, 6.126752632717665, 0.15643267446192116, 4.416729710958001, 1.8664555962215852, 5.008048249811378, 1.2751370573682081, 1.2939466911973065, 4.98923861598228, 1.8476459623924866, 4.4355393447871, 6.229344859512386, 0.05384044766720016, 3.195433101256993, 3.0877522059225933, 0.20788181519614665, 6.07530349198344, 2.9337108383936465, 3.3494744687859397, 4.887513180254119, 1.3956721269254664, 4.53726478051526, 1.7459205266643267]
weights: [0.8394383394376632, 0.8394383394376632, 0.8394383394376632, 0.8394383394376632, 0.8394383394376632, 0.8394383394376632, 0.8394383394376632, 0.8394383394376632, 0.8394383394376632, 0.8394383394376632, 0.8394383394376632, 0.8394383394376632, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791]

symmetries =  [1.         0.         0.         0.         0.         0.
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 0.         0.         0.         0.         0.         0.
 0.         0.         0.02482586 0.06994034 0.02790389]
symmetry error = 0.0
minimal relative intensity = 0.999234 rms = 0.00014760

N = 192, l_max = 23, 4 different weights;

192 points (theta,phi) 4 different weights:
theta: [1.5707963267935952, 1.5707963267961982, 1.5707963267961982, 1.5707963267935952, 1.0172219678993604, 2.124370685690433, 2.124370685690433, 1.0172219678993604, 0.5535743588955363, 2.588018294694257, 2.588018294694257, 0.5535743588955363, 2.1464652349718594, 0.9951274186179337, 0.9951274186179337, 2.1464652349718594, 2.503549185843261, 0.6380434677465324, 0.6380434677465324, 2.503549185843261, 1.3267147292639176, 1.8148779243258755, 1.8148779243258755, 1.3267147292639176, 0.7992449725143074, 2.342347681075486, 2.342347681075486, 0.7992449725143074, 2.7306273587309704, 0.41096529485882277, 0.41096529485882277, 2.7306273587309704, 1.7431839770582105, 1.3984086765315829, 1.3984086765315829, 1.7431839770582105, 1.93991423789508, 1.2016784156947133, 1.2016784156947133, 1.93991423789508, 1.6066884025715498, 1.5349042510182436, 1.5349042510182436, 1.6066884025715498, 2.3687531618924687, 0.7728394916973245, 0.7728394916973245, 2.3687531618924687, 1.7920854421396617, 1.3495072114501316, 1.3495072114501316, 1.7920854421396617, 0.35133711226740955, 2.7902555413223835, 2.7902555413223835, 0.35133711226740955, 1.23988132212481, 1.9017113314649832, 1.9017113314649832, 1.23988132212481, 1.4570784237609438, 1.6845142298288494, 1.6845142298288494, 1.4570784237609438, 0.9242189870895275, 2.217373666500266, 2.217373666500266, 0.9242189870895275, 0.6960540389542376, 2.4455386146355558, 2.4455386146355558, 0.6960540389542376, 1.0658765344148418, 2.0757161191749516, 2.0757161191749516, 1.0658765344148418, 0.7984031295940905, 2.3431895239957026, 2.3431895239957026, 0.7984031295940905, 1.6593112355362603, 1.482281418053533, 1.482281418053533, 1.6593112355362603, 2.026179888972515, 1.1154127646172782, 1.1154127646172782, 2.026179888972515, 0.8933947658043098, 2.2481978877854836, 2.2481978877854836, 0.8933947658043098, 1.2207977393465268, 1.9207949142432663, 1.9207949142432663, 1.2207977393465268, 0.24693194716432712, 2.894660706425466, 2.894660706425466, 0.24693194716432712, 1.7309605063133866, 1.4106321472764067, 1.4106321472764067, 1.7309605063133866, 1.800701446577565, 1.3408912070122283, 1.3408912070122283, 1.800701446577565, 1.0143097139670758, 2.1272829396227175, 2.1272829396227175, 1.0143097139670758, 2.0782228735666113, 1.063369780023182, 1.063369780023182, 2.0782228735666113, 0.8721032725558162, 2.269489381033977, 2.269489381033977, 0.8721032725558162, 0.5133531538291254, 2.6282394997606677, 2.6282394997606677, 0.5133531538291254, 1.641937236346669, 1.4996554172431242, 1.4996554172431242, 1.641937236346669, 0.3878178477303071, 2.753774805859486, 2.753774805859486, 0.3878178477303071, 2.513339623039212, 0.6282530305505813, 0.6282530305505813, 2.513339623039212, 1.1442981742373886, 1.9972944793524046, 1.9972944793524046, 1.1442981742373886, 1.1853770147850173, 1.956215638804776, 1.956215638804776, 1.1853770147850173, 0.5153586774838607, 2.6262339761059326, 2.6262339761059326, 0.5153586774838607, 2.3925957691743966, 0.7489968844153966, 0.7489968844153966, 2.3925957691743966, 2.232478002231715, 0.9091146513580781, 0.9091146513580781, 2.232478002231715, 1.2730055011948624, 1.868587152394931, 1.868587152394931, 1.2730055011948624, 1.2464523149230375, 1.8951403386667556, 1.8951403386667556, 1.2464523149230375, 1.1401204520635007, 2.0014722015262927, 2.0014722015262927, 1.1401204520635007, 2.1854884420180656, 0.9561042115717276, 0.9561042115717276, 2.1854884420180656, 2.487224681924875, 0.6543679716649181, 0.6543679716649181, 2.487224681924875, 1.766657176594097, 1.3749354769956963, 1.3749354769956963, 1.766657176594097, 3.002861200035678, 0.1387314535541154, 0.1387314535541154, 3.002861200035678, 1.6318948459458, 1.509697807643993, 1.509697807643993, 1.6318948459458, 1.6951929803788715, 1.4463996732109219, 1.4463996732109219, 1.6951929803788715]
phi: [1.0172219678993604, 5.265963339280226, 2.1243706856904327, 4.158814621489153, 1.5299634691768186e-12, 6.283185307178056, 3.1415926535882632, 3.141592653591323, 1.570796326792421, 4.712388980387165, 1.5707963267973721, 4.712388980382214, 2.849352027925563, 3.4338332792540234, 0.2922406256642304, 5.990944681515356, 5.130170634970153, 1.153014672209434, 4.294607325799227, 1.988577981380359, 0.9751728569186264, 5.30801245026096, 2.1664197966711667, 4.1167655105084195, 4.662308233394719, 1.6208770737848672, 4.76246972737466, 1.5207155798049259, 4.268582333271671, 2.014602973907915, 5.156195627497708, 1.1269896796818777, 4.3374443354580725, 1.9457409717215137, 5.087333625311307, 1.1958516818682794, 3.3265687698783544, 2.956616537301232, 6.0982091908910245, 0.1849761162885614, 2.369414246526438, 3.9137710606531484, 0.7721784070633554, 5.5110069001162305, 6.231764964906783, 0.05142034227280329, 3.1930129958625963, 3.09017231131699, 4.046808171765665, 2.2363771354139206, 5.377969789003713, 0.9052155181758728, 1.2347934816904786, 5.048391825489108, 1.9067991718993147, 4.3763861352802715, 0.12027232207262839, 6.162912985106958, 3.021320331517165, 3.261864975662421, 1.2376470929987975, 5.045538214180789, 1.9039455605909956, 4.37923974658859, 2.8630080281880113, 3.420177278991575, 0.278584625401782, 6.004600681777804, 5.061763552496586, 1.2214217546829995, 4.363014408272792, 1.9201708989067938, 4.064484193445679, 2.218701113733907, 5.3602937673237, 0.9228915398558862, 3.9709066668995647, 2.3122786402800215, 5.453871293869814, 0.8293140133097718, 6.052360233988297, 0.23082507319128928, 3.3724177267810824, 2.910767580398504, 0.7724482906637435, 5.510737016515843, 2.3691443629260496, 3.9140409442535367, 2.170540033780949, 4.112645273398638, 0.9710526198088443, 5.312132687370742, 4.882988388751892, 1.4001969184276937, 4.541789572017487, 1.7413957351620994, 0.3700424099972232, 5.913142897182363, 2.77155024359257, 3.511635063587016, 0.35473520865554653, 5.928450098524039, 2.7868574449342467, 3.4963278622453395, 1.661709698651784, 4.621475608527803, 1.4798829549380093, 4.803302352241577, 3.747727383293556, 2.5354579238860304, 5.677050577475823, 0.6061347297037629, 4.793807377859979, 1.489377929319607, 4.6309705829094, 1.6522147242701863, 2.5297245107419446, 3.7534607964376416, 0.6118681428478487, 5.671317164331738, 3.2868407271084434, 2.996344580071143, 6.137937233660936, 0.1452480735186506, 5.774348305997246, 0.5088370011823405, 3.6504296547721333, 2.632755652407453, 5.847847492544365, 0.43533781463522137, 3.5769304682250143, 2.706254838954572, 0.789969831074441, 5.493215476105146, 2.351622822515352, 3.931562484664234, 5.1887666277894535, 1.0944186793901325, 4.236011332979926, 2.047173974199661, 4.361304345638666, 1.9218809615409203, 5.063473615130713, 1.219711692048873, 3.8447440299537394, 2.438441277225847, 5.58003393081564, 0.7031513763639463, 2.696090965078092, 3.5870943421014942, 0.4455016885117012, 5.837683618667885, 1.9518441392733714, 4.331341167906215, 1.189748514316422, 5.093436792863164, 0.698060543496394, 5.585124763683192, 2.443532110093399, 3.839653197086187, 3.5494331847962552, 2.733752122383331, 5.875344775973124, 0.40784053120646213, 2.6689311491357337, 3.6142541580438525, 0.4726615044540596, 5.810523802725527, 1.3302675340200247, 4.9529177731595615, 1.8113251195697686, 4.471860187609818, 0.3254528556082703, 5.9577324515713155, 2.816139797981523, 3.4670455091980634, 3.7701127359494704, 2.513072571230116, 5.654665224819909, 0.6285200823596774, 4.25506394828045, 2.028121358899136, 5.169714012488929, 1.1134712946906573, 4.587758566515891, 1.6954267406636951, 4.837019394253488, 1.446165912926098, 3.2031675753706463, 3.08001773180894, 6.221610385398733, 0.06157492178085346]
weights: [0.7996569902556139, 0.7996569902556139, 0.7996569902556139, 0.7996569902556139, 0.7996569902556139, 0.7996569902556139, 0.7996569902556139, 0.7996569902556139, 0.7996569902556139, 0.7996569902556139, 0.7996569902556139, 0.7996569902556139, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585]

symmetries =  [1.         0.         0.         0.         0.         0.
 0.         0.         0.         0.         0.         0.
 0.         0.         0.         0.         0.         0.
 0.         0.         0.         0.         0.         0.
 0.00228945 0.05805168 0.03208777]
symmetry error = 0.0
minimal relative intensity = 0.999974 rms = 0.00000467

In the table below all solutions N >= 100 are created using this method and also show a reduced number of distinct weights.

Lebedev Spherical Designs

Some Lebedev spherical designs show relatively smooth weight distribution and could be viable alternatives:

N = 50, l_max = 11
https://github.com/ifilot/pylebedev/blob/master/pylebedev/data/lebedev_011.txt
weigths = [0.6349206349206488, 1.008666776895948, 1.054687499999998, 1.1287477954144478]
minimal relative intensity = 0.984713 rms = 0.00330522

N = 86, l_max = 15
https://github.com/ifilot/pylebedev/blob/master/pylebedev/data/lebedev_015.txt
weigths = [0.9555077911118807, 0.9927849927850555, 1.0158581222337681, 1.0213791113302062, 1.0271761813836404]
minimal relative intensity = 0.998664 rms = 0.00025879

N = 110, l_max = 17
https://github.com/ifilot/pylebedev/blob/master/pylebedev/data/lebedev_017.txt
weigths = [0.4211097544430708, 0.9032911011510117, 1.055501846967812, 1.066449599782932, 1.0773111263736819, 1.093709638029582]
minimal relative intensity = 0.999217 rms = 0.00014577

N = 170, l_max = 21
https://github.com/ifilot/pylebedev/blob/master/pylebedev/data/lebedev_021.txt
weigths = [0.8811758899171654, 0.9311143754732957, 0.9426232933462958, 1.0146252779057763, 1.0321265710140766, 1.0542839011201364, 1.0740479316683866, 1.0852247114975568]
minimal relative intensity = 0.999955 rms = 0.00000756

Variation of Beam Strength

There are unavoidable fluctuations in laser energy in each beam, roughly 1% variation in peak power. Are perfect t-designs of any value at all? We can check by randomly varying the weight for each beam and check the resulting error for each l ⇐ l_max:

    errors = []
    syms = []
    for i in range(1000):
        # max 1 % strength variation for each laser
        variation = np.random.uniform(low=0.995, high=1.005, size=N)
        randomized_weight = variation * weights
        errors.append(t_design_error(points, l_max, randomized_weight))
        syms.append(symmetry(points, l_max, randomized_weight))
    syms = np.array(syms)
    print(f'mean error = {np.mean(errors)}')
    print(f'standard dev error = {np.std(errors)}')
    print(f'mean syms = {np.mean(syms, axis=0)}')
    print(f'standard dev syms = {np.std(syms, axis=0)}')

We see that the resulting error is evenly distributed over all l ⇐ l_max and it is ⇐ 1.5E7. So using a perfect design helps to manage the error for all l ⇐ l_max when we consider a 1% beam strength variation.

Uniform t-designs for Laser Reactors

From Alexander Shvydky from Rochester I got the following hint:

"It is much simpler to make all the beamlines the same (close enough) than to make one laser intensity be certain percentage different from another and maintain it throughout the whole time varying laser pulse (which has a thoroughly optimized shape)"

So let’s have a closer look at uniform designs. There seems not much progress in this area recently. Sloanes 108/14 design is really not bad. How can we improve it?

The idea is to clarify the requirements:

As we have to deal with small beam strength fluctuations it makes no sense to search for exact zero values for the symmetries. The fluctuations will anyway spoil our result. Alternatively we can tolerate a symmetry error eps - which could be a vector if we want to define the acceptable error separately for each symmetry.
On the other hand we are interested in a uniform intensity distribution, which means a low intensity distribution rms value.

We could trade a small relaxation regarding the symmetries for enhanced uniformity.

How to do this? We implement the eps-relaxation as follows:

def t_design_error_eps(points, l_max, weights=None, eps=1E-7):
    syms = symmetry(points, l_max, weights)
    if not eps is None:
        syms[np.abs(syms) < eps] = 0.
    return sum(syms[1:l_max+1])

and adapt the objective function using a weighted sum approach for both objectives:

    def fit(x):
        pts = x_to_points(x)
        td_err = t_design_error_eps(pts, l_max, eps)
        dis_err = distribution_rms(pts[:, 0], pts[:, 1])
        return dis_err + 1E6*td_err

The objective-weights prioritize the relaxed symmetry (t-design) error. But as soon as all symmetries are lower than eps, the optimizer is free to target the uniformity-rms. Note that there are no more weights for the points, as we are optimizing a uniform design. By the way, it is a bad idea to seed the optimization with Sloanes 108/14-design as this is a very strong local minimum the optimizer struggles to escape. So we seed as usual with the Fibonacci/Spiral design.

Sloanes Uniform 108/14-design:

Sloanes 108/14-design has a uniformity rms = 0.00035970.

108 points (theta,phi) 1 different weights:
theta = [1.4293263903585374, 2.2467486734030717, 1.4290667444413534, 2.445466740819859, 0.8948127028349676, 0.6960821506187987, 2.246584388414859, 2.4453418464927843, 0.8950395376219203, 0.696294560418363, 1.7123919093525397, 1.7124002606267177, 2.548434524513169, 0.9799965601559985, 2.5486188410905863, 1.6157213193320548, 2.1617687306252673, 1.5260363846715133, 0.9798131774014407, 1.6156526832172242, 2.161606837751843, 1.5257749192439805, 0.5929599287656716, 0.5931720087279596, 1.1153181665367216, 1.8294924371763588, 1.115427219369692, 0.5338523321016375, 1.312021553449747, 2.6075684728934525, 1.8297481677196767, 0.533937238881064, 1.3119231525224517, 2.607827288582799, 2.0263311571779252, 2.0261087686938892, 0.21902348817560122, 1.7871525037910794, 0.21880114861906086, 1.6030180488219035, 1.3542673537872052, 1.5383804085857005, 1.7873187248018294, 1.6031750967110983, 1.3544467245488565, 1.5386117528289158, 2.9227657834094547, 2.922594864252192, 1.7530727222047986, 2.131578575814294, 1.7528197715345382, 2.5451970305773797, 1.0100510597136296, 0.5964597486908426, 2.1313473575330972, 2.5449692574489555, 1.0102083186292243, 0.5965592859160455, 1.388627444930057, 1.3886653713893893, 1.9731777719907917, 2.291944622180968, 1.9729146776430384, 2.266779619483447, 0.8497381483205383, 0.8749144651744073, 2.291684814638696, 2.266529487168296, 0.8498177354788391, 0.874961747950756, 1.1685575356305518, 1.1685353292247256, 1.2108835482568447, 2.663339718276588, 1.2106402196721338, 1.8714711911971076, 0.478120570304277, 1.2700481184482912, 2.6633441274418304, 1.8713838335366724, 0.4783808584926781, 1.2702821603434686, 1.93089772773883, 1.9307638076276166, 1.6298615362230826, 1.8363348860556417, 1.6296424699396344, 2.8694854651553765, 1.305268324496202, 0.2721507774454247, 1.8361258601486155, 2.8692732016554174, 1.3054562370495975, 0.2722758892568215, 1.5117833697993552, 1.5118979317314447, 1.1271756793620396, 2.3681823596521765, 1.126912650917313, 2.1545034364062676, 0.7733040335859169, 0.9869869057954318, 2.3681317525914274, 2.1544574925099553, 0.7735671442801316, 0.987237462451619, 2.0145745006964146, 2.0145224683037113]
phi =  [4.028473040650788, 2.959704903533529, 0.8868867822202753, 4.934499243071394, 0.1819325175047432, 1.3490992994220066, 6.101462744794258, 1.7926289935692952, 3.3236326748973006, 4.490505401356661, 2.2548714307708235, 5.396501627411351, 6.203037164005931, 4.658351700129297, 3.061165139460878, 2.5501093838420745, 4.766613675922243, 3.7332520705306105, 1.516885975625664, 5.691713465144661, 1.6248815582424554, 0.5916579514094831, 0.08035291601723347, 3.2221775663000463, 1.8599434279467801, 0.4724689908242624, 5.001418603974276, 5.7560616441570485, 2.669345550802886, 0.5271355975172249, 3.6140448357695467, 2.614890936866441, 5.810873488571133, 3.668644641811759, 4.423516367277582, 1.281854499555609, 3.2912624951233025, 1.6039365665790282, 0.14903427658050905, 6.066725244044186, 1.5378359709278275, 0.2166425673381977, 4.7455741750077785, 2.9251257541109124, 4.679386183746633, 3.3582392963240153, 2.991970695113703, 6.13446473037117, 4.141222217729651, 3.3571998163373777, 0.9996159732227217, 4.384250341176422, 6.0676532200212145, 1.8993933238220304, 0.21568650099492528, 1.242462600952465, 2.926193312169186, 5.040626684167283, 2.1421750341577064, 5.283719659964159, 3.9124023989342063, 3.6899183167775558, 0.7708035813620293, 4.177234383498687, 5.73491108734092, 2.106274913010375, 0.5483642594030087, 1.0355725397105189, 2.593539193953008, 5.247651060978968, 2.3710229732835786, 5.512503923184724, 3.4636447488122277, 2.2696015862517926, 0.3220140396465419, 5.090288971364255, 0.8719576185564184, 1.1931347741446376, 5.41170246018617, 1.948619247798896, 4.013471237414443, 4.334689899060276, 2.819697997226762, 5.961376074121109, 4.4465766666228115, 3.202759340077593, 1.3049753707894722, 4.491816068170362, 6.222152830118369, 1.7922455873638954, 0.061210298256301726, 1.3496641382647239, 3.080610389416716, 4.933007497997477, 1.8368011034440916, 4.978379751513337, 3.7980020391210325, 2.4797206903276123, 0.6564019511203719, 5.253007696886229, 0.6619133976550754, 1.0304713567421488, 5.621578193318431, 2.111243686058617, 3.803520583123075, 4.17201015290721, 2.4853067182238275, 5.627022162612651]

symmetries =  [1.         0.         0.         0.         0.         0.
 0.         0.         0.         0.         0.         0.
 0.         0.         0.         0.01251775 0.00509248 0.01286083
 0.0358571  0.006429   0.02579122 0.02190473 0.00258595 0.01573529]
symmetry error = 0.09854836940179791
minimal relative intensity = 0.997935 rms = 0.00035970

Symmetry Relaxation to eps = 1E-8 / 1E-5 Uniform 108/14-design

Accepting symmetry tolerances of 1E-8 and 1E-5 for the 108/14-design results in uniformity rms = 0.00016276 and 0.00015247 which is > factor 2 improvement. A tolerance of 1E-8 is sufficient to generate a much better uniformity. There is not much to gain lowering accuracy even further. Note that only for higher l the accuracy is lowered, although the objective function applies the same limit to all l ⇐ l_max.

108 points (theta,phi) 1 different weights, symmetry tolerance 1E-8:
theta = [1.7879791959036195, 1.3536195337939336, 1.3535809718786596, 1.787986980110825, 1.0928777011544104, 2.048698076603844, 2.0486709306584086, 1.092900074814112, 0.5326911409424016, 2.608948099176327, 2.6089596465308884, 0.532673724916103, 0.7574328914851715, 2.3842045533433134, 2.384198390225391, 0.7573800397414605, 2.1209121331659904, 1.0206648518386685, 1.020653422590061, 2.1209053999936334, 1.108804815637042, 2.0327647094741175, 2.032819154780888, 1.108829105612335, 0.1562692590337622, 2.9853745449102815, 2.9853489902562647, 0.15624807627376777, 1.5509547726729598, 1.5905829772681637, 1.5906083991044166, 1.550980232330044, 1.7257636459712236, 1.4158730048949952, 1.4158164613301238, 1.7257839999906646, 1.9476748441201808, 1.1938997337880508, 1.1938829703023017, 1.9476683405211328, 1.4369745025535317, 1.7046261048241076, 1.7045727060762204, 1.4370178160682723, 0.40224160449634605, 2.7393361382435515, 2.739346731225384, 0.40221309355426027, 2.6471658787691354, 0.4943926649839936, 0.49439079749890674, 2.647204433528753, 1.434135086285351, 1.707431533262893, 1.707456999826795, 1.4341472182954333, 2.0425618460666306, 1.0990432941150137, 1.0990687536070058, 2.0425700685745856, 1.3914080060731764, 1.7501856935777587, 1.7501514644509701, 1.3913774766252616, 2.7320572240197722, 0.40954595259801707, 0.4095423024198564, 2.7320374933988276, 1.9347826058361897, 1.2068257641791214, 1.2068552237888754, 1.934773697564281, 1.7120522463411114, 1.4295823848777853, 1.4295438591241971, 1.7120461618473812, 2.385122347401441, 0.7564475387665189, 0.7564370213660752, 2.385103621224422, 2.30738704964027, 0.8342030004955093, 0.8342076307536913, 2.307399362605794, 0.8476021092115015, 2.293941158891961, 2.293932664007723, 0.8476468570893965, 1.3818451794233164, 1.7597814618515515, 1.7597990805476134, 1.3818160347545987, 0.7585330196246826, 2.3830327797678477, 2.3830659497676967, 0.7585183869744094, 1.0611693527497057, 2.0804263949891797, 2.0803884024419887, 1.0611759999972574, 2.397342024504223, 0.7442500421910102, 0.7443019056142848, 2.3973615426252994, 2.0600856269524304, 1.0815437210695191, 1.081531535993027, 2.0600867458294654]
phi = [1.0803751321529764, 5.202794037366813, 2.0612115006195726, 4.221944662009171, 6.038001130639576, 0.24511782990767977, 3.386789238433795, 2.8964167587027863, 2.0090226837895884, 4.274142211454655, 1.13261760708689, 5.150560994890467, 2.435652877939931, 3.8475773782152385, 0.7060391351148126, 5.57719531204838, 1.02063710785862, 5.262529638945017, 2.1209610376857535, 4.162235198689187, 5.659523035097491, 0.6236633925905859, 3.765262392311009, 2.5179395162335676, 4.8400382272405995, 1.4431286851590328, 4.584948683302275, 1.6983983539853869, 1.725745701065679, 4.557374604499867, 1.4158654368288164, 4.867334046122429, 0.020009256871379884, 6.263164475853891, 3.1215511085021888, 3.161575994984207, 1.426817754168694, 4.856350303317676, 1.714758351354698, 4.568460679669029, 5.902727294116728, 0.3804475793261526, 3.5220089793805647, 2.7611465913983495, 2.793735526217619, 3.489423061450126, 0.3477732396487166, 5.9354237402936665, 5.003613598997184, 1.2795305388407854, 4.421103144435088, 1.8620710790401067, 4.23577749457089, 2.0473219000910206, 5.188908017349277, 1.0942708312967644, 2.9880790225764176, 3.295136799958593, 0.15347545957489175, 6.129617627939121, 3.511768630687868, 2.7713334020240623, 5.912970855938136, 0.3701873598369511, 2.6771068527204096, 3.6060244940107755, 0.46449010746818176, 5.8187838244004375, 4.90450329368826, 1.3786373123064273, 4.5202599583387535, 1.762925125786767, 3.8873092708637786, 2.3958352407761234, 5.537459715731312, 0.7457186239875067, 3.3481167047546516, 2.935125815471038, 6.076698952942277, 0.2065693972411621, 4.521170905186799, 1.7619957160566708, 4.9036419471976345, 1.379569478785172, 1.317493433420961, 4.96567920480791, 1.8240431866663733, 4.4591467890439445, 0.7392017452571301, 5.543957282912564, 2.402349947592391, 3.880750528752595, 0.276738928157249, 6.006505287204018, 2.864863748516786, 3.418303546428841, 3.7101849581691058, 2.573052464750095, 5.714566595045439, 0.5685353263477717, 2.3375143735881756, 3.945615636342476, 0.804078944745536, 5.479145211170264, 5.297991168739409, 0.9851444900435141, 4.126832051590671, 2.156409199544971]

symmetries =  [1.00000000e+00 5.43925835e-14 3.49987389e-14 3.08009863e-13
 6.12773248e-12 2.15290630e-11 9.99844652e-09 3.25187779e-10
 8.43024133e-10 6.77427803e-09 9.99505906e-09 9.99800951e-09
 9.99981932e-09 9.99863060e-09 9.99956636e-09 5.06778600e-04
 5.89343233e-03 2.43932412e-03]
symmetry error = 6.796007551155183e-08
minimal relative intensity = 0.999122 rms = 0.00016276

108 points (theta,phi) 1 different weights, symmetry tolerance 1E-5:
theta = [1.2005488868440366, 1.9414338453308406, 1.9406091531796288, 1.2006330243644041, 0.41147606816470916, 2.7306214086393874, 2.730545251981607, 0.4111470266894151, 1.3998932277289484, 1.7415290211954688, 1.7419836704759981, 1.3997963527959874, 1.09902318468138, 2.04308543155557, 2.0432054546533474, 1.0985015455887879, 1.7162637679450463, 1.4248324250983688, 1.4242347654274068, 1.7170190963873944, 2.6437832130480623, 0.4979334089356483, 0.49778950566793545, 2.643563382480946, 1.4322147990829024, 1.709498083722549, 1.7091144049508245, 1.4318798011518723, 1.946167050635876, 1.1953165547597375, 1.1952846490832203, 1.9465058617576065, 0.4032791545436394, 2.7377143692300843, 2.7382653069339344, 0.4031345095931194, 1.7270373473513332, 1.415319040094052, 1.415744503118833, 1.7269535802294729, 1.580999886879063, 1.560791165031166, 1.560927987202181, 1.5802545650288535, 2.985875794923778, 0.155849681439474, 0.1556586747277587, 2.9860330914515387, 1.0217524282686428, 2.1193478438084483, 2.1193472777286457, 1.0222027542078298, 2.3838453676494047, 0.7575734958016055, 0.7576209209745388, 2.383768518581805, 1.1077881901134388, 2.03324282941977, 2.033469268412262, 1.1085986021803784, 0.8306627484904092, 2.3110630242221655, 2.3109643828592232, 0.8306548299755715, 2.3804084112427426, 0.7606911588656334, 0.7606621597467581, 2.381068557530198, 1.4239228542676412, 1.7174871945969254, 1.7173648188970665, 1.4244989226098397, 1.7539467261773949, 1.3878449079616828, 1.387644266927554, 1.7538336487094952, 2.300448453015362, 0.8416097458868494, 0.8409972566231015, 2.3000133382676045, 0.7616912339341632, 2.379902650538294, 2.3800856888773345, 0.7618929841392214, 2.070759737166792, 1.0710637149418765, 1.0705174045273673, 2.071213865789679, 1.0721540972432089, 2.0696564019857377, 2.06996766899578, 1.0711054596099083, 2.3960664049632006, 0.7461963387471835, 0.745540975033007, 2.395837789342185, 0.5253071611071948, 2.6164651884786414, 2.6167071272417397, 0.5248323913579201, 2.0443938623495232, 1.096919482531799, 1.096777858553783, 2.0448854959184106, 1.3600670513402948, 1.7818870361597445, 1.7818739951613596, 1.358728533780874]
phi = [0.1828371938818712, 6.100110536965051, 2.9583668997427415, 3.324865770533919, 1.1319993818675746, 5.151639602372664, 2.0099361066419017, 4.273918980155486, 1.1953582477708997, 5.088114193219525, 1.9463749636793806, 4.336776743748126, 4.547791819809902, 1.7361674419617723, 4.8771897769000345, 1.4059770252514578, 2.664422138388656, 3.617902652848034, 0.4767670115946208, 5.806798185334502, 5.9722292237896095, 0.3102871875660064, 3.4533112708151323, 2.8310117944614324, 1.9497130697344027, 4.332852200659205, 1.1908117501018103, 5.0919950039827935, 0.14889322251329615, 6.133362102506607, 2.992642885212376, 3.290941591557037, 5.075388317549237, 1.2085339256613792, 4.349376626278145, 1.9339930163699195, 4.702655714679251, 1.5801851790514507, 4.722134223351966, 1.5611020131608464, 3.2962990290759544, 2.9859971430930563, 6.128063575194578, 0.155118608268755, 3.1959991449199654, 3.087037195829123, 6.230499751019017, 0.05695292910751386, 2.591697872646602, 3.691098354298781, 0.5496037300890891, 5.733370174950931, 0.8623055172935407, 5.420778332227444, 2.2789160670316364, 4.003872414865462, 5.333537356438701, 0.9491988218830275, 4.091052962115068, 2.192277863057854, 2.943067320309811, 3.340244539833864, 0.19947094589377365, 6.084305143122687, 1.3575094903299776, 4.925026318200578, 1.7842584266864467, 4.499914058098887, 5.460899526363481, 0.8212024439345312, 3.9635330033166722, 2.3195436469003488, 2.3145881925767227, 3.9684139267218344, 0.8265942121948753, 5.456780446198132, 6.036693861087215, 0.24618035922884327, 3.388423495814293, 2.8952094222554354, 4.444215380301397, 1.838491453191527, 4.980700838173393, 1.3030466292354688, 5.290422137427588, 0.9928468220158547, 4.134182495514766, 2.149365955475639, 3.719558794991368, 2.5638390025299294, 5.70481834087845, 0.5782600443282879, 2.3571247179819306, 3.9262977466297384, 0.784548397942109, 5.498727560755089, 2.7091088366825957, 3.5739392823115623, 0.43255395758737974, 5.8511013238688845, 1.3331922934447709, 4.9494538042367076, 1.8082002172860496, 4.474622796524546, 5.796741316547201, 0.4853959294561085, 3.627518196695667, 2.655624109746674]

symmetries =  [1.00000000e+00 2.49768845e-13 2.97490121e-13 6.13618410e-13
 3.08702136e-12 9.47816835e-12 6.65917723e-07 1.19002950e-10
 2.86470457e-10 2.19431629e-09 2.53182221e-08 3.76110469e-06
 5.33449390e-06 6.78139456e-07 9.99950648e-06 1.52974845e-04
 7.65920649e-03 1.71527408e-03]
symmetry error = 2.0467093985035933e-05
minimal relative intensity = 0.999163 rms = 0.00015247

Improved Uniform 108/14-design

Further investigations revealed that, starting with the relaxed eps = 1E-8 solution it is possible to lower eps even further leading to an exact solution thereby preserving the improved uniformity:

108 points (theta,phi), l_max = 14:
theta = [1.706258830293821, 1.4353338232959723, 1.4353338232959723, 1.706258830293821, 0.493410941036837, 2.6481817125529563, 2.6481817125529563, 0.493410941036837, 1.0995791012810194, 2.0420135523087737, 2.0420135523087737, 1.0995791012810194, 2.0484487623934915, 1.0931438911963016, 1.0931438911963016, 2.0484487623934915, 1.7889961490961024, 1.3525965044936907, 1.3525965044936907, 1.7889961490961024, 0.5329942822272317, 2.608598371362562, 2.608598371362562, 0.5329942822272317, 0.7590384888285923, 2.382554164761201, 2.382554164761201, 0.7590384888285923, 1.7602457310190778, 1.3813469225707156, 1.3813469225707156, 1.7602457310190778, 0.8473678967795097, 2.2942247568102836, 2.2942247568102836, 0.8473678967795097, 2.0596464314079723, 1.0819462221818208, 1.0819462221818208, 2.0596464314079723, 0.7442675500191338, 2.3973251035706595, 2.3973251035706595, 0.7442675500191338, 1.0607209225713132, 2.08087173101848, 2.08087173101848, 1.0607209225713132, 2.1213673096492127, 1.0202253439405806, 1.0202253439405806, 2.1213673096492127, 2.384106376084974, 0.7574862775048196, 0.7574862775048196, 2.384106376084974, 1.109162191782581, 2.032430461807212, 2.032430461807212, 1.109162191782581, 0.7565413115823003, 2.385051342007493, 2.385051342007493, 0.7565413115823003, 1.4300128688821834, 1.71157978470761, 1.71157978470761, 1.4300128688821834, 0.8340049050003967, 2.3075877485893965, 2.3075877485893965, 0.8340049050003967, 0.1559518378527657, 2.9856408157370273, 2.9856408157370273, 0.1559518378527657, 1.416295523904419, 1.725297129685374, 1.725297129685374, 1.416295523904419, 1.5497406494173571, 1.591852004172436, 1.591852004172436, 1.5497406494173571, 0.40253002412947403, 2.739062629460319, 2.739062629460319, 0.40253002412947403, 1.4380151811633661, 1.7035774724264272, 1.7035774724264272, 1.4380151811633661, 1.1931878645240486, 1.9484047890657445, 1.9484047890657445, 1.1931878645240486, 1.9344447568191627, 1.2071478967706306, 1.2071478967706306, 1.9344447568191627, 2.7317430441088373, 0.40984960948095595, 0.40984960948095595, 2.7317430441088373, 1.7514057341801088, 1.3901869194096843, 1.3901869194096843, 1.7514057341801088]
phi = [0.4759335887003198, 5.807251718479266, 2.6656590648894736, 3.6175262422901127, 5.994039525262884, 0.28914578191670237, 3.4307384355064956, 2.8524468716730906, 1.7229496260502266, 4.56023568112936, 1.4186430275395665, 4.86454227964002, 1.817031377917313, 4.466153929262273, 1.32456127567248, 4.958624031507107, 5.792919583356634, 0.49026572382295286, 3.6318583774127458, 2.6513269297668405, 3.581693670544203, 2.7014916366353834, 5.8430842902251765, 0.4401010169544099, 1.8479604641564233, 4.435224843023163, 1.29363218943337, 4.989553117746216, 0.8311620611539188, 5.452023246025668, 2.3104305924358743, 3.972754714743712, 6.029219234254537, 0.2539660729250495, 3.3955587265148424, 2.887626580664744, 0.5859796953942574, 5.6972056117853285, 2.555612958195536, 3.7275723489840504, 5.517236576695909, 0.7659487304836774, 3.9075413840734705, 2.3756439231061157, 2.1389871205529127, 4.1441981866266735, 1.0026055330368806, 5.280579774142706, 2.5916419678038247, 3.6915433393757615, 0.5499506857859685, 5.733234621393618, 5.4176747637460005, 0.865510543433586, 4.007103197023379, 2.2760821101562074, 4.088323527509269, 2.1948617796703167, 5.336454433260109, 0.9467308739194766, 1.3649209100988406, 4.918264397080746, 1.7766717434909527, 4.5065135636886335, 0.8249017676634591, 5.4582835395161275, 2.3166908859263344, 3.966494421253252, 0.19061137877018797, 6.092573928409398, 2.9509812748196054, 3.332204032359981, 0.13597138383448135, 6.147213923345105, 3.005621269755312, 3.2775640374242743, 1.549486782477249, 4.733698524702337, 1.5921058711125444, 4.691079436067042, 0.1545353328099902, 6.128649974369596, 2.987057320779803, 3.2961279863997834, 1.2260576725533572, 5.057127634626229, 1.9155349810364362, 4.36765032614315, 1.1896629541652977, 5.093522353014288, 1.9519296994244955, 4.331255607755091, 0.14291127931076092, 6.140274027868825, 2.998681374279032, 3.284503932900554, 3.3349938753686654, 2.948191431810921, 6.089784085400714, 0.19340122177887265, 4.244732683088433, 2.038452624091154, 5.1800452776809465, 1.1031400294986393, 4.3424403423543865, 1.9407449648252002, 5.082337618414993, 1.2008476887645931]

symmetries =  [1.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
 0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
 0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
 0.00000000e+00 0.00000000e+00 0.00000000e+00 5.57703952e-04
 5.90610848e-03 2.56996114e-03]
symmetry error = 0.0
minimal relative intensity = 0.999117 rms = 0.00016444

The weighted 132/19 design mentioned above achieves an even lower intensity rms = 0.00014760 and has 5 symmetries more. If it is possible to construct three different lasers whose beam strengths match the three given weight values precisely, this would be an even better solution. The weighted 192/23 above is even better. Additionally, the Gauss-Quadratures page presents smaller solutions (130/19 and 186/23), although they involve a greater number of weights.

Table of Uniform Designs Optimized for RMS

Not all Sloanes designs are easy to improve - I failed for 120/15, 156/17 and 180/18. But we can mitigate this issue by just adding 12 lasers more. The resulting design no longer has minimal size, but improves significantly regarding uniformity rms. Graef focusses on lowering the number of points, but some of his designs have exceptional low uniformity rms:

108/14: rms = 0.00016444 (same as ours)
156/17: rms = 0.00003266
216/20: rms = 0.00000573

Combining everything we have, including good designs from Sloane and Graef, we finally get the following rms values:

60/10-design has rms = 0.00339499
64/10-design has rms = 0.00259430
72/11-design has rms = 0.00113089
84/12-design has rms = 0.00144015
96/13-design has rms = 0.00040338
96/10-design has rms = 0.00026632
108/14-design has rms = 0.00016444
108/13-design has rms = 0.00015962
120/14-design has rms = 0.00007921
132/15-design has rms = 0.00003909
144/16-design has rms = 0.00005939
156/17-design has rms = 0.00003266
168/17-design has rms = 0.00002226
180/18-design has rms = 0.00001566
192/18-design has rms = 0.00000915
204/19-design has rms = 0.00000907
216/20-design has rms = 0.00000573

The 132/15-design looks quite beautiful.

60 points (theta,phi), l_max = 10: # from Graef:
theta = [1.8506658161647254, 1.6914200773212014, 1.1297740862167645, 2.1483211987319866, 1.536700171015863, 0.2908588224971378, 0.36755130003338415, 0.6707496275008731, 0.7169266926924625, 0.7768132988782437, 2.8507338310926555, 2.774041353556409, 2.4708430260889203, 2.4246659608973307, 2.3647793547115494, 0.2908588224971378, 0.36755130003338415, 0.6707496275008731, 0.7169266926924625, 0.7768132988782437, 2.8507338310926555, 2.774041353556409, 2.4708430260889203, 2.4246659608973307, 2.3647793547115494, 1.647923491215885, 1.2253843391866721, 1.1019929360281577, 1.1964962129498298, 0.7951453208796827, 1.493669162373908, 1.916208314403121, 2.0395997175616354, 1.9450964406399636, 2.3464473327101105, 1.493669162373908, 1.916208314403121, 2.0395997175616354, 1.9450964406399636, 2.3464473327101105, 1.647923491215885, 1.2253843391866721, 1.1019929360281577, 1.1964962129498298, 0.7951453208796827, 1.290926837425068, 1.450172576268592, 2.011818567373029, 0.9932714548578065, 1.6048924825739304, 1.290926837425068, 1.450172576268592, 2.011818567373029, 0.9932714548578065, 1.6048924825739304, 1.8506658161647254, 1.6914200773212014, 1.1297740862167645, 2.1483211987319866, 1.536700171015863]
phi = [6.20292916330498, 0.34804704120231844, 0.5231662407931559, 0.45159110354310444, 0.7762214932086381, 4.440366630326179, 5.941711677446939, 0.7570053597054287, 5.302485100749999, 0.048648352010954736, 1.8428186768534074, 0.3414736297326474, 5.526179947474158, 0.9807002064295871, 6.234536955168632, 1.298773976736386, 2.800119023857146, 3.8985980132952216, 2.1608924471602062, 3.1902410056007477, 4.9844113304432005, 3.48306628332244, 2.3845872938843646, 4.12229286001938, 3.0929443015788385, 1.8515228982955914, 1.6990323337739885, 1.071863838878197, 2.1976053421846853, 1.523031382647103, 1.290069755294202, 1.4425603198158048, 2.0697288147115964, 0.9439873114051077, 1.6185612709426902, 4.431662408883994, 4.5841529734055975, 5.2113214683013895, 4.085579964994901, 4.760153924532483, 4.993115551885384, 4.840624987363782, 4.21345649246799, 5.339197995774478, 4.664624036236896, 0.08025614387460586, 5.935138265977268, 5.76001906638643, 5.831594203636482, 5.506963813970948, 3.221848797464399, 2.793545612387475, 2.6184264127966372, 2.6900015500466887, 2.365371160381155, 3.0613365097151872, 3.4896396947921113, 3.664758894382949, 3.5931837571328975, 3.917814146798431]

symmetries =  [1.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
 0.00000000e+00 0.00000000e+00 5.97990845e-24 0.00000000e+00
 2.53313915e-23 1.73581748e-23 1.17458458e-23 8.85549983e-04
 1.37926578e-02 4.63722277e-02]
symmetry error = 6.041532050275254e-23
minimal relative intensity = 0.985449 rms = 0.00339499


64 points (theta,phi), l_max = 10:
theta = [1.632176538586544, 1.541688911680978, 1.3300416716745278, 1.2641992397676671, 1.2884281792889045, 1.974168561187641, 2.553136588305605, 0.672573799572858, 1.5756886612978627, 1.6355461816045842, 1.2212501465321162, 1.553888724571982, 1.9368527951139987, 1.7880716040330944, 0.8284134646292552, 2.1349081773154723, 1.7127532106010617, 2.715907708012393, 1.2172939132962395, 1.6830575105705072, 2.183235068444593, 2.606407466975482, 2.090473195179284, 0.6683753273776721, 0.5834368117475511, 0.20742727460890387, 0.39945998560604573, 2.2865597439227323, 2.088087438893532, 0.7847942372570159, 0.6410993263054289, 2.263863699247259, 1.7983250149328056, 2.380945374350676, 1.6063539272905882, 0.907176398038652, 1.7168667256452923, 2.3932642411533727, 2.7242184610868456, 2.4332332267728987, 3.0164620429373903, 1.0449966977074867, 1.7486624786503266, 2.0632405648891092, 1.502646700621933, 1.2931782612810376, 0.5183724923358085, 1.0555009762338166, 0.8640670638021295, 2.5089766326077334, 0.2275479031902284, 1.1253845327889098, 1.1541839152535034, 1.7694120093027463, 2.0453942474542073, 2.1407429069189754, 1.4994097070404924, 1.202043855551731, 1.261315669427466, 0.6246538593118189, 2.0678337198881094, 0.9892068571582587, 1.02246052950625, 2.773938088216067]
phi = [5.7167224227804265, 3.146743639529362, 2.281850739624058, 1.798249121235623, 4.155246739562434, 0.7355192986988576, 4.8697731360856, 4.541943610316178, 1.506785270083141, 1.1044095784152694, 5.457172832724657, 0.6158729469179233, 3.3328997252426347, 2.318287769225026, 2.8463853483461383, 2.8337739143068417, 4.171410112404639, 2.9478470919551225, 2.7013217913424685, 2.757785710496127, 1.7568527785194248, 1.937482725040812, 5.903158714864713, 0.6916071762978273, 6.169637169400199, 0.7803472476934571, 5.088298308350221, 5.498723555865031, 4.9616537441856945, 5.620681074615373, 3.491897809223288, 2.3794153866977483, 5.329592653984892, 3.3662888283029377, 4.6518521666345665, 2.107009013433988, 3.702680117356994, 0.9503700321813728, 5.855654356816859, 0.12226206919162971, 4.1678228317315655, 0.012734436621896822, 6.192971982356003, 1.313455035251271, 0.2165191495585899, 3.6675496023440544, 2.327013139147196, 0.5225444638837036, 3.97025721423691, 4.146760537598115, 3.466988420314265, 3.281895544895507, 4.589346642719668, 1.9168312306551611, 4.506273918042864, 3.928514005708311, 5.043852473400535, 0.986668894572328, 5.9558525133380815, 1.4654702425206896, 0.2813558335139145, 5.050506645657774, 1.3774035612670399, 0.9925567841368828]

symmetries =  [1.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
 0.00000000e+00 0.00000000e+00 6.96020276e-24 1.03278318e-23
 4.89375333e-24 2.87689251e-24 0.00000000e+00 1.37545052e-02
 1.30029367e-02 2.13081134e-02]
symmetry error = 2.5058680429656315e-23
minimal relative intensity = 0.983049 rms = 0.00261470


72 points (theta,phi), l_max = 11: # similar but not equivalent to Sloanes:
theta = [1.716347423153848, 1.4252452304359453, 1.4252452304359453, 1.716347423153848, 1.4265942197311197, 1.7149984338586735, 1.7149984338586735, 1.4265942197311197, 0.20561843586241377, 2.9359742177273795, 2.9359742177273795, 0.20561843586241377, 2.12724052710617, 1.0143521264836233, 1.0143521264836233, 2.12724052710617, 0.556445221693057, 2.5851474318967362, 2.5851474318967362, 0.556445221693057, 1.5698391648092676, 1.5717534887805258, 1.5717534887805258, 1.5698391648092676, 2.0852360750346177, 1.0563565785551754, 1.0563565785551754, 2.0852360750346177, 2.4389152668645857, 0.7026773867252075, 0.7026773867252075, 2.4389152668645857, 2.0031198259586938, 1.1384728276310996, 1.1384728276310996, 2.0031198259586938, 1.2142702030108135, 1.9273224505789797, 1.9273224505789797, 1.2142702030108135, 2.3137677151644604, 0.8278249384253328, 0.8278249384253328, 2.3137677151644604, 2.2764166984375693, 0.8651759551522241, 0.8651759551522241, 2.2764166984375693, 1.6834503389025683, 1.458142314687225, 1.458142314687225, 1.6834503389025683, 2.5358740194204668, 0.6057186341693267, 0.6057186341693267, 2.5358740194204668, 2.162944460087873, 0.9786481935019203, 0.9786481935019203, 2.162944460087873, 2.7123666772615422, 0.4292259763282511, 0.4292259763282511, 2.7123666772615422, 1.9033276509291959, 1.2382650026605972, 1.2382650026605972, 1.9033276509291959, 1.3097041378949388, 1.8318885156948543, 1.8318885156948543, 1.3097041378949388]
phi = [1.4250422028846832, 4.8581431042949035, 1.7165504507051101, 4.566634856474476, 6.136096628499513, 0.14708867868007328, 3.2886813322698663, 2.99450397490972, 2.360817488745908, 3.922367818433678, 0.7807751648438851, 5.502410142335701, 0.0011272154003525233, 6.282058091779234, 3.1404654381894406, 3.1427198689901457, 4.714201200827698, 1.5689841063518886, 4.710576759941682, 1.5726085472379046, 2.127240812035012, 4.155944495144574, 1.014351841554781, 5.268833465624805, 3.6436997308849195, 2.6394855762946667, 5.78107822988446, 0.5021070772951267, 4.007020638971039, 2.276164668208547, 5.41775732179834, 0.8654279853812465, 4.139686430662897, 2.1434988765166896, 5.285091530106483, 0.9980937770731036, 3.9058820056807573, 2.377303301498829, 5.518895955088622, 0.764289352090964, 2.647860163415648, 3.6353251437639384, 0.49373249017414556, 5.789452817005441, 5.18870506681834, 1.094480240361246, 4.236072893951039, 2.0471124132285476, 3.738038110054841, 2.545147197124745, 5.686739850714538, 0.596445456465048, 3.3403435398462973, 2.942841767333289, 6.084434420923082, 0.1987508862565042, 4.576488371291892, 1.7066969358876942, 4.848289589477488, 1.434895717702099, 2.472506082232052, 3.8106792249475343, 0.6690865713577413, 5.614098735821845, 4.988999256997134, 1.2941860501824518, 4.435778703772245, 1.8474066034073413, 3.486264907506691, 2.7969203996728953, 5.938513053262689, 0.3446722539168978]

symmetries =  [1.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
 0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
 0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
 2.25363274e-03 1.60685565e-05 4.71086885e-06]
symmetry error = 0.0
minimal relative intensity = 0.995147 rms = 0.00113089


84 points (theta,phi) l_max = 12:
theta = [1.6508919029820703, 1.490700750607723, 1.490700750607723, 1.6508919029820703, 0.5333063652716179, 2.608286288318175, 2.608286288318175, 0.5333063652716179, 1.044831313412919, 2.096761340176874, 2.096761340176874, 1.044831313412919, 1.1513389488377743, 1.990253704752019, 1.990253704752019, 1.1513389488377743, 2.400219297091892, 0.7413733564979015, 0.7413733564979015, 2.400219297091892, 1.0019364555401693, 2.139656198049624, 2.139656198049624, 1.0019364555401693, 1.3655591969233047, 1.7760334566664886, 1.7760334566664886, 1.3655591969233047, 2.9128677557940685, 0.22872489779572466, 0.22872489779572466, 2.9128677557940685, 1.4712591756345508, 1.6703334779552423, 1.6703334779552423, 1.4712591756345508, 2.4042622755825525, 0.7373303780072408, 0.7373303780072408, 2.4042622755825525, 1.7333484303052105, 1.4082442232845827, 1.4082442232845827, 1.7333484303052105, 0.8598581319414899, 2.2817345216483034, 2.2817345216483034, 0.8598581319414899, 2.4388799427277115, 0.7027127108620819, 0.7027127108620819, 2.4388799427277115, 1.0870786133611214, 2.0545140402286717, 2.0545140402286717, 1.0870786133611214, 2.0361885081438467, 1.1054041454459467, 1.1054041454459467, 2.0361885081438467, 1.303256894587972, 1.8383357590018212, 1.8383357590018212, 1.303256894587972, 1.3718440804446392, 1.7697485731451539, 1.7697485731451539, 1.3718440804446392, 0.3363810775159133, 2.80521157607388, 2.80521157607388, 0.3363810775159133, 0.5764284755727427, 2.5651641780170507, 2.5651641780170507, 0.5764284755727427, 1.4081757248998346, 1.7334169286899588, 1.7334169286899588, 1.4081757248998346, 2.1181512506414517, 1.0234414029483416, 1.0234414029483416, 2.1181512506414517]
phi = [0.527833089711558, 5.755352217468028, 2.613759563878235, 3.669425743301351, 6.125147093582933, 0.158038213596654, 3.299630867186447, 2.983554439993139, 1.663443002872254, 4.619742304307332, 1.4781496507175393, 4.805035656462047, 2.510779401995272, 3.772405905184314, 0.6308132515945212, 5.652372055585065, 0.6473643576220225, 5.635820949557564, 2.494228295967771, 3.7889570112118154, 5.216913795592463, 1.0662715115871235, 4.207864165176916, 2.07532114200267, 3.039914396593268, 3.2432709105863182, 0.10167825699652523, 6.181507050183061, 1.1171165610556897, 5.166068746123896, 2.024476092534104, 4.258709214645482, 4.918661724703433, 1.3645235824761524, 4.506116236065946, 1.7770690711136408, 1.8139004689626967, 4.46928483821689, 1.3276921846270966, 4.95549312255249, 5.43488892336605, 0.8482963838135363, 3.989889037403329, 2.293296269776257, 3.3568265884831066, 2.9263587186964797, 6.067951372286273, 0.21523393489331358, 5.515622226041898, 0.7675630811376888, 3.9091557347274817, 2.3740295724521046, 4.180766387297085, 2.102418919882501, 5.244011573472294, 1.0391737337072922, 2.5942466338700156, 3.6889386733095706, 0.5473460197197776, 5.735839287459808, 1.3644010826236461, 4.9187842245559406, 1.777191570966147, 4.505993736213439, 0.27305949903723054, 6.010125808142356, 2.8685331545525625, 3.4146521526270237, 0.6419813181002244, 5.6412039890793615, 2.499611335489569, 3.7835739716900174, 5.013995943482552, 1.2691893636970344, 4.410782017286827, 1.8724032898927587, 2.126320323580144, 4.1568649835994425, 1.0152723300096498, 5.267912977169937, 0.190760193741466, 6.092425113438121, 2.950832459848327, 3.332352847331259]

symmetries =  [1.         0.         0.         0.         0.         0.
 0.         0.         0.         0.         0.         0.
 0.         0.01179593 0.00850964 0.02879536]
symmetry error = 0.0
minimal relative intensity = 0.992326 rms = 0.00144015


96 points (theta,phi), l_max = 13:
theta = [1.0488516157783208, 2.0927410378114724, 2.0927410378114724, 1.0488516157783208, 2.573107135678921, 0.5684855179108724, 0.5684855179108724, 2.573107135678921, 1.7753420279912235, 1.3662506255985698, 1.3662506255985698, 1.7753420279912235, 1.3912227636806267, 1.7503698899091664, 1.7503698899091664, 1.3912227636806267, 2.9555379096336067, 0.18605474395618668, 0.18605474395618668, 2.9555379096336067, 1.5226399954968812, 1.618952658092912, 1.618952658092912, 1.5226399954968812, 1.8860578671977475, 1.2555347863920456, 1.2555347863920456, 1.8860578671977475, 1.4139698639699234, 1.7276227896198697, 1.7276227896198697, 1.4139698639699234, 2.7870302681977797, 0.35456238539201373, 0.35456238539201373, 2.7870302681977797, 1.5122538748797063, 1.629338778710087, 1.629338778710087, 1.5122538748797063, 0.6984818613351144, 2.4431107922546786, 2.4431107922546786, 0.6984818613351144, 0.8757923043458723, 2.265800349243921, 2.265800349243921, 0.8757923043458723, 0.9903862092499145, 2.151206444339879, 2.151206444339879, 0.9903862092499145, 2.000571332655021, 1.1410213209347722, 1.1410213209347722, 2.000571332655021, 2.381888709181271, 0.7597039444085223, 0.7597039444085223, 2.381888709181271, 1.9665137474062429, 1.1750789061835505, 1.1750789061835505, 1.9665137474062429, 0.8316394835377792, 2.309953170052014, 2.309953170052014, 0.8316394835377792, 0.888540002270664, 2.253052651319129, 2.253052651319129, 0.888540002270664, 1.994277449371569, 1.1473152042182244, 1.1473152042182244, 1.994277449371569, 1.7856576501475616, 1.3559350034422317, 1.3559350034422317, 1.7856576501475616, 2.660265534045337, 0.4813271195444561, 0.4813271195444561, 2.660265534045337, 2.5727074735620548, 0.5688851800277384, 0.5688851800277384, 2.5727074735620548, 1.3944966604786826, 1.7470959931111105, 1.7470959931111105, 1.3944966604786826, 1.0363761537058764, 2.105216499883917, 2.105216499883917, 1.0363761537058764]
phi = [3.3781137864054243, 2.905071520774162, 6.046664174363955, 0.23652113281563145, 1.9576763846058225, 4.325508922573764, 1.1839162689839708, 5.099269038195615, 5.246622578006132, 1.0365627291734536, 4.178155382763247, 2.1050299244163395, 3.0926486868381997, 3.1905366203413865, 0.04894396675159353, 6.234241340427992, 1.3075384517855468, 4.975646855394039, 1.8340542018042465, 4.44913110537534, 4.892173236915219, 1.3910120702643676, 4.532604723854161, 1.7505805833254255, 4.8774179846383925, 1.405767322541194, 4.547359976130987, 1.7358253310485992, 3.460909017653525, 2.8222762895260614, 5.9638689431158545, 0.31931636406373176, 2.674977594329814, 3.6082077128497723, 0.46661505925997937, 5.816570247919607, 0.696435700175368, 5.586749607004219, 2.445156953414425, 3.838028353765161, 0.09111191692325762, 6.192073390256328, 3.0504807366665356, 3.2327045705130506, 1.4945439082096512, 4.788641398969935, 1.647048745380142, 4.636136561799445, 4.190794881896958, 2.0923904252826286, 5.233983078872422, 1.0492022283071645, 2.49405484748909, 3.7891304596904964, 0.6475378061007036, 5.635647501078883, 5.633421898155788, 0.6497634090237983, 3.791356062613591, 2.491829244565995, 0.7523487168518461, 5.53083659032774, 2.389243936737947, 3.893941370441639, 5.7344781025996365, 0.5487072045799493, 3.690299858169742, 2.5928854490098443, 2.0905228221544894, 4.192662485025097, 1.0510698314353037, 5.2321154757442825, 4.476331570545134, 1.8068537366344526, 4.9484463902242455, 1.3347389169553405, 3.5757080582107226, 2.7074772489688637, 5.849069902558656, 0.4341154046209295, 3.620201524853621, 2.6629837823259654, 5.804576435915759, 0.4786088712638278, 1.2391710606338548, 5.044014246545731, 1.9024215929559385, 4.380763714223647, 5.256153904675299, 1.0270314025042873, 4.16862405609408, 2.1145612510855063, 2.9363490811343884, 3.346836226045198, 0.2052435724554049, 6.077941734724181]

symmetries =  [1.         0.         0.         0.         0.         0.
 0.         0.         0.         0.         0.         0.
 0.         0.         0.00135019 0.00553928 0.00483174]
symmetry error = 0.0
minimal relative intensity = 0.997621 rms = 0.00040338


96 points (theta,phi), l_max=10: # reduced l_max to lower rms:
theta = [1.6163080771408236, 1.5252845764489698, 1.5252845764489698, 1.6163080771408236, 2.4998885381584524, 0.6417041154313411, 0.6417041154313411, 2.4998885381584524, 2.210340645178749, 0.9312520084110443, 0.9312520084110443, 2.210340645178749, 1.9672377162557928, 1.1743549373340005, 1.1743549373340005, 1.9672377162557928, 2.3649308862314826, 0.7766617673583107, 0.7766617673583107, 2.3649308862314826, 2.195609305709714, 0.9459833478800792, 0.9459833478800792, 2.195609305709714, 1.0004100803821756, 2.141182573207618, 2.141182573207618, 1.0004100803821756, 2.5622673062771355, 0.5793253473126576, 0.5793253473126576, 2.5622673062771355, 1.661241939729434, 1.4803507138603593, 1.4803507138603593, 1.661241939729434, 1.8481855490196166, 1.2934071045701767, 1.2934071045701767, 1.8481855490196166, 1.692294349919484, 1.4492983036703093, 1.4492983036703093, 1.692294349919484, 2.8374583216106726, 0.3041343319791206, 0.3041343319791206, 2.8374583216106726, 2.2600975034426174, 0.8814951501471758, 0.8814951501471758, 2.2600975034426174, 2.306961724830524, 0.8346309287592691, 0.8346309287592691, 2.306961724830524, 1.1806307273272802, 1.960961926262513, 1.960961926262513, 1.1806307273272802, 1.2962474776548982, 1.845345175934895, 1.845345175934895, 1.2962474776548982, 2.6161834277903155, 0.5254092257994775, 0.5254092257994775, 2.6161834277903155, 2.006424915496522, 1.135167738093271, 1.135167738093271, 2.006424915496522, 1.308552963758806, 1.8330396898309873, 1.8330396898309873, 1.308552963758806, 2.0639266063400727, 1.0776660472497204, 1.0776660472497204, 2.0639266063400727, 0.5701106715583705, 2.5714819820314228, 2.5714819820314228, 0.5701106715583705, 1.4905902957775565, 1.6510023578122368, 1.6510023578122368, 1.4905902957775565, 1.7718973385439922, 1.369695315045801, 1.369695315045801, 1.7718973385439922, 2.9246780487583854, 0.21691460483140762, 0.21691460483140762, 2.9246780487583854]
phi = [3.7819082194390443, 2.501277087740542, 5.642869741330335, 0.6403155658492512, 3.2176750065800035, 3.0655103005995827, 6.207102954189376, 0.0760823529902103, 4.655656326346129, 1.627528980833457, 4.76912163442325, 1.5140636727563364, 3.828472859130337, 2.454712448049249, 5.596305101639042, 0.686880205540544, 3.72505388799098, 2.5581314191886064, 5.699724072778399, 0.5834612344011869, 4.216193357532221, 2.066991949647365, 5.208584603237158, 1.0746007039424281, 3.2491100755224105, 3.0340752316571757, 6.175667885246969, 0.1075174219326177, 1.7365386586432408, 4.5466466485363455, 1.4050539949465526, 4.878131312233034, 5.285410352502411, 0.9977749546771755, 4.139367608266968, 2.143817698912618, 4.58603657696334, 1.6971487302162462, 4.838741383806039, 1.4444439233735469, 3.4210970762700006, 2.8620882309095856, 6.003680884499379, 0.2795044226802075, 3.5582612398143922, 2.724924067365194, 5.866516720954987, 0.4166685862245991, 2.626209890293443, 3.656975416886143, 0.51538276329635, 5.767802543883236, 5.251353658103977, 1.03183164907561, 4.173424302665403, 2.109761004514183, 3.954100457614544, 2.3290848495650422, 5.470677503154835, 0.8125078040247511, 3.5954097775923253, 2.687775529587261, 5.829368183177054, 0.4538171240025325, 2.570524150962248, 3.7126611562173384, 0.5710685026275455, 5.712116804552041, 5.016077243608352, 1.267108063571234, 4.408700717161027, 1.8744845900185594, 2.0830502642804847, 4.200135042899102, 1.0585423893093087, 5.224642917870278, 0.29873741687973276, 5.984447890299854, 2.8428552367100606, 3.4403300704695257, 5.213423388558628, 1.069761918620959, 4.211354572210752, 2.071830734968834, 4.510630457012318, 1.7725548501672688, 4.914147503757062, 1.3690378034225246, 3.0597333488277667, 3.2234519583518195, 0.08185930476202671, 6.201326002417559, 5.093847720768554, 1.1893375864110323, 4.330930240000825, 1.9522550671787609]

symmetries =  [1.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
 0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
 0.00000000e+00 0.00000000e+00 0.00000000e+00 4.82400429e-05
 1.91813676e-04 6.64968057e-05]
symmetry error = 0.0
minimal relative intensity = 0.998601 rms = 0.00026632


108 points (theta,phi), l_max = 14:
theta = [1.706258830293821, 1.4353338232959723, 1.4353338232959723, 1.706258830293821, 0.493410941036837, 2.6481817125529563, 2.6481817125529563, 0.493410941036837, 1.0995791012810194, 2.0420135523087737, 2.0420135523087737, 1.0995791012810194, 2.0484487623934915, 1.0931438911963016, 1.0931438911963016, 2.0484487623934915, 1.7889961490961024, 1.3525965044936907, 1.3525965044936907, 1.7889961490961024, 0.5329942822272317, 2.608598371362562, 2.608598371362562, 0.5329942822272317, 0.7590384888285923, 2.382554164761201, 2.382554164761201, 0.7590384888285923, 1.7602457310190778, 1.3813469225707156, 1.3813469225707156, 1.7602457310190778, 0.8473678967795097, 2.2942247568102836, 2.2942247568102836, 0.8473678967795097, 2.0596464314079723, 1.0819462221818208, 1.0819462221818208, 2.0596464314079723, 0.7442675500191338, 2.3973251035706595, 2.3973251035706595, 0.7442675500191338, 1.0607209225713132, 2.08087173101848, 2.08087173101848, 1.0607209225713132, 2.1213673096492127, 1.0202253439405806, 1.0202253439405806, 2.1213673096492127, 2.384106376084974, 0.7574862775048196, 0.7574862775048196, 2.384106376084974, 1.109162191782581, 2.032430461807212, 2.032430461807212, 1.109162191782581, 0.7565413115823003, 2.385051342007493, 2.385051342007493, 0.7565413115823003, 1.4300128688821834, 1.71157978470761, 1.71157978470761, 1.4300128688821834, 0.8340049050003967, 2.3075877485893965, 2.3075877485893965, 0.8340049050003967, 0.1559518378527657, 2.9856408157370273, 2.9856408157370273, 0.1559518378527657, 1.416295523904419, 1.725297129685374, 1.725297129685374, 1.416295523904419, 1.5497406494173571, 1.591852004172436, 1.591852004172436, 1.5497406494173571, 0.40253002412947403, 2.739062629460319, 2.739062629460319, 0.40253002412947403, 1.4380151811633661, 1.7035774724264272, 1.7035774724264272, 1.4380151811633661, 1.1931878645240486, 1.9484047890657445, 1.9484047890657445, 1.1931878645240486, 1.9344447568191627, 1.2071478967706306, 1.2071478967706306, 1.9344447568191627, 2.7317430441088373, 0.40984960948095595, 0.40984960948095595, 2.7317430441088373, 1.7514057341801088, 1.3901869194096843, 1.3901869194096843, 1.7514057341801088]
phi = [0.4759335887003198, 5.807251718479266, 2.6656590648894736, 3.6175262422901127, 5.994039525262884, 0.28914578191670237, 3.4307384355064956, 2.8524468716730906, 1.7229496260502266, 4.56023568112936, 1.4186430275395665, 4.86454227964002, 1.817031377917313, 4.466153929262273, 1.32456127567248, 4.958624031507107, 5.792919583356634, 0.49026572382295286, 3.6318583774127458, 2.6513269297668405, 3.581693670544203, 2.7014916366353834, 5.8430842902251765, 0.4401010169544099, 1.8479604641564233, 4.435224843023163, 1.29363218943337, 4.989553117746216, 0.8311620611539188, 5.452023246025668, 2.3104305924358743, 3.972754714743712, 6.029219234254537, 0.2539660729250495, 3.3955587265148424, 2.887626580664744, 0.5859796953942574, 5.6972056117853285, 2.555612958195536, 3.7275723489840504, 5.517236576695909, 0.7659487304836774, 3.9075413840734705, 2.3756439231061157, 2.1389871205529127, 4.1441981866266735, 1.0026055330368806, 5.280579774142706, 2.5916419678038247, 3.6915433393757615, 0.5499506857859685, 5.733234621393618, 5.4176747637460005, 0.865510543433586, 4.007103197023379, 2.2760821101562074, 4.088323527509269, 2.1948617796703167, 5.336454433260109, 0.9467308739194766, 1.3649209100988406, 4.918264397080746, 1.7766717434909527, 4.5065135636886335, 0.8249017676634591, 5.4582835395161275, 2.3166908859263344, 3.966494421253252, 0.19061137877018797, 6.092573928409398, 2.9509812748196054, 3.332204032359981, 0.13597138383448135, 6.147213923345105, 3.005621269755312, 3.2775640374242743, 1.549486782477249, 4.733698524702337, 1.5921058711125444, 4.691079436067042, 0.1545353328099902, 6.128649974369596, 2.987057320779803, 3.2961279863997834, 1.2260576725533572, 5.057127634626229, 1.9155349810364362, 4.36765032614315, 1.1896629541652977, 5.093522353014288, 1.9519296994244955, 4.331255607755091, 0.14291127931076092, 6.140274027868825, 2.998681374279032, 3.284503932900554, 3.3349938753686654, 2.948191431810921, 6.089784085400714, 0.19340122177887265, 4.244732683088433, 2.038452624091154, 5.1800452776809465, 1.1031400294986393, 4.3424403423543865, 1.9407449648252002, 5.082337618414993, 1.2008476887645931]

symmetries =  [1.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
 0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
 0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
 0.00000000e+00 0.00000000e+00 0.00000000e+00 5.57703952e-04
 5.90610848e-03 2.56996114e-03]
symmetry error = 0.0
minimal relative intensity = 0.999117 rms = 0.00016444


108 points (theta,phi), l_max=13: # reduced l_max to lower rms
theta = [1.5654199299532323, 1.576172723636561, 1.576172723636561, 1.5654199299532323, 2.9858587337439455, 0.15573391984584778, 0.15573391984584778, 2.9858587337439455, 1.4151567559618714, 1.726435897627922, 1.726435897627922, 1.4151567559618714, 2.037883801224506, 1.103708852365287, 1.103708852365287, 2.037883801224506, 1.0268347386401584, 2.114757914949635, 2.114757914949635, 1.0268347386401584, 0.7559772910149308, 2.385615362574862, 2.385615362574862, 0.7559772910149308, 1.7196206278292465, 1.4219720257605468, 1.4219720257605468, 1.7196206278292465, 1.9387965748348095, 1.2027960787549838, 1.2027960787549838, 1.9387965748348095, 2.741928360757486, 0.3996642928323076, 0.3996642928323076, 2.741928360757486, 1.3879307619138288, 1.7536618916759643, 1.7536618916759643, 1.3879307619138288, 2.2964227448326193, 0.8451699087571738, 0.8451699087571738, 2.2964227448326193, 2.3827677498911046, 0.7588249036986886, 0.7588249036986886, 2.3827677498911046, 1.1964358067090146, 1.9451568468807787, 1.9451568468807787, 1.1964358067090146, 2.7292729572665, 0.4123196963232931, 0.4123196963232931, 2.7292729572665, 1.735457452359589, 1.4061352012302042, 1.4061352012302042, 1.735457452359589, 2.066442210411369, 1.075150443178424, 1.075150443178424, 2.066442210411369, 2.3961255135866963, 0.7454671400030967, 0.7454671400030967, 2.3961255135866963, 1.0659776158425567, 2.0756150377472364, 2.0756150377472364, 1.0659776158425567, 1.772688372760188, 1.3689042808296052, 1.3689042808296052, 1.772688372760188, 2.612386262061149, 0.5292063915286442, 0.5292063915286442, 2.612386262061149, 2.0525305166942687, 1.0890621368955244, 1.0890621368955244, 2.0525305166942687, 1.7195239708363697, 1.4220686827534235, 1.4220686827534235, 1.7195239708363697, 2.3048364807228117, 0.8367561728669815, 0.8367561728669815, 2.3048364807228117, 2.3855217554271135, 0.7560708981626796, 0.7560708981626796, 2.3855217554271135, 1.4182006478519606, 1.7233920057378325, 1.7233920057378325, 1.4182006478519606, 0.49757612102101767, 2.6440165325687754, 2.6440165325687754, 0.49757612102101767, 1.1012903966402177, 2.0403022569495755, 2.0403022569495755, 1.1012903966402177]
phi = [2.9859508149572194, 3.297234492222367, 0.1556418386325739, 6.127543468547012, 0.034669687909634796, 6.248515619269951, 3.1069229656801585, 3.1762623414994278, 4.717831159707585, 1.565354147472001, 4.706946801061794, 1.5762385061177924, 0.9525386058096127, 5.330646701369973, 2.1890540477801803, 4.094131259399406, 5.729009280306456, 0.5541760268731302, 3.695768680462923, 2.587416626716663, 2.2868277969619264, 3.99635751021766, 0.854764856627867, 5.4284204505517195, 4.340075500059234, 1.9431098071203527, 5.084702460710146, 1.1984828464694404, 3.301184363750163, 2.9820009434294232, 6.123593597019216, 0.15959171016036994, 4.321441966044153, 1.961743341135433, 5.103335994725226, 1.1798493124543603, 3.971619006928263, 2.3115663002513234, 5.4531589538411165, 0.8300263533384701, 2.8960474833859613, 3.387137823793625, 0.24554517020383212, 6.037640136975754, 4.9798540739252655, 1.303331233254321, 4.444923886844114, 1.8382614203354721, 3.318631538981429, 2.964553768198157, 6.10614642178795, 0.17703888539163637, 1.9922011696373136, 4.290984137542273, 1.1493914839524795, 5.133793823227107, 5.092142327313098, 1.1910429798664874, 4.33263563345628, 1.950549673723306, 2.5594532211394094, 3.723732086040177, 0.5821394324503839, 5.7010458747292025, 5.506177746311073, 0.7770075608685139, 3.9186002144583068, 2.3645850927212795, 4.137930889034239, 2.1452544181453477, 5.28684707173514, 0.9963382354444456, 3.6341970726174484, 2.648988234562138, 5.790580888151931, 0.49260441902765517, 3.550051083278211, 2.7331342239013754, 5.8747268774911685, 0.40845842968841806, 4.4841371531032195, 1.7990481540763663, 4.940640807666159, 1.342544499513427, 3.9682311611115137, 2.3149541460680725, 5.456546799657866, 0.8266385075217205, 3.34251776463112, 2.940667542548466, 6.082260196138259, 0.20092511104132696, 4.494689187310351, 1.7884961198692346, 4.930088773459028, 1.3530965337205585, 0.4754797464807979, 5.807705560698788, 2.6661129071089955, 3.6170724000705907, 0.32411366167659955, 5.959071645502987, 2.8174789919131937, 3.4657063152663925, 1.3995120151711895, 4.883673292008397, 1.7420806384186038, 4.541104668760982]

symmetries =  [1.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
 0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
 0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
 0.00000000e+00 0.00000000e+00 2.23336396e-05 3.38152217e-04
 6.38880688e-03]
symmetry error = 0.0
minimal relative intensity = 0.999085 rms = 0.00015962


120 points (theta,phi), l_max = 15: # from Sloane
theta = [1.546605035284434, 1.5949876183053593, 1.5949876183053593, 1.546605035284434, 1.340586901281069, 1.8010057523087244, 1.8010057523087244, 1.340586901281069, 2.910069860783339, 0.23152279280645455, 0.23152279280645455, 2.910069860783339, 1.3027781978055917, 1.8388144557842017, 1.8388144557842017, 1.3027781978055917, 0.48685809041265327, 2.65473456317714, 2.65473456317714, 0.48685809041265327, 1.966749140573332, 1.1748435130164612, 1.1748435130164612, 1.966749140573332, 2.0343019711119834, 1.10729068247781, 1.10729068247781, 2.0343019711119834, 1.0022476168820607, 2.1393450367077325, 2.1393450367077325, 1.0022476168820607, 0.7751688288073486, 2.3664238247824447, 2.3664238247824447, 0.7751688288073486, 0.7751688288062561, 2.366423824783537, 2.366423824783537, 0.7751688288062561, 2.139345036700983, 1.0022476168888104, 1.0022476168888104, 2.139345036700983, 2.0343019711182757, 1.1072906824715179, 1.1072906824715179, 2.0343019711182757, 0.7309323419373923, 2.410660311652401, 2.410660311652401, 0.7309323419373923, 2.270317342877302, 0.8712753107124914, 0.8712753107124914, 2.270317342877302, 1.3935248336315598, 1.7480678199582336, 1.7480678199582336, 1.3935248336315598, 1.8010057523063616, 1.3405869012834315, 1.3405869012834315, 1.8010057523063616, 1.5466050352848295, 1.5949876183049636, 1.5949876183049636, 1.5466050352848295, 2.910069860785731, 0.23152279280406193, 0.23152279280406193, 2.910069860785731, 2.410660311642648, 0.730932341947145, 0.730932341947145, 2.410660311642648, 1.748067819961221, 1.3935248336285724, 1.3935248336285724, 1.748067819961221, 0.8712753107037043, 2.2703173428860888, 2.2703173428860888, 0.8712753107037043, 0.486858090412392, 2.654734563177401, 2.654734563177401, 0.486858090412392, 1.3027781978097839, 1.8388144557800092, 1.8388144557800092, 1.3027781978097839, 1.1748435130137567, 1.9667491405760367, 1.9667491405760367, 1.1748435130137567, 1.0408659651537104, 2.100726688436083, 2.100726688436083, 1.0408659651537104, 1.4494968742736056, 1.6920957793161875, 1.6920957793161875, 1.4494968742736056, 0.5465635694479958, 2.5950290841417973, 2.5950290841417973, 0.5465635694479958, 2.5950290841362924, 0.5465635694535008, 0.5465635694535008, 2.5950290841362924, 1.449496874278012, 1.6920957793117812, 1.6920957793117812, 1.449496874278012, 1.0408659651468923, 2.100726688442901, 2.100726688442901, 1.0408659651468923]
phi = [4.942667000342016, 1.3405183068375701, 4.482110960427363, 1.801074346752223, 3.116745736722445, 3.166439570457141, 0.02484691686734809, 6.258338390312238, 1.465183307116186, 4.8180020000634, 1.6764093464736074, 4.606775960705979, 5.871704209440523, 0.4114810977390635, 3.5530737513288564, 2.73011155585073, 2.539902897336728, 3.743282409842858, 0.6016897562530653, 5.681495550926521, 1.2796730946224675, 5.003512212557119, 1.8619195589673256, 4.42126574821226, 0.9248961122278925, 5.358289194951694, 2.216696541361901, 4.066488765817685, 5.72393460239225, 0.5592507047873368, 3.7008433583771296, 2.5823419488024566, 2.263791337597923, 4.019393969581664, 0.8778013159918705, 5.4053839911877155, 3.8345876644041983, 2.448597642775388, 5.590190296365181, 0.6929950108144055, 2.130047031587414, 4.153138275592172, 1.0115456220023797, 5.271639685177206, 5.637285092618175, 0.6459002145614118, 3.7874928681512046, 2.4956924390283817, 2.8742586027074046, 3.4089267044721816, 0.26733405088238865, 6.015851256297197, 1.338235385018275, 4.944949922161311, 1.8033572685715182, 4.479828038608068, 5.570190466746898, 0.7129948404326882, 3.854587494022481, 2.428597813157105, 4.737235897251618, 1.5459494099279685, 4.687542063517761, 1.5956432436618246, 3.3718706735447537, 2.9113146336348326, 6.052907287224626, 0.23027801995496078, 3.0359796339117393, 3.247205673267847, 0.10561301967805412, 6.177572287501532, 1.8381303776788733, 4.445054929500713, 1.30346227591092, 4.979723031268666, 5.4253838208268705, 0.8578014863527155, 3.9993941399425084, 2.283791167237078, 3.3741535953721153, 2.909031711807471, 6.050624365397264, 0.23256094178232234, 0.9691065705519735, 5.3140787366276125, 2.17248608303782, 4.110699224141767, 1.159315229053512, 5.123870078126075, 1.982277424536281, 4.300907882643305, 0.29112323216819386, 5.992062075011392, 2.8504694214215993, 3.432715885757987, 1.4300957828269596, 4.853089524352627, 1.7114968707628337, 4.5716884364167525, 0.534272251572582, 5.748913055607004, 2.607320402017211, 3.675864905162375, 0.23496245552539977, 6.048222851654186, 2.9066301980643936, 3.3765551091151926, 1.3358338712803162, 4.94735143589927, 1.8057587823094772, 4.4774265248701095, 5.24666123196384, 1.0365240752157463, 4.17811672880554, 2.105068578374047, 3.0008921096264105, 3.2822931975531757, 0.14070054396338286, 6.142484763216204]

symmetries =  [1.         0.         0.         0.         0.         0.
 0.         0.         0.         0.         0.         0.
 0.         0.         0.         0.         0.00399324 0.00209691]
symmetry error = 0.0
minimal relative intensity = 0.999536 rms = 0.00007921


132 points (theta,phi) l_max = 15:
theta = [0.4684854499462691, 2.673107203643524, 2.673107203643524, 0.4684854499462691, 1.2984000324791398, 1.8431926211106533, 1.8431926211106533, 1.2984000324791398, 1.1997062041033923, 1.9418864494864008, 1.9418864494864008, 1.1997062041033923, 1.6971916873806943, 1.444400966209099, 1.444400966209099, 1.6971916873806943, 2.6147097369440075, 0.5268829166457858, 0.5268829166457858, 2.6147097369440075, 1.0623920521572776, 2.0792006014325155, 2.0792006014325155, 1.0623920521572776, 2.9500746674699263, 0.19151798611986703, 0.19151798611986703, 2.9500746674699263, 1.7510213357820383, 1.390571317807755, 1.390571317807755, 1.7510213357820383, 1.5067058127215087, 1.6348868408682846, 1.6348868408682846, 1.5067058127215087, 1.0702528992141347, 2.0713397543756584, 2.0713397543756584, 1.0702528992141347, 2.380531039113545, 0.7610616144762488, 0.7610616144762488, 2.380531039113545, 2.0890283823254805, 1.0525642712643126, 1.0525642712643126, 2.0890283823254805, 0.44267912479580357, 2.6989135287939896, 2.6989135287939896, 0.44267912479580357, 1.9648909780216046, 1.1767016755681885, 1.1767016755681885, 1.9648909780216046, 1.3797438522389833, 1.7618488013508098, 1.7618488013508098, 1.3797438522389833, 2.3455308549094585, 0.7960617986803348, 0.7960617986803348, 2.3455308549094585, 1.3326055095055083, 1.8089871440842848, 1.8089871440842848, 1.3326055095055083, 2.3111263145627796, 0.8304663390270137, 0.8304663390270137, 2.3111263145627796, 2.325974761813453, 0.81561789177634, 0.81561789177634, 2.325974761813453, 0.9311846029364954, 2.2104080506532977, 2.2104080506532977, 0.9311846029364954, 1.140598953017898, 2.0009937005718954, 2.0009937005718954, 1.140598953017898, 2.588175425302436, 0.5534172282873572, 0.5534172282873572, 2.588175425302436, 1.5704912593173153, 1.571101394272478, 1.571101394272478, 1.5704912593173153, 2.1242134510147577, 1.0173792025750357, 1.0173792025750357, 2.1242134510147577, 1.253872294287062, 1.8877203593027314, 1.8877203593027314, 1.253872294287062, 2.066802266278826, 1.0747903873109672, 1.0747903873109672, 2.066802266278826, 0.6051379601771035, 2.53645469341269, 2.53645469341269, 0.6051379601771035, 0.25211102053930556, 2.8894816330504876, 2.8894816330504876, 0.25211102053930556, 1.500472465531943, 1.64112018805785, 1.64112018805785, 1.500472465531943, 1.8124904759398213, 1.3291021776499718, 1.3291021776499718, 1.8124904759398213, 0.854627866855748, 2.2869647867340452, 2.2869647867340452, 0.854627866855748, 1.6943061622544453, 1.4472864913353478, 1.4472864913353478, 1.6943061622544453, 0.7314621748848394, 2.4101304787049536, 2.4101304787049536, 0.7314621748848394]
phi = [0.9324926694052448, 5.350692637774341, 2.2090999841845482, 4.074085322995038, 1.18476579661747, 5.098419510562116, 1.9568268569723233, 4.3263584502072625, 0.2928588910331448, 5.990326416146441, 2.8487337625566482, 3.434451544622938, 2.62870155169399, 3.6544837554855962, 0.5128911018958032, 5.770294205283783, 6.0297887368395875, 0.25339657033999863, 3.394989223929792, 2.8881960832497944, 4.567572106145851, 1.7156132010337346, 4.857205854623528, 1.4259794525560587, 2.7984279436224266, 3.4847573635571596, 0.3431647099673666, 5.94002059721222, 4.777536116640596, 1.50564919053899, 4.647241844128783, 1.6359434630508034, 3.32219252282822, 2.960992784351366, 6.102585437941159, 0.18059986923842722, 3.7415538448913335, 2.5416314622882528, 5.683224115878046, 0.5999611913015407, 2.372027850968037, 3.911157456211549, 0.7695648026217561, 5.51362050455783, 5.297677338666395, 0.9855079685131913, 4.127100622102985, 2.156084685076602, 2.6823171046474776, 3.6008682025321086, 0.4592755489423156, 5.823909758237271, 1.3636612607090903, 4.919524046470496, 1.7779313928807028, 4.5052539142988834, 5.881371552962978, 0.4018137542166078, 3.5434064078064007, 2.7397788993731855, 5.048877386633201, 1.2343079205463854, 4.375900574136178, 1.9072847330434077, 3.9451793330564024, 2.338005974123184, 5.479598627712977, 0.8035866794666096, 2.816283213477276, 3.46690209370231, 0.3253094401125173, 5.957875867067069, 0.6098612522210275, 5.673324054958559, 2.5317314013687655, 3.7514539058108207, 5.259006278776421, 1.0241790284031655, 4.165771681992958, 2.117413625186628, 2.4251285832107508, 3.8580567239688355, 0.7164640703790421, 5.566721236800544, 4.7129694008079985, 1.570215906371588, 4.711808559961381, 1.5713767472182052, 4.158971827415911, 2.1242134797636756, 5.265806133353468, 1.0173791738261178, 3.1412340601014996, 3.1419512470780866, 0.0003585934882937185, 6.282826713691293, 2.0953886415403997, 4.187796665639187, 1.0462040120493934, 5.236981295130192, 0.3622162368073112, 5.920969070372275, 2.779376416782482, 3.5038088903971043, 5.292153301071323, 0.9910320061082626, 4.132624659698056, 2.1505606474815306, 4.997938440933997, 1.2852468662455891, 4.426839519835382, 1.8563457873442042, 1.813101338993801, 4.470083968185785, 1.328491314595992, 4.954693992583595, 0.07243272970932178, 6.210752577470265, 3.0691599238804717, 3.2140253832991146, 1.734850463254161, 4.548334843925425, 1.406742190335632, 4.876443116843954, 0.7228686640854746, 5.5603166430941116, 2.4187239895043184, 3.864461317675268, 6.097686613020973, 0.18549869415861314, 3.3270913477484063, 2.95609395943118]

symmetries =  [1.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
 0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
 0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
 0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
 3.20657832e-04 1.04728369e-07 2.88753205e-03]
symmetry error = 0.0
minimal relative intensity = 0.999810 rms = 0.00003909


132 points (theta,phi) l_max = 15:
theta = [1.6982810859781827, 1.4433115676116106, 1.4433115676116106, 1.6982810859781827, 2.4063630030294765, 0.7352296505603169, 0.7352296505603169, 2.4063630030294765, 0.8518424358245617, 2.2897502177652314, 2.2897502177652314, 0.8518424358245617, 1.0729918056218453, 2.068600847967948, 2.068600847967948, 1.0729918056218453, 2.380381888616176, 0.7612107649736171, 0.7612107649736171, 2.380381888616176, 1.049720784533563, 2.0918718690562303, 2.0918718690562303, 1.049720784533563, 1.3304079966469187, 1.8111846569428747, 1.8111846569428747, 1.3304079966469187, 2.8902563016460077, 0.2513363519437855, 0.2513363519437855, 2.8902563016460077, 1.642753969591353, 1.4988386839984404, 1.4988386839984404, 1.642753969591353, 1.0699942300738259, 2.0715984235159675, 2.0715984235159675, 1.0699942300738259, 2.535713311560512, 0.6058793420292811, 0.6058793420292811, 2.535713311560512, 1.8820477440525814, 1.2595449095372118, 1.2595449095372118, 1.8820477440525814, 1.502943913656219, 1.6386487399335743, 1.6386487399335743, 1.502943913656219, 2.9492758858034454, 0.19231676778634796, 0.19231676778634796, 2.9492758858034454, 1.3911272600803817, 1.7504653935094114, 1.7504653935094114, 1.3911272600803817, 2.0045739806414806, 1.1370186729483125, 1.1370186729483125, 2.0045739806414806, 2.332435200581649, 0.8091574530081442, 0.8091574530081442, 2.332435200581649, 0.9407906686163091, 2.2008019849734843, 2.2008019849734843, 0.9407906686163091, 1.7635767611997357, 1.3780158923900574, 1.3780158923900574, 1.7635767611997357, 2.700085394290309, 0.4415072592994843, 0.4415072592994843, 2.700085394290309, 1.1788972111524263, 1.9626954424373668, 1.9626954424373668, 1.1788972111524263, 1.6894775339583656, 1.4521151196314277, 1.4521151196314277, 1.6894775339583656, 2.6140045468896878, 0.5275881067001057, 0.5275881067001057, 2.6140045468896878, 2.0821173149905507, 1.0594753385992424, 1.0594753385992424, 2.0821173149905507, 1.2018079238650614, 1.9397847297247317, 1.9397847297247317, 1.2018079238650614, 2.6713308346904756, 0.4702618188993178, 0.4702618188993178, 2.6713308346904756, 1.2929486420128462, 1.8486440115769471, 1.8486440115769471, 1.2929486420128462, 2.1219213663580567, 1.0196712872317364, 1.0196712872317364, 2.1219213663580567, 2.5904155916388296, 0.5511770619509634, 0.5511770619509634, 2.5904155916388296, 1.5776093770084958, 1.5639832765812973, 1.5639832765812973, 1.5776093770084958, 1.3326625478762133, 1.8089301057135798, 1.8089301057135798, 1.3326625478762133, 2.311870649964783, 0.8297220036250101, 0.8297220036250101, 2.311870649964783, 0.7967771828701535, 2.34481547071964, 2.34481547071964, 0.7967771828701535]
phi = [2.415455450506549, 3.867729856673037, 0.7261372030832441, 5.557048104096342, 6.09248508183496, 0.1907002253446263, 3.3322928789344193, 2.950892428245167, 4.54261707807068, 1.7405682291089069, 4.8821608826987, 1.4010244244808863, 2.5392468847838683, 3.743938422395718, 0.6023457688059249, 5.680839538373661, 0.7645713497727326, 5.518613957406854, 2.3770213038170604, 3.906164003362526, 5.2954316930444065, 0.98775361413518, 4.129346267724973, 2.1538390394546134, 3.2156845030940198, 3.0675008040855665, 6.209093457675359, 0.07409184950422698, 1.864069490812535, 4.419115816367051, 1.277523162777258, 5.005662144402328, 4.953413357499623, 1.3297719496799634, 4.471364603269756, 1.81182070390983, 3.498228020477547, 2.7849572867020393, 5.9265499402918325, 0.35663536688775366, 2.1385823233480954, 4.144602983831491, 1.003010330241698, 5.2801749769378885, 5.241033562936116, 1.0421517442434702, 4.183744397833263, 2.099440909346323, 2.9615046650446613, 3.321680642134925, 0.18008798854513197, 6.103097318634454, 0.3626229625571805, 5.920562344622406, 2.778969691032613, 3.5042156161469733, 4.781353239921465, 1.5018320672581211, 4.643424720847914, 1.6397605863316722, 2.434951315533737, 3.8482339916458494, 0.7066413380560563, 5.57654396912353, 5.663519802243567, 0.6196655049360197, 3.7612581585258127, 2.5219271486537735, 4.165351103312332, 2.117834203867254, 5.259426857457047, 1.0237584497225392, 2.7418805329291533, 3.541304774250433, 0.39971212066063994, 5.8834731865189465, 5.818247367645055, 0.4649379395345311, 3.606530593124324, 2.676654714055262, 4.503569329365923, 1.7796159778136635, 4.921208631403457, 1.3619766757761298, 3.656892990407634, 2.626292316771952, 5.767884970361745, 0.5153003368178409, 3.3789989011142616, 2.9041864060653246, 6.045779059655118, 0.23740624752446846, 4.576200757208659, 1.7069845499709269, 4.84857720356072, 1.4346081036188665, 2.8430995574347095, 3.4400857497448767, 0.2984930961550837, 5.984692211024503, 0.9206171356910273, 5.362568171488559, 2.220975517898766, 4.06220978928082, 5.096846443346836, 1.1863388638327501, 4.3279315174225435, 1.9552537897570432, 3.149589816071623, 3.1335954911079633, 6.275188144697756, 0.007997162481830123, 4.699379021688214, 1.5838062854913717, 4.725398939081165, 1.5577863680984214, 4.161249675079813, 2.121935632099773, 5.263528285689566, 1.01965702149002, 2.3387781665484173, 3.944407140631169, 0.8028144870413759, 5.48037082013821, 0.3254597936423098, 5.957725513537277, 2.8161328599474835, 3.4670524472321027, 5.048550596427225, 1.2346347107523619, 4.376227364342155, 1.9069579428374313]

symmetries =  [1.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
 0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
 0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
 0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
 3.39858759e-04 3.85275935e-06 3.08190054e-03]
symmetry error = 0.0
minimal relative intensity = 0.999702 rms = 0.00005220


144 points (theta,phi), l_max = 16:
theta = [1.5204272431737225, 1.6211654104160709, 1.6211654104160709, 1.5204272431737225, 0.5809733797162132, 2.56061927387358, 2.56061927387358, 0.5809733797162132, 0.9925888721078561, 2.1490037814819374, 2.1490037814819374, 0.9925888721078561, 1.250815546977299, 1.8907771066124943, 1.8907771066124943, 1.250815546977299, 1.426797788291818, 1.7147948652979754, 1.7147948652979754, 1.426797788291818, 0.3530229112468267, 2.7885697423429665, 2.7885697423429665, 0.3530229112468267, 0.5108492874007279, 2.6307433661890656, 2.6307433661890656, 0.5108492874007279, 1.7499789545325977, 1.3916136990571957, 1.3916136990571957, 1.7499789545325977, 1.0981134369246763, 2.043479216665117, 2.043479216665117, 1.0981134369246763, 1.733447046373552, 1.408145607216241, 1.408145607216241, 1.733447046373552, 2.8349022029980424, 0.3066904505917507, 0.3066904505917507, 2.8349022029980424, 1.8284390469102922, 1.313153606679501, 1.313153606679501, 1.8284390469102922, 1.381051412804554, 1.7605412407852394, 1.7605412407852394, 1.381051412804554, 2.317895430424477, 0.8236972231653162, 0.8236972231653162, 2.317895430424477, 0.7827096575425341, 2.3588829960472593, 2.3588829960472593, 0.7827096575425341, 1.1493712804405205, 1.9922213731492726, 1.9922213731492726, 1.1493712804405205, 2.5341437492575967, 0.6074489043321963, 0.6074489043321963, 2.5341437492575967, 1.9801985104514088, 1.1613941431383843, 1.1613941431383843, 1.9801985104514088, 1.0882231303028715, 2.053369523286922, 2.053369523286922, 1.0882231303028715, 2.135402910551702, 1.006189743038091, 1.006189743038091, 2.135402910551702, 0.7870642541345525, 2.3545283994552406, 2.3545283994552406, 0.7870642541345525, 2.016280166330063, 1.12531248725973, 1.12531248725973, 2.016280166330063, 0.45966380417429076, 2.6819288494155025, 2.6819288494155025, 0.45966380417429076, 1.6765985349413663, 1.4649941186484268, 1.4649941186484268, 1.6765985349413663, 2.2956970482087744, 0.8458956053810188, 0.8458956053810188, 2.2956970482087744, 2.393894861364627, 0.7476977922251663, 0.7476977922251663, 2.393894861364627, 1.4196046373092255, 1.7219880162805676, 1.7219880162805676, 1.4196046373092255, 1.997721811034199, 1.1438708425555943, 1.1438708425555943, 1.997721811034199, 0.8328531516142843, 2.3087395019755093, 2.3087395019755093, 0.8328531516142843, 2.2308108327521787, 0.9107818208376146, 0.9107818208376146, 2.2308108327521787, 2.07280223508056, 1.068790418509233, 1.068790418509233, 2.07280223508056, 1.2895258192082024, 1.8520668343815907, 1.8520668343815907, 1.2895258192082024, 0.5889721316093266, 2.5526205219804665, 2.5526205219804665, 0.5889721316093266, 0.15086425561842967, 2.9907283979713637, 2.9907283979713637, 0.15086425561842967, 1.5944186228571593, 1.5471740307326338, 1.5471740307326338, 1.5944186228571593, 1.7197717139776274, 1.421820939612166, 1.421820939612166, 1.7197717139776274]
phi = [0.579036404683078, 5.704148902496508, 2.562556248906715, 3.720629058272871, 0.09186441983349446, 6.191320887346092, 3.0497282337562988, 3.2334570734232875, 1.5106392436228504, 4.772546063556736, 1.630953409966943, 4.652231897212643, 1.419039684817746, 4.86414562236184, 1.7225529687720473, 4.5606323384075385, 0.32344826832776297, 5.959737038851824, 2.8181443852620305, 3.4650409219175557, 0.42800961678650756, 5.855175690393079, 2.7135830368032856, 3.5696022703763006, 1.9439242755844535, 4.339261031595132, 1.1976683780053399, 5.085516929174246, 1.0897755616518883, 5.193409745527698, 2.051817091937905, 4.231368215241681, 6.081649314982551, 0.2015359921970353, 3.343128645786828, 2.940056661392758, 3.40276132426993, 2.8804239829096563, 6.022016636499449, 0.26116867068013716, 3.7077292058761477, 2.5754561013034385, 5.717048754893232, 0.5661365522863548, 4.5441343630731925, 1.7390509441063937, 4.880643597696187, 1.4025417094833994, 2.3349578222067295, 3.9482274849728567, 0.8066348313830638, 5.476550475796523, 0.25999698953974565, 6.023188317639841, 2.8815956640500477, 3.4015896431295385, 4.98313729030585, 1.3000480168737365, 4.44164067046353, 1.8415446367160568, 3.5929950645753244, 2.690190242604262, 5.8317828961940545, 0.4514024109855315, 2.342566362511282, 3.940618944668304, 0.7990262910785112, 5.484159016101075, 5.174581727580268, 1.108603579599318, 4.250196233189111, 2.0329890739904752, 2.2193890843384576, 4.063796222841129, 0.9222035692513356, 5.360981737928251, 0.5815431239711614, 5.701642183208425, 2.560049529618632, 3.7231357775609544, 5.426822188344829, 0.8563631188347572, 3.9979557724245502, 2.285229534755036, 6.165890093233814, 0.11729521394577284, 3.2588878675355657, 3.0242974396440205, 4.472043416958574, 1.8111418902210124, 4.952734543810806, 1.3304507633687808, 2.0189669928967713, 4.2642183142828145, 1.1226256606930218, 5.160559646486565, 2.939003008514942, 3.344182298664644, 0.20258964507485114, 6.080595662104735, 4.935751745371648, 1.3474335618079385, 4.489026215397732, 1.7941590917818548, 3.977220107931676, 2.30596519924791, 5.447557852837703, 0.835627454341883, 5.5441334825935, 0.7390518245860866, 3.8806444781758795, 2.4025408290037067, 3.735580489706133, 2.547604817473453, 5.689197471063246, 0.5939878361163398, 2.1225196427886397, 4.1606656643909465, 1.0190730108011532, 5.264112296378433, 1.248606665366325, 5.034578641813261, 1.8929859882234683, 4.3901993189561175, 5.758584854795334, 0.5246004523842518, 3.666193105974045, 2.6169922012055413, 2.6183614728286226, 3.6648238343509636, 0.5232311807611707, 5.759954126418416, 4.554573985892574, 1.7286113212870122, 4.870203974876805, 1.4129813323027811, 1.7198135991879517, 4.563371707991634, 1.4217790544018416, 4.861406252777744, 6.259298382276848, 0.023886924902738235, 3.165479578492531, 3.117705728687055]

symmetries =  [1.         0.         0.         0.         0.         0.
 0.         0.         0.         0.         0.         0.
 0.         0.         0.         0.         0.         0.00245441
 0.00768922 0.01190206]
symmetry error = 0.0
minimal relative intensity = 0.999611 rms = 0.00005939


156 points (theta,phi), l_max = 17: # from Graef
theta = [1.7179293249553946, 1.6330493609666368, 1.3358327274592987, 1.9802056470997538, 1.8875758442793638, 1.5565639903635076, 2.0906715379474257, 1.2403338056674795, 2.1973364826070405, 1.8362688327507553, 2.3326663464393684, 2.1288678488373804, 2.2227958421139746, 0.1483752557503681, 0.2751870731756657, 0.38356594491914753, 0.4117180185321424, 0.4716883586705256, 0.5511871716524928, 0.5756942632790084, 0.6572353676535431, 0.6677888592429058, 0.6928332594513524, 0.7673533087258756, 0.7883813412122189, 0.8338331982787693, 2.993217397839425, 2.8664055804141277, 2.758026708670646, 2.729874635057651, 2.6699042949192675, 2.5904054819373004, 2.565898390310785, 2.48435728593625, 2.4738037943468876, 2.448759394138441, 2.3742393448639176, 2.3532113123775744, 2.307759455311024, 0.1483752557503681, 0.2751870731756657, 0.38356594491914753, 0.4117180185321424, 0.4716883586705256, 0.5511871716524928, 0.5756942632790084, 0.6572353676535431, 0.6677888592429058, 0.6928332594513524, 0.7673533087258756, 0.7883813412122189, 0.8338331982787693, 2.993217397839425, 2.8664055804141277, 2.758026708670646, 2.729874635057651, 2.6699042949192675, 2.5904054819373004, 2.565898390310785, 2.48435728593625, 2.4738037943468876, 2.448759394138441, 2.3742393448639176, 2.3532113123775744, 2.307759455311024, 1.589817930328685, 1.303101373855824, 1.2734342218361077, 1.5296779521325727, 1.2336366976911282, 1.0198361787023777, 1.7954085481512094, 1.0267091024586803, 1.3702793700742224, 0.9491842108438746, 1.6448811293423002, 1.079509225450186, 2.0091808995077507, 1.551774723261108, 1.8384912797339694, 1.8681584317536857, 1.6119147014572206, 1.907955955898665, 2.1217564748874156, 1.3461841054385837, 2.114883551131113, 1.7713132835155707, 2.1924084427459185, 1.4967115242474929, 2.062083428139607, 1.1324117540820424, 1.551774723261108, 1.8384912797339694, 1.8681584317536857, 1.6119147014572206, 1.907955955898665, 2.1217564748874156, 1.3461841054385837, 2.114883551131113, 1.7713132835155707, 2.1924084427459185, 1.4967115242474929, 2.062083428139607, 1.1324117540820424, 1.589817930328685, 1.303101373855824, 1.2734342218361077, 1.5296779521325727, 1.2336366976911282, 1.0198361787023777, 1.7954085481512094, 1.0267091024586803, 1.3702793700742224, 0.9491842108438746, 1.6448811293423002, 1.079509225450186, 2.0091808995077507, 1.4236633286343987, 1.5085432926231563, 1.8057599261304944, 1.1613870064900396, 1.2540168093104296, 1.5850286632262858, 1.050921115642368, 1.9012588479223136, 0.9442561709827528, 1.305323820839038, 0.808926307150425, 1.012724804752413, 0.9187968114758186, 1.4236633286343987, 1.5085432926231563, 1.8057599261304944, 1.1613870064900396, 1.2540168093104296, 1.5850286632262858, 1.050921115642368, 1.9012588479223136, 0.9442561709827528, 1.305323820839038, 0.808926307150425, 1.012724804752413, 0.9187968114758186, 1.7179293249553946, 1.6330493609666368, 1.3358327274592987, 1.9802056470997538, 1.8875758442793638, 1.5565639903635076, 2.0906715379474257, 1.2403338056674795, 2.1973364826070405, 1.8362688327507553, 2.3326663464393684, 2.1288678488373804, 2.2227958421139746]
phi = [6.263955913561296, 0.2682273256109725, 0.30603197075503485, 0.044825076711233654, 0.3555752345415193, 0.5510223835231352, 6.023646738046173, 0.5790721735130016, 0.248425107704688, 0.6478675951089605, 6.18070846011311, 0.5897338477396912, 5.719837494768065, 4.583368185292142, 6.052182854329472, 0.6714196940233752, 4.815290108119063, 5.527825686011537, 0.027178989355958486, 4.290906891660774, 0.5599279842957277, 5.039849925162913, 5.85988972781435, 4.605568890074337, 5.440136695941028, 4.101957883736364, 1.6998171218874443, 0.2310024528501144, 5.611765613156211, 1.4678951990605227, 0.7553596211680493, 6.256006317823628, 1.9922784155188122, 5.723257322883859, 1.2433353820166733, 0.4232955793652366, 1.6776164171052494, 0.8430486112385576, 2.1812274234432225, 1.4417755317023488, 2.9105902007396787, 3.813012347613168, 1.6736974545292707, 2.386233032421744, 3.1687716429457513, 1.149314238070981, 3.7015206378855208, 1.89825727157312, 2.718297074224557, 1.4639762364845437, 2.298544042351236, 0.9603652301465709, 4.841409775477238, 3.3725951064399076, 2.470172959566418, 4.609487852650315, 3.8969522747578424, 3.114413664233835, 5.133871069108605, 2.5816646692940655, 4.384928035606466, 3.5648882329550293, 4.819209070695043, 3.9846412648283502, 5.322820077033016, 1.7179561407383188, 1.6353516472359468, 1.3248298053339111, 1.9805727614289983, 1.907198204233442, 1.5540919303118854, 2.105483626885339, 1.181818034751335, 2.2122165089361787, 1.8994151041368894, 2.3352937321525946, 2.2150284006319505, 2.3052032146045685, 1.4236365128514745, 1.5062410063538463, 1.816762848255882, 1.161019892160795, 1.2343944493563512, 1.5875007232779077, 1.0361090267044542, 1.9597746188384582, 0.9293761446536145, 1.242177549452904, 0.8062989214371985, 0.9265642529578427, 0.8363894389852249, 4.565229166441267, 4.64783365994364, 4.958355501845675, 4.302612545750588, 4.375987102946144, 4.7290933768677, 4.177701680294247, 5.101367272428251, 4.070968798243408, 4.383770203042697, 3.9478915750269916, 4.068156906547635, 3.977982092575018, 4.859548794328112, 4.77694430082574, 4.466422458923704, 5.122165415018792, 5.048790857823235, 4.695684583901679, 5.247076280475132, 4.323410688341128, 5.353809162525971, 5.041007757726682, 5.476886385742388, 5.356621054221743, 5.446795868194362, 0.019229393618289844, 6.014957981568614, 5.977153336424552, 6.238360230468353, 5.927610072638067, 5.7321629236564515, 0.25953856913341333, 5.704113133666585, 6.034760199474898, 5.635317712070625, 0.1024768470664758, 5.6934514594398955, 0.5633478124115211, 3.160822047208083, 2.8733653279788207, 2.8355606828347586, 3.0967675768785594, 2.786017419048274, 2.590570270066658, 3.4011312227232064, 2.562520480076792, 2.893167545885105, 2.4937250584808326, 3.244069500656269, 2.551858805850102, 3.704940466001314, 3.1223632599715034, 3.4098199792007655, 3.4476246243448276, 3.186417730301027, 3.497167888131312, 3.6926150371129283, 2.88205408445638, 3.7206648271027944, 3.390017761294481, 3.7894602486987536, 3.0391158065233173, 3.7313265013294843, 2.578244841178272]

symmetries =  [1.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
 1.76666416e-24 5.26310936e-25 1.56303263e-23 8.38061016e-25
 1.81560938e-23 2.08535400e-24 9.89453539e-23 2.31632157e-24
 1.73759774e-22 2.03901913e-23 1.70337419e-22 3.13921884e-23
 5.67008997e-23 2.23662998e-23 2.34856353e-03 3.30707391e-03
 1.91004264e-03]
symmetry error = 6.15211257753062e-22
minimal relative intensity = 0.999795 rms = 0.00003266


168 points (theta,phi), l_max = 17:
theta = [0.31420079678640944, 2.827391856803384, 2.827391856803384, 0.31420079678640944, 1.5232305969064692, 1.6183620566833241, 1.6183620566833241, 1.5232305969064692, 1.260461992011232, 1.8811306615785612, 1.8811306615785612, 1.260461992011232, 0.7479293579641823, 2.393663295625611, 2.393663295625611, 0.7479293579641823, 1.8337719973179343, 1.307820656271859, 1.307820656271859, 1.8337719973179343, 0.8911962640131885, 2.2503963895766046, 2.2503963895766046, 0.8911962640131885, 1.0951713718334462, 2.046421281756347, 2.046421281756347, 1.0951713718334462, 0.805620472563803, 2.33597218102599, 2.33597218102599, 0.805620472563803, 2.161890547428608, 0.9797021061611851, 0.9797021061611851, 2.161890547428608, 2.3399767193060343, 0.8016159342837589, 0.8016159342837589, 2.3399767193060343, 2.0483280479871344, 1.093264605602659, 1.093264605602659, 2.0483280479871344, 0.9857211151158223, 2.155871538473971, 2.155871538473971, 0.9857211151158223, 1.7910607179943634, 1.3505319355954297, 1.3505319355954297, 1.7910607179943634, 0.3102883677056841, 2.831304285884109, 2.831304285884109, 0.3102883677056841, 1.3558576977208256, 1.7857349558689677, 1.7857349558689677, 1.3558576977208256, 1.2358788916598764, 1.905713761929917, 1.905713761929917, 1.2358788916598764, 1.2862824232224568, 1.8553102303673366, 1.8553102303673366, 1.2862824232224568, 0.4469674885106166, 2.694625165079177, 2.694625165079177, 0.4469674885106166, 2.5868549583045772, 0.5547376952852159, 0.5547376952852159, 2.5868549583045772, 1.5042221497578645, 1.6373705038319288, 1.6373705038319288, 1.5042221497578645, 1.0210131241339009, 2.1205795294558922, 2.1205795294558922, 1.0210131241339009, 0.9757308032509054, 2.165861850338888, 2.165861850338888, 0.9757308032509054, 1.8739300075501037, 1.2676626460396896, 1.2676626460396896, 1.8739300075501037, 2.4534663179596223, 0.6881263356301708, 0.6881263356301708, 2.4534663179596223, 3.0092290913897584, 0.1323635622000351, 0.1323635622000351, 3.0092290913897584, 1.445634649948744, 1.695958003641049, 1.695958003641049, 1.445634649948744, 1.6136372160250945, 1.5279554375646989, 1.5279554375646989, 1.6136372160250945, 1.4233906412469963, 1.7182020123427968, 1.7182020123427968, 1.4233906412469963, 0.6154010641790297, 2.5261915894107636, 2.5261915894107636, 0.6154010641790297, 2.1631195176789655, 0.9784731359108279, 0.9784731359108279, 2.1631195176789655, 1.9468130809205935, 1.1947795726691997, 1.1947795726691997, 1.9468130809205935, 1.1840468800215582, 1.957545773568235, 1.957545773568235, 1.1840468800215582, 0.5543800139310461, 2.5872126396587474, 2.5872126396587474, 0.5543800139310461, 2.3840105948565777, 0.7575820587332156, 0.7575820587332156, 2.3840105948565777, 0.8197935788833441, 2.321799074706449, 2.321799074706449, 0.8197935788833441, 1.6519194392103052, 1.4896732143794882, 1.4896732143794882, 1.6519194392103052, 1.9856914439210012, 1.1559012096687922, 1.1559012096687922, 1.9856914439210012, 1.591219554299304, 1.5503730992904892, 1.5503730992904892, 1.591219554299304, 0.4154600949247283, 2.726132558665065, 2.726132558665065, 0.4154600949247283, 2.524752453796336, 0.6168401997934573, 0.6168401997934573, 2.524752453796336, 1.3393775480280135, 1.8022151055617799, 1.8022151055617799, 1.3393775480280135, 2.1306323731509282, 1.010960280438865, 1.010960280438865, 2.1306323731509282]
phi = [1.4163346229171783, 4.866850684262408, 1.725258030672615, 4.557927276506971, 1.2600988411945844, 5.023086465985002, 1.881493812395209, 4.4016914947843775, 0.04995379293516168, 6.233231514244425, 3.0916388606546317, 3.1915464465249546, 1.962991851828763, 4.320193455350823, 1.1786008017610303, 5.104584505418556, 0.8620822622435986, 5.421103044935988, 2.2795103913461947, 4.0036749158333915, 5.942420494872314, 0.34076481230727224, 3.4823574658970653, 2.800827841282521, 5.60568732228072, 0.6774979848988666, 3.8190906384886594, 2.464094668690927, 2.4537749543056857, 3.8294103528739005, 0.6878176992841076, 5.595367607895478, 0.9866835404292221, 5.296501766750364, 2.154909113160571, 4.128276194019016, 2.264861004859861, 4.018324302319725, 0.876731648729932, 5.406453658449654, 5.383458974780157, 0.8997263323994293, 4.041318985989222, 2.241866321190364, 3.7254939192268797, 2.5576913879527066, 5.6992840415425, 0.5839012656370869, 0.220346888934941, 6.062838418244645, 2.921245764654852, 3.361939542524734, 5.485744039076992, 0.7974412681025944, 3.9390339216923875, 2.3441513854871987, 1.7963371875160448, 4.4868481196635415, 1.3452554660737484, 4.937929841105838, 1.2690329527954738, 5.014152354384112, 1.8725597007943195, 4.410625606385267, 0.3495325332852431, 5.933652773894343, 2.79206012030455, 3.491125186875036, 0.7067917242821383, 5.576393582897448, 2.434800929307655, 3.8483843778719313, 1.4441576693597726, 4.839027637819814, 1.6974349842300207, 4.585750322949566, 5.263533280824287, 1.0196520263552993, 4.161244679945092, 2.121940627234494, 3.0634906858248883, 3.219694621354698, 0.07810196776490498, 6.205083339414681, 4.343613350629301, 1.9395719565502856, 5.081164610140078, 1.2020206970395078, 2.5138213291683793, 3.769363978011207, 0.6277713244214138, 5.655413982758173, 5.793859393679513, 0.4893259135000731, 3.630918567089866, 2.65226674008972, 5.952693255214349, 0.3304920519652373, 3.47208470555503, 2.811100601624556, 4.6692101167669655, 1.6139751904126205, 4.755567844002414, 1.5276174631771726, 3.0163154267682213, 3.266869880411365, 0.12527722682157189, 6.157908080358014, 5.6834660550034215, 0.5997192521761651, 3.741311905765958, 2.5418734014136284, 2.884345930331246, 3.3988393768483403, 0.2572467232585471, 6.0259385839210395, 1.392828074616775, 4.8903572325628115, 1.7487645789730184, 4.534420728206568, 1.1532584449854597, 5.129926862194127, 1.9883342086043336, 4.294851098575252, 5.875479484765604, 0.40770582241398245, 3.5492984760037753, 2.733886831175811, 2.342813329857608, 3.9403719773219783, 0.7987793237321853, 5.484405983447401, 6.164985241903832, 0.11820006527575468, 3.2597927188655476, 3.0233925883140387, 4.60130758848558, 1.6818777186940064, 4.823470372283799, 1.459714934895787, 2.387505447204078, 3.895679859975508, 0.7540872063857149, 5.529098100793871, 1.593113226771314, 4.690072080408273, 1.5484794268184794, 4.734705880361107, 5.868198311756053, 0.4149869954235332, 3.556579649013326, 2.7266056581662603, 3.192212025222163, 3.090973281957423, 6.232565935547216, 0.05061937163237002, 5.120088476006512, 1.163096831173074, 4.304689484762867, 1.9784958224167193, 4.1352940062844, 2.1478913008951865, 5.28948395448498, 0.9937013526946065, 2.867493167350754, 3.415692139828832, 0.2740994862390392, 6.009085820940547]

symmetries =  [1.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
 0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
 0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
 0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
 0.00000000e+00 0.00000000e+00 9.59554463e-04 1.56153262e-03
 1.22134729e-03]
symmetry error = 0.0
minimal relative intensity = 0.999863 rms = 0.00002226


168 points (theta,phi) l_max = 17:
theta = [2.5119744731026863, 0.629618180487107, 0.629618180487107, 2.5119744731026863, 1.9999037049904345, 1.1416889485993587, 1.1416889485993587, 1.9999037049904345, 1.1410055391749054, 2.0005871144148877, 2.0005871144148877, 1.1410055391749054, 1.938193871298644, 1.2033987822911492, 1.2033987822911492, 1.938193871298644, 2.733079654716209, 0.40851299887358444, 0.40851299887358444, 2.733079654716209, 1.7412907040131653, 1.4003019495766278, 1.4003019495766278, 1.7412907040131653, 0.461175307255749, 2.6804173463340444, 2.6804173463340444, 0.461175307255749, 2.0030017698008478, 1.1385908837889454, 1.1385908837889454, 2.0030017698008478, 1.4199921571950493, 1.7216004963947438, 1.7216004963947438, 1.4199921571950493, 2.5121817885029367, 0.6294108650868565, 0.6294108650868565, 2.5121817885029367, 2.000065960585932, 1.1415266930038617, 1.1415266930038617, 2.000065960585932, 2.0001644070176194, 1.1414282465721737, 1.1414282465721737, 2.0001644070176194, 2.2333764317577276, 0.9082162218320656, 0.9082162218320656, 2.2333764317577276, 2.0128824823646343, 1.1287101712251588, 1.1287101712251588, 2.0128824823646343, 0.8470022855073879, 2.2945903680824054, 2.2945903680824054, 0.8470022855073879, 1.1317360243997225, 2.009856629190071, 2.009856629190071, 1.1317360243997225, 2.674000083609836, 0.46759256997995746, 0.46759256997995746, 2.674000083609836, 1.721245014894374, 1.4203476386954192, 1.4203476386954192, 1.721245014894374, 0.9580136896352619, 2.183578963954531, 2.183578963954531, 0.9580136896352619, 1.4753533910941279, 1.6662392624956652, 1.6662392624956652, 1.4753533910941279, 0.6224011325876416, 2.519191521002152, 2.519191521002152, 0.6224011325876416, 0.9078227241037431, 2.23376992948605, 2.23376992948605, 0.9078227241037431, 2.2941746541949315, 0.8474179993948618, 0.8474179993948618, 2.2941746541949315, 1.1286703251168557, 2.0129223284729374, 2.0129223284729374, 1.1286703251168557, 1.5058027956814222, 1.6357898579083712, 1.6357898579083712, 1.5058027956814222, 2.94807486425244, 0.19351778933735342, 0.19351778933735342, 2.94807486425244, 1.3887797815153402, 1.752812872074453, 1.752812872074453, 1.3887797815153402, 2.734603656228376, 0.4069889973614173, 0.4069889973614173, 2.734603656228376, 1.9354548877511861, 1.2061377658386072, 1.2061377658386072, 1.9354548877511861, 1.3981584462057537, 1.7434342073840394, 1.7434342073840394, 1.3981584462057537, 1.5050171464923237, 1.6365755070974695, 1.6365755070974695, 1.5050171464923237, 1.3846396252904036, 1.7569530282993897, 1.7569530282993897, 1.3846396252904036, 0.19769429462752788, 2.943898358962265, 2.943898358962265, 0.19769429462752788, 1.4730932234616605, 1.6684994301281328, 1.6684994301281328, 1.4730932234616605, 2.1877109989568275, 0.9538816546329655, 0.9538816546329655, 2.1877109989568275, 0.6269625673076249, 2.5146300862821684, 2.5146300862821684, 0.6269625673076249, 0.8702781092146479, 2.2713145443751452, 2.2713145443751452, 0.8702781092146479, 1.7609575930858683, 1.380635060503925, 1.380635060503925, 1.7609575930858683, 0.7365752900092108, 2.4050173635805825, 2.4050173635805825, 0.7365752900092108, 1.3811713996350266, 1.7604212539547668, 1.7604212539547668, 1.3811713996350266, 2.276288116691221, 0.8653045368985719, 0.8653045368985719, 2.276288116691221, 2.400289606993134, 0.7413030465966591, 0.7413030465966591, 2.400289606993134]
phi = [2.355448335852147, 3.9277369713274393, 0.7861443177376463, 5.49704098944194, 5.188385477928834, 1.0947998292507528, 4.236392482840546, 2.0467928243390405, 3.6169816167321547, 2.6662036904474316, 5.807796344037225, 0.4753889631423615, 3.32441142759299, 2.9587738795865963, 6.100366533176389, 0.18281877400319668, 4.2710886626466635, 2.0120966445329223, 5.153689298122715, 1.129496009056871, 4.3393228638990475, 1.9438624432805383, 5.085455096870332, 1.1977302103092549, 2.7972244528941372, 3.485960854285449, 0.34436820069565593, 5.93881710648393, 1.4045850390305141, 4.878600268149072, 1.7370076145592792, 4.546177692620307, 5.845677848197204, 0.43750745898238175, 3.5791001125721746, 2.7040851946074116, 3.927098338944758, 2.356086968234828, 5.497679621824621, 0.7855056853549652, 4.236829657887462, 2.0463556492921247, 5.187948302881917, 1.0952370042976685, 3.617064451615631, 2.666120855563955, 5.807713509153748, 0.47547179802583794, 2.1443799427284786, 4.138805364451107, 0.9972127108613145, 5.285972596318272, 5.534626402259917, 0.7485589049196687, 3.8901515585094617, 2.3930337486701245, 3.7492778820393955, 2.5339074251401907, 5.675500078729984, 0.6076852284496026, 3.307946348124588, 2.975238959054998, 6.116831612644791, 0.16635369453479532, 1.9097758077280054, 4.373409499451581, 1.2318168458617877, 5.051368461317798, 5.156821705807943, 1.1263636013716432, 4.267956254961437, 2.01522905221815, 1.4540371041784037, 4.829148203001182, 1.6875555494113896, 4.595629757768196, 0.6160007343775776, 5.667184572802008, 2.5255919192122156, 3.7575933879673706, 0.16420190084227546, 6.1189834063373105, 2.977390752747518, 3.3057945544320684, 2.5677561554584716, 3.7154291517211147, 0.5738364981313218, 5.709348809048264, 0.9633079313691247, 5.319877375810462, 2.1782847222206687, 4.1049005849589175, 5.461433923632744, 0.8217513835468423, 3.9633440371366353, 2.319841270042951, 2.959186673089024, 3.3239986340905623, 0.1824059805007693, 6.100779326678817, 0.34449410937230135, 5.938691197807285, 2.797098544217492, 3.486086762962094, 4.7784757737341295, 1.5047095334454565, 4.64630218703525, 1.6368831201443366, 2.6927083545033974, 3.590476952676189, 0.4488842990863958, 5.83430100809319, 4.897313162744606, 1.3858721444349804, 4.5274647980247735, 1.755720509154813, 3.5120176229192577, 2.7711676842603286, 5.912760337850122, 0.37042496932946467, 1.3842314152866309, 4.898953891892956, 1.7573612383031625, 4.525824068876424, 0.066937354557176, 6.21624795262241, 3.0746552990326173, 3.208530008146969, 1.2295438310555942, 5.0536414761239925, 1.9120488225341992, 4.371136484645387, 2.1911138972643625, 4.092071409915224, 0.9504787563254308, 5.332706550854155, 0.11987948364940954, 6.163305823530177, 3.0217131699403836, 3.2614721372392026, 4.8794333403385295, 1.4037519668410572, 4.54534462043085, 1.737840686748736, 1.8206274339029538, 4.462557873276633, 1.3209652196868396, 4.962220087492747, 0.7160994755991998, 5.567085831580386, 2.4254931779905933, 3.857692129188993, 5.997955169406079, 0.28523013777350753, 3.4268227913633007, 2.8563625158162855, 3.9912465043974157, 2.2919388027821705, 5.433531456371964, 0.8496538508076225, 2.8913978144805776, 3.3917874926990086, 0.2501948391092155, 6.032990468070371, 4.995289510436498, 1.287895796743088, 4.429488450332881, 1.8536968568467052]

symmetries =  [1.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
 0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
 0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
 0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
 0.00000000e+00 0.00000000e+00 4.72777376e-04 1.77991377e-03
 1.29382110e-02]
symmetry error = 0.0
minimal relative intensity = 0.999910 rms = 0.00001470


180 points (theta,phi), l_max = 18: # from Sloane:
theta = [1.1079603899455988, 2.0336322636441944, 2.0336322636441944, 1.1079603899455988, 2.618696894589635, 0.522895759000158, 0.522895759000158, 2.618696894589635, 1.3451903481609402, 1.796402305428853, 1.796402305428853, 1.3451903481609402, 1.9481745542611166, 1.1934180993286767, 1.1934180993286767, 1.9481745542611166, 2.509587592108825, 0.6320050614809681, 0.6320050614809681, 2.509587592108825, 2.0507730998538776, 1.0908195537359155, 1.0908195537359155, 2.0507730998538776, 0.14112499905840847, 3.0004676545313846, 3.0004676545313846, 0.14112499905840847, 1.7060515674058379, 1.4355410861839555, 1.4355410861839555, 1.7060515674058379, 1.610828280830695, 1.530764372759098, 1.530764372759098, 1.610828280830695, 2.8534899701131016, 0.28810268347669155, 0.28810268347669155, 2.8534899701131016, 1.5209815975556338, 1.6206110560341593, 1.6206110560341593, 1.5209815975556338, 1.8543158729381242, 1.2872767806516692, 1.2872767806516692, 1.8543158729381242, 0.28570346892079457, 2.8558891846689987, 2.8558891846689987, 0.28570346892079457, 1.3638504732399657, 1.7777421803498274, 1.7777421803498274, 1.3638504732399657, 1.7649142876535184, 1.376678365936275, 1.376678365936275, 1.7649142876535184, 2.1787130949485407, 0.9628795586412526, 0.9628795586412526, 2.1787130949485407, 1.8534499850783401, 1.2881426685114532, 1.2881426685114532, 1.8534499850783401, 0.6888087820319936, 2.4527838715577994, 2.4527838715577994, 0.6888087820319936, 1.6850063104204558, 1.4565863431693375, 1.4565863431693375, 1.6850063104204558, 0.42177665137370784, 2.7198160022160853, 2.7198160022160853, 0.42177665137370784, 1.9749058339414323, 1.166686819648361, 1.166686819648361, 1.9749058339414323, 2.127953313831016, 1.013639339758777, 1.013639339758777, 2.127953313831016, 0.5573426567118547, 2.5842499968779387, 2.5842499968779387, 0.5573426567118547, 1.5837069096999146, 1.5578857438898788, 1.5578857438898788, 1.5837069096999146, 0.8142145998093022, 2.327378053780491, 2.327378053780491, 0.8142145998093022, 2.3404018197892436, 0.8011908338005498, 0.8011908338005498, 2.3404018197892436, 1.358029949196552, 1.7835627043932412, 1.7835627043932412, 1.358029949196552, 0.6916376093935847, 2.4499550441962086, 2.4499550441962086, 0.6916376093935847, 1.8355989373288508, 1.3059937162609423, 1.3059937162609423, 1.8355989373288508, 2.1915250779452276, 0.9500675756445659, 0.9500675756445659, 2.1915250779452276, 1.044389717807896, 2.0972029357818975, 2.0972029357818975, 1.044389717807896, 1.0400980695721798, 2.1014945840176136, 2.1014945840176136, 1.0400980695721798, 0.7940074987722813, 2.347585154817512, 2.347585154817512, 0.7940074987722813, 0.8424761420307482, 2.299116511559045, 2.299116511559045, 0.8424761420307482, 2.027228361932103, 1.1143642916576901, 1.1143642916576901, 2.027228361932103, 0.9244900914981595, 2.217102562091634, 2.217102562091634, 0.9244900914981595, 1.215079532322274, 1.926513121267519, 1.926513121267519, 1.215079532322274, 2.668708048921193, 0.47288460466860005, 0.47288460466860005, 2.668708048921193, 1.8687027403141334, 1.2728899132756597, 1.2728899132756597, 1.8687027403141334, 0.7244423143865756, 2.417150339203218, 2.417150339203218, 0.7244423143865756, 0.8473468207249365, 2.2942458328648567, 2.2942458328648567, 0.8473468207249365, 1.5394013760017096, 1.6021912775880836, 1.6021912775880836, 1.5394013760017096, 2.0515291199987287, 1.0900635335910647, 1.0900635335910647, 2.0515291199987287, 1.6968622849128643, 1.444730368676929, 1.444730368676929, 1.6968622849128643, 2.6418299197729858, 0.4997627338168075, 0.4997627338168075, 2.6418299197729858]
phi = [2.8889129135290244, 3.394272393650562, 0.2526797400607689, 6.0305055671188175, 1.1063368551532082, 5.176848452026378, 2.0352557984365847, 4.247929508743002, 5.18824125891223, 1.0949440482673565, 4.236536701857149, 2.046648605322437, 3.661396752293582, 2.6217885548860043, 5.763381208475797, 0.5198040987037892, 3.8151143599716577, 2.4680709472079285, 5.609663600797721, 0.6735217063818648, 4.283978843659645, 1.9992064635199414, 5.140799117109735, 1.142386190069852, 3.430109681341256, 2.85307562583833, 5.994668279428123, 0.2885170277514634, 1.6111974647487424, 4.671987842430844, 1.530395188841051, 4.7527901183385355, 6.14782095000454, 0.13536435717504597, 3.276957010764839, 3.006228296414747, 4.888547644398781, 1.394637662780805, 4.536230316370598, 1.746954990808988, 4.428507524121805, 1.8546777830577814, 4.996270436647574, 1.2869148705320117, 3.0897046399798356, 3.1934806671997507, 0.05188801360995779, 6.231297293569629, 5.529331889179266, 0.7538534180003204, 3.895446071590113, 2.387739235589473, 1.769202274081465, 4.513983033098121, 1.3723903795083283, 4.910794927671258, 0.21096631034721153, 6.072218996832374, 2.9306263432425816, 3.3525589639370046, 1.917479410977376, 4.36570589620221, 1.2241132426124173, 5.059072064567169, 5.646218817378333, 0.6369664898012531, 3.778559143391046, 2.50462616378854, 3.595848109413244, 2.6873371977663423, 5.828929851356135, 0.4542554558234508, 5.876269784608444, 0.4069155225711428, 3.5485081761609356, 2.7346771310186506, 3.4236947930842745, 2.8594905140953117, 6.001083167685104, 0.2821021394944814, 1.69506117870977, 4.588124128469817, 1.446531474880023, 4.836653832299563, 6.267974064991125, 0.015211242188461396, 3.1568038957782543, 3.126381411401332, 4.687978498402597, 1.59520680877699, 4.736799462366783, 1.5463858448128032, 2.1280052397509506, 4.155180067428636, 1.0135874138388428, 5.269597893340744, 2.8469626504656085, 3.4362226567139778, 0.2946300031241848, 5.988555304055401, 1.2723606432425, 5.010824663937086, 1.869232010347293, 4.413953296832293, 5.490978103025325, 0.7922072041542606, 3.9337998577440536, 2.3493854494355326, 4.289554914832778, 1.9936303923468082, 5.135223045936601, 1.147962261242985, 2.2175932679048325, 4.065592039274754, 0.9239993856849608, 5.359185921494626, 5.9556215166658895, 0.32756379051369683, 3.46915644410349, 2.8140288630760963, 0.9454397338456204, 5.337745573333966, 2.196152919744173, 4.087032387435413, 0.6218727421262578, 5.661312565053328, 2.519719911463535, 3.763465395716051, 0.7890726715493422, 5.494112635630244, 2.3525199820404508, 3.9306653251391355, 2.202577276244872, 4.080608030934714, 0.9390153773449211, 5.344169929834665, 0.8353426081270091, 5.447842699052577, 2.306250045462784, 3.976935261716802, 5.698305758818236, 0.58487954836135, 3.726472201951143, 2.556713105228443, 3.4600710273904953, 2.823114279789091, 5.964706933378884, 0.31847837380070254, 2.271102173028187, 4.0120831341514, 0.8704904805616064, 5.4126948266179795, 5.085279752811319, 1.1979055543682677, 4.33949820795806, 1.9436870992215256, 0.047382969367557565, 6.235802337812029, 3.0942096842222355, 3.1889756229573507, 1.528904588645423, 4.754280718534163, 1.6126880649443704, 4.670497242235216, 0.7238850234015844, 5.5593002837780015, 2.417707630188209, 3.8654776769913775, 4.570104367826299, 1.7130809393532875, 4.85467359294308, 1.4285117142365058, 3.626501970433355, 2.656683336746231, 5.798275990336024, 0.4849093168435622, 3.4070702094376575, 2.8761150977419287, 6.017707751331722, 0.26547755584786425]

symmetries =  [1.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
 0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
 0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
 0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
 0.00000000e+00 0.00000000e+00 0.00000000e+00 1.00666633e-03
 8.23389521e-04 2.17889338e-03]
symmetry error = 0.0
minimal relative intensity = 0.999907 rms = 0.00001566


192 points (theta,phi), l_max = 18:
theta = [1.4358714179903287, 1.7057212355994646, 1.7057212355994646, 1.4358714179903287, 2.7250646124324622, 0.41652804115733116, 0.41652804115733116, 2.7250646124324622, 1.9622920843084117, 1.1793005692813814, 1.1793005692813814, 1.9622920843084117, 1.8237358910566561, 1.317856762533137, 1.317856762533137, 1.8237358910566561, 1.1181764020736336, 2.02341625151616, 2.02341625151616, 1.1181764020736336, 0.5280642431350854, 2.613528410454708, 2.613528410454708, 0.5280642431350854, 2.476340173806544, 0.6652524797832494, 0.6652524797832494, 2.476340173806544, 1.7875311226035124, 1.354061530986281, 1.354061530986281, 1.7875311226035124, 0.9537998307945434, 2.1877928227952497, 2.1877928227952497, 0.9537998307945434, 2.3426966882109355, 0.7988959653788578, 0.7988959653788578, 2.3426966882109355, 2.369542012883751, 0.7720506407060422, 0.7720506407060422, 2.369542012883751, 1.5830532703301508, 1.5585393832596426, 1.5585393832596426, 1.5830532703301508, 2.7864992267937723, 0.35509342679602107, 0.35509342679602107, 2.7864992267937723, 1.8672785306009647, 1.2743141229888286, 1.2743141229888286, 1.8672785306009647, 1.7604081574056505, 1.3811844961841429, 1.3811844961841429, 1.7604081574056505, 0.16003200882583427, 2.981560644763959, 2.981560644763959, 0.16003200882583427, 1.4145027833254928, 1.7270898702643005, 1.7270898702643005, 1.4145027833254928, 1.5366877177305833, 1.6049049358592098, 1.6049049358592098, 1.5366877177305833, 1.7905876279030137, 1.3510050256867796, 1.3510050256867796, 1.7905876279030137, 2.4590163501974693, 0.682576303392324, 0.682576303392324, 2.4590163501974693, 0.9373608582380594, 2.204231795351734, 2.204231795351734, 0.9373608582380594, 1.9978848377134544, 1.143707815876339, 1.143707815876339, 1.9978848377134544, 2.5130772379119923, 0.6285154156778009, 0.6285154156778009, 2.5130772379119923, 1.1403780012939948, 2.0012146522957983, 2.0012146522957983, 1.1403780012939948, 2.257257749205009, 0.884334904384784, 0.884334904384784, 2.257257749205009, 2.2574713164141835, 0.8841213371756098, 0.8841213371756098, 2.2574713164141835, 1.1116951011287146, 2.0298975524610787, 2.0298975524610787, 1.1116951011287146, 0.21094711739647568, 2.9306455361933175, 2.9306455361933175, 0.21094711739647568, 1.6058714743319085, 1.5357211792578846, 1.5357211792578846, 1.6058714743319085, 1.3628727541018786, 1.7787198994879148, 1.7787198994879148, 1.3628727541018786, 0.39032726269674145, 2.7512653908930518, 2.7512653908930518, 0.39032726269674145, 1.373663841847831, 1.7679288117419623, 1.7679288117419623, 1.373663841847831, 1.238504729960403, 1.90308792362939, 1.90308792362939, 1.238504729960403, 1.2373200506325512, 1.904272602957242, 1.904272602957242, 1.2373200506325512, 2.50296712244851, 0.6386255311412833, 0.6386255311412833, 2.50296712244851, 1.0493000498924125, 2.0922926036973806, 2.0922926036973806, 1.0493000498924125, 1.6383034150301672, 1.5032892385596262, 1.5032892385596262, 1.6383034150301672, 2.597096153593405, 0.5444964999963882, 0.5444964999963882, 2.597096153593405, 2.110144039533087, 1.0314486140567058, 1.0314486140567058, 2.110144039533087, 2.063814969682167, 1.0777776839076263, 1.0777776839076263, 2.063814969682167, 2.6478792398762723, 0.4937134137135211, 0.4937134137135211, 2.6478792398762723, 1.5948734848813777, 1.5467191687084156, 1.5467191687084156, 1.5948734848813777, 2.351207210367343, 0.7903854432224501, 0.7903854432224501, 2.351207210367343, 1.827074440929064, 1.3145182126607293, 1.3145182126607293, 1.827074440929064, 2.2967889060348172, 0.8448037475549758, 0.8448037475549758, 2.2967889060348172, 0.8096363209848896, 2.3319563326049035, 2.3319563326049035, 0.8096363209848896, 1.078688638538209, 2.062904015051584, 2.062904015051584, 1.078688638538209, 2.151512511363398, 0.990080142226395, 0.990080142226395, 2.151512511363398]
phi = [3.536877595288609, 2.746307711890977, 5.88790036548077, 0.39528494169881595, 2.802664489080142, 3.480520818099444, 0.3389281645096512, 5.944257142669935, 4.858434131675365, 1.4247511755042217, 4.566343829094015, 1.7168414780855716, 1.102131306216136, 5.18105400096345, 2.039461347373657, 4.243723959805929, 6.001190932237626, 0.2819943749419606, 3.423587028531754, 2.8595982786478324, 2.0905492197642834, 4.192636087415303, 1.05104343382551, 5.2321418733540765, 1.9266415262024597, 4.356543780977127, 1.2149511273873337, 5.0682341797922525, 5.346483756719106, 0.9367015504604799, 4.078294204050273, 2.2048911031293135, 3.4084015796466955, 2.8747837275328907, 6.016376381122684, 0.26680892605690254, 3.158697685469871, 3.124487621709715, 6.266080275299508, 0.01710503188007838, 4.694818506256902, 1.5883668009226843, 4.7299594545124775, 1.5532258526671088, 3.9404154963810503, 2.342769810798536, 5.484362464388329, 0.7988228427912574, 3.7145318583551656, 2.5686534488244206, 5.710246102414214, 0.5729392047653724, 4.51401435253109, 1.7691709546484966, 4.910763608238289, 1.3724216989412965, 3.4436546420653142, 2.839530665114272, 5.981123318704065, 0.3020619884755213, 0.21567530491754147, 6.067510002262045, 2.9259173486722516, 3.3572679585073346, 1.5362666716992595, 4.746918635480327, 1.6053259818905339, 4.677859325289052, 0.15638525191550973, 6.126800055264076, 2.9852074016742836, 3.2979779055053027, 2.48993147328546, 3.793253833894126, 0.6516611803043332, 5.631524126875253, 5.930267540357818, 0.35291776682176856, 3.4945104204115616, 2.7886748867680247, 4.438472389887243, 1.8447129172923433, 4.986305570882136, 1.2968797362974498, 2.665365809772564, 3.617819497407022, 0.4762268438172291, 5.806958463362357, 5.501429352784082, 0.7817559543955043, 3.9233486079852975, 2.3598366991942887, 4.239123706878271, 2.0440616003013155, 5.185654253891109, 1.0975310532884779, 2.531540200152817, 3.751645107026769, 0.6100524534369761, 5.67313285374261, 5.322563799418114, 0.9606215077614726, 4.102214161351266, 2.1809711458283205, 3.9271211071657675, 2.3560642000138188, 5.497656853603612, 0.7855284535759746, 1.7390691852642841, 4.544116121915302, 1.402523468325509, 4.880661838854078, 1.3627429095330454, 4.920442397646541, 1.778849744056748, 4.504335563122838, 6.247337742247277, 0.035847564932309124, 3.177440218522102, 3.105745088657484, 1.0300791462005798, 5.253106160979007, 2.1115135073892133, 4.1716717997903725, 1.2316810701201568, 5.051504237059429, 1.9099115834696363, 4.37327372370995, 0.20870393619917757, 6.074481370980409, 2.9328887173906155, 3.3502965897889707, 2.586263888776435, 3.6969214184031514, 0.5553287648133584, 5.7278565423662275, 0.5813179391690413, 5.701867368010545, 2.560274714420752, 3.722910592758834, 5.099495307306874, 1.1836899998727115, 4.325282653462504, 1.9579026537170818, 3.6823073633780865, 2.6008779438014997, 5.742470597391293, 0.5407147097882937, 3.272190330175932, 3.010994977003654, 6.152587630593447, 0.130597676586139, 4.633691949440656, 1.6494933577389308, 4.791086011328724, 1.4920992958508623, 3.1689256152945435, 3.1142596918850427, 6.255852345474835, 0.027332961704750736, 4.661565505169506, 1.62161980201008, 4.763212455599874, 1.519972851579713, 4.219214562539231, 2.063970744640355, 5.205563398230148, 1.077621908949438, 4.347652493558869, 1.935532813620717, 5.07712546721051, 1.2060598399690763, 3.9560099933797064, 2.32717531379988, 5.468767967389673, 0.8144173397899134, 3.4873951563192755, 2.7957901508603107, 5.937382804450103, 0.3458025027294826, 5.423360401003239, 0.8598249061763472, 4.00141755976614, 2.281767747413446, 2.2427139477507234, 4.040471359428863, 0.8988787058390697, 5.3843066013405165, 0.6005854761567402, 5.682599831022846, 2.541007177433053, 3.742178129746533]

symmetries =  [1.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
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 0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
 0.00000000e+00 0.00000000e+00 0.00000000e+00 2.78230434e-04
 1.00916092e-03 1.06383452e-03]
symmetry error = 0.0
minimal relative intensity = 0.999948 rms = 0.00000915


192 points (theta,phi), l_max = 18:
theta = [0.11856413292853131, 3.0230285206612617, 3.0230285206612617, 0.11856413292853131, 1.6386588772471251, 1.502933776342668, 1.502933776342668, 1.6386588772471251, 1.4737241557459189, 1.6678684978438745, 1.6678684978438745, 1.4737241557459189, 2.417816425581302, 0.7237762280084913, 0.7237762280084913, 2.417816425581302, 2.2930448655082865, 0.8485477880815067, 0.8485477880815067, 2.2930448655082865, 1.5318530455888495, 1.6097396080009436, 1.6097396080009436, 1.5318530455888495, 2.319817719799065, 0.8217749337907284, 0.8217749337907284, 2.319817719799065, 2.181404589350839, 0.9601880642389541, 0.9601880642389541, 2.181404589350839, 1.0977195695397122, 2.043873084050081, 2.043873084050081, 1.0977195695397122, 2.0355357994694696, 1.1060568541203237, 1.1060568541203237, 2.0355357994694696, 2.2224849020692767, 0.9191077515205166, 0.9191077515205166, 2.2224849020692767, 2.287223375751041, 0.8543692778387522, 0.8543692778387522, 2.287223375751041, 2.170169782059297, 0.9714228715304963, 0.9714228715304963, 2.170169782059297, 1.7963233466116817, 1.3452693069781114, 1.3452693069781114, 1.7963233466116817, 2.4895258218918075, 0.6520668316979855, 0.6520668316979855, 2.4895258218918075, 1.907518766601974, 1.2340738869878192, 1.2340738869878192, 1.907518766601974, 1.9406417370227045, 1.2009509165670886, 1.2009509165670886, 1.9406417370227045, 0.511764975790594, 2.6298276777991996, 2.6298276777991996, 0.511764975790594, 1.8920392070146765, 1.2495534465751166, 1.2495534465751166, 1.8920392070146765, 0.9525746318083235, 2.18901802178147, 2.18901802178147, 0.9525746318083235, 2.4207571555895715, 0.7208354980002215, 0.7208354980002215, 2.4207571555895715, 1.2481059482982562, 1.8934867052915372, 1.8934867052915372, 1.2481059482982562, 2.8096785510053635, 0.33191410258443, 0.33191410258443, 2.8096785510053635, 1.6458049085993642, 1.4957877449904289, 1.4957877449904289, 1.6458049085993642, 2.0027981330143136, 1.1387945205754795, 1.1387945205754795, 2.0027981330143136, 1.150121701539359, 1.9914709520504343, 1.9914709520504343, 1.150121701539359, 0.6247189968421768, 2.5168736567476166, 2.5168736567476166, 0.6247189968421768, 2.5310987195925794, 0.6104939339972137, 0.6104939339972137, 2.5310987195925794, 1.6532400264411784, 1.488352627148615, 1.488352627148615, 1.6532400264411784, 0.9675405155798266, 2.1740521380099667, 2.1740521380099667, 0.9675405155798266, 2.864330473649743, 0.2772621799400506, 0.2772621799400506, 2.864330473649743, 1.8178593422668854, 1.323733311322908, 1.323733311322908, 1.8178593422668854, 1.4475361711195858, 1.6940564824702076, 1.6940564824702076, 1.4475361711195858, 1.5649970062811378, 1.5765956473086555, 1.5765956473086555, 1.5649970062811378, 2.6919388544033156, 0.4496537991864777, 0.4496537991864777, 2.6919388544033156, 1.121185487178981, 2.0204071664108123, 2.0204071664108123, 1.121185487178981, 1.7790636820268462, 1.3625289715629472, 1.3625289715629472, 1.7790636820268462, 0.8938140373861783, 2.247778616203615, 2.247778616203615, 0.8938140373861783, 0.7203986855425768, 2.4211939680472163, 2.4211939680472163, 0.7203986855425768, 0.3560100040012956, 2.7855826495884974, 2.7855826495884974, 0.3560100040012956, 1.854522632264572, 1.2870700213252213, 1.2870700213252213, 1.854522632264572, 1.3616339099304433, 1.77995874365935, 1.77995874365935, 1.3616339099304433, 1.4243583248958573, 1.717234328693936, 1.717234328693936, 1.4243583248958573, 1.1296738009768104, 2.011918852612983, 2.011918852612983, 1.1296738009768104, 2.673482859409027, 0.46810979418076615, 0.46810979418076615, 2.673482859409027, 1.8154861328707954, 1.326106520718998, 1.326106520718998, 1.8154861328707954, 2.5103318554956875, 0.6312607980941055, 0.6312607980941055, 2.5103318554956875, 1.0025564324703362, 2.139036221119457, 2.139036221119457, 1.0025564324703362]
phi = [2.1812911113920577, 4.101894195787528, 0.9603015421977356, 5.32288376498185, 1.4734994929769247, 4.809685814202662, 1.6680931606128686, 4.615092146566718, 6.215001264726791, 0.06818404245279527, 3.209776696042588, 3.073408611136998, 3.0827663356962036, 3.2004189714833826, 0.058826317893589615, 6.224358989285997, 4.764301324151482, 1.5188839830281042, 4.660476636617897, 1.622708670561689, 3.8645099058378873, 2.418675401341699, 5.560268054931492, 0.7229172522480944, 2.470120236932482, 3.813065070247104, 0.671472416657311, 5.611712890522275, 5.302096726845951, 0.9810885803336348, 4.122681233923428, 2.1605040732561585, 3.841449072584682, 2.441736234594904, 5.583328888184697, 0.6998564189948893, 3.9666814541184054, 2.316503853061181, 5.458096506650974, 0.8250888005286122, 3.7404727494288466, 2.5427125577507397, 5.684305211340533, 0.5988800958390533, 4.076003078345953, 2.2071822288336334, 5.3487748824234265, 0.93441042475616, 4.438135523488299, 1.8450497836912876, 4.9866424372810805, 1.2965428698985058, 3.758827085446181, 2.5243582217334053, 5.665950875323198, 0.6172344318563878, 3.518992986183961, 2.764192320995625, 5.905784974585418, 0.3774003325941683, 1.963814656044084, 4.319370651135502, 1.1777779975457094, 5.105407309633877, 5.920960274067385, 0.3622250331122018, 3.5038176867019946, 2.7793676204775917, 3.9718767127692822, 2.311308594410304, 5.452901248000097, 0.8302840591794892, 5.369503714032178, 0.9136815931474083, 4.055274246737201, 2.227911060442385, 3.5394686361629977, 2.7437166710165886, 5.885309324606381, 0.39787598257320483, 2.0696213544828788, 4.213563952696708, 1.0719712991069144, 5.211214008072671, 3.2206917664860235, 3.0624935406935627, 6.204086194283356, 0.07909911289623071, 1.8028488333163506, 4.480336473863236, 1.3387438202734425, 4.944441486906143, 5.036022371016072, 1.2471629361635141, 4.388755589753307, 1.894429717426279, 1.1043785497415417, 5.1788067574380445, 2.0372141038482514, 4.245971203331335, 5.806675942883487, 0.47650936429609925, 3.6181020178858923, 2.665083289293694, 2.368663450384597, 3.9145218567949893, 0.7729292032051964, 5.51025610397439, 1.7149446266098627, 4.5682406805697235, 1.4266480269799304, 4.856537280199656, 5.317994651394682, 0.9651906557849044, 4.1067833093746975, 2.1764019978048887, 3.241761355577503, 3.0414239516020833, 6.1830166051918765, 0.10016870198771007, 2.67575681373584, 3.6074284934437464, 0.4658358398539533, 5.8173494673256325, 4.839529768235241, 1.443655538944345, 4.585248192534138, 1.6979371146454483, 3.3905843091602894, 2.892600998019297, 6.03419365160909, 0.2489916555704963, 2.691973698831334, 3.591211608348252, 0.44961895475845914, 5.833566352421127, 0.013342711945049183, 6.269842595234537, 3.1282499416447442, 3.154935365534842, 4.718828268734901, 1.5643570384446848, 4.705949692034478, 1.5772356151451086, 0.8759339725717133, 5.4072513346078726, 2.26565868101808, 4.017526626161507, 6.014706847182685, 0.26847845999690145, 3.4100711135866946, 2.8731141935928917, 1.8895994452883016, 4.393585861891284, 1.2519932083014917, 5.031192098878094, 2.5033949601930874, 3.779790346986499, 0.6381976933967058, 5.64498761378288, 1.3527853998771733, 4.930399907302413, 1.7888072537126198, 4.494378053466966, 5.992955972479009, 0.2902293347005774, 3.43182198829037, 2.851363318889216, 5.158625774962134, 1.1245595322174522, 4.266152185807245, 2.017033121372341, 2.979522185685508, 3.303663121494078, 0.162070467904285, 6.121114839275301, 1.2414828914564124, 5.041702415723174, 1.9001097621333807, 4.383075545046205, 2.5536243569077373, 3.729560950271849, 0.5879682966820561, 5.69521701049753, 5.860195153043922, 0.42299015413566476, 3.5645828077254578, 2.7186024994541285, 4.4208524547186006, 1.8623328524609852, 5.003925506050778, 1.279259801128808]

symmetries =  [1.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
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 0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
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 0.00000000e+00 0.00000000e+00 0.00000000e+00 8.91348407e-04
 2.14866635e-03 4.12772200e-03]
symmetry error = 0.0
minimal relative intensity = 0.999950 rms = 0.00000804


204 points (theta,phi), l_max = 19:
theta = [1.1792814129106908, 1.9623112406791026, 1.9623112406791026, 1.1792814129106908, 2.721989537412195, 0.41960311617759816, 0.41960311617759816, 2.721989537412195, 1.4276089547672681, 1.7139836988225252, 1.7139836988225252, 1.4276089547672681, 1.3384307937585356, 1.8031618598312578, 1.8031618598312578, 1.3384307937585356, 2.4296282636113915, 0.7119643899784018, 0.7119643899784018, 2.4296282636113915, 2.228615645411002, 0.9129770081787913, 0.9129770081787913, 2.228615645411002, 1.6883718712370273, 1.453220782352766, 1.453220782352766, 1.6883718712370273, 2.2548317077077105, 0.8867609458820827, 0.8867609458820827, 2.2548317077077105, 0.6980449834241197, 2.4435476701656738, 2.4435476701656738, 0.6980449834241197, 1.4475454716301615, 1.6940471819596319, 1.6940471819596319, 1.4475454716301615, 2.6507190833613437, 0.49087357022844946, 0.49087357022844946, 2.6507190833613437, 2.043261877380644, 1.0983307762091497, 1.0983307762091497, 2.043261877380644, 1.315176001287845, 1.826416652301948, 1.826416652301948, 1.315176001287845, 2.692392935077001, 0.4491997185127923, 0.4491997185127923, 2.692392935077001, 1.2099774085834771, 1.9316152450063162, 1.9316152450063162, 1.2099774085834771, 1.6414647651206589, 1.5001278884691345, 1.5001278884691345, 1.6414647651206589, 2.044384897627104, 1.097207755962689, 1.097207755962689, 2.044384897627104, 0.47970361038051373, 2.6618890432092797, 2.6618890432092797, 0.47970361038051373, 1.68809989744225, 1.4534927561475433, 1.4534927561475433, 1.68809989744225, 2.340388605308278, 0.8012040482815154, 0.8012040482815154, 2.340388605308278, 2.3583006192936073, 0.783292034296186, 0.783292034296186, 2.3583006192936073, 2.1071771836205686, 1.0344154699692247, 1.0344154699692247, 2.1071771836205686, 2.455534121331824, 0.6860585322579693, 0.6860585322579693, 2.455534121331824, 1.1870651652717799, 1.9545274883180133, 1.9545274883180133, 1.1870651652717799, 1.4697843042959462, 1.6718083492938471, 1.6718083492938471, 1.4697843042959462, 2.5487082994232844, 0.5928843541665088, 0.5928843541665088, 2.5487082994232844, 0.9889341552682702, 2.152658498321523, 2.152658498321523, 0.9889341552682702, 1.9103073327811262, 1.2312853208086671, 1.2312853208086671, 1.9103073327811262, 2.6590932820442497, 0.4824993715455437, 0.4824993715455437, 2.6590932820442497, 1.2418075497783008, 1.8997851038114923, 1.8997851038114923, 1.2418075497783008, 1.962922380468806, 1.178670273120987, 1.178670273120987, 1.962922380468806, 2.457583045617837, 0.684009607971956, 0.684009607971956, 2.457583045617837, 2.0981549866888223, 1.0434376669009708, 1.0434376669009708, 2.0981549866888223, 2.0219988707421135, 1.1195937828476796, 1.1195937828476796, 2.0219988707421135, 2.247561836538655, 0.8940308170511384, 0.8940308170511384, 2.247561836538655, 2.2734595725209097, 0.8681330810688833, 0.8681330810688833, 2.2734595725209097, 1.771252025548796, 1.370340628040997, 1.370340628040997, 1.771252025548796, 2.898843437854438, 0.24274921573535546, 0.24274921573535546, 2.898843437854438, 1.4357323696151043, 1.705860283974689, 1.705860283974689, 1.4357323696151043, 1.4418030630299072, 1.6997895905598859, 1.6997895905598859, 1.4418030630299072, 2.9944938387820192, 0.14709881480777384, 0.14709881480777384, 2.9944938387820192, 1.5004869785230113, 1.6411056750667818, 1.6411056750667818, 1.5004869785230113, 1.5910250432750956, 1.5505676103146977, 1.5505676103146977, 1.5910250432750956, 2.8676474328968777, 0.27394522069291555, 0.27394522069291555, 2.8676474328968777, 1.843955056622709, 1.2976375969670844, 1.2976375969670844, 1.843955056622709, 2.034958988454155, 1.1066336651356383, 1.1066336651356383, 2.034958988454155, 2.310234367013419, 0.8313582865763742, 0.8313582865763742, 2.310234367013419, 0.9424864975527023, 2.199106156037091, 2.199106156037091, 0.9424864975527023, 1.3037559285609162, 1.8378367250288772, 1.8378367250288772, 1.3037559285609162, 2.1471058163871306, 0.9944868372026627, 0.9944868372026627, 2.1471058163871306, 0.6503457592557067, 2.4912468943340866, 2.4912468943340866, 0.6503457592557067]
phi = [2.9865925285553887, 3.2965927786241975, 0.15500012503440455, 6.128185182145182, 1.2129388663800154, 5.070246440799571, 1.9286537872097778, 4.354531519969808, 5.108176149102898, 1.1750091580766877, 4.316601811666481, 1.9665834955131054, 3.8209305262057702, 2.462254780973816, 5.603847434563609, 0.6793378726159773, 2.7813775213625056, 3.5018077858170806, 0.3602151322272875, 5.922970174952299, 5.007665985930496, 1.275519321249091, 4.417111974838884, 1.8660733323407022, 2.2605132984505563, 4.02267200872903, 0.881079355139237, 5.402105952040349, 6.13124595889429, 0.1519393482852966, 3.2935320018750898, 2.9896533053044965, 4.528847411948962, 1.7543378952306241, 4.8959305488204174, 1.387254758359169, 3.6179686299467644, 2.665216677232822, 5.806809330822615, 0.47637597635697154, 2.8777444097331606, 3.4054408974464256, 0.26384824385663264, 6.019337063322953, 4.850895521290696, 1.43228978588889, 4.573882439478683, 1.709302867700903, 2.768070181952797, 3.515115125226789, 0.37352247163699615, 5.90966283554259, 0.6215121486481061, 5.66167315853148, 2.5200805049416872, 3.763104802237899, 4.986038952575601, 1.2971463546039852, 4.438739008193778, 1.8444462989858081, 2.045667679369895, 4.237517627809691, 1.0959249742198984, 5.187260332959688, 6.203759572029271, 0.07942573515031563, 3.2210183887401085, 3.0621669184394777, 4.558790836577918, 1.7243944706016683, 4.865987124191461, 1.417198182988125, 3.93607036053377, 2.3471149466458163, 5.488707600235609, 0.7944777069439767, 3.305279526028771, 2.9779057811508154, 6.119498434740608, 0.16368687243897787, 4.545757122346607, 1.7374281848329796, 4.879020838422773, 1.4041644687568138, 2.690943638657601, 3.5922416685219853, 0.4506490149321924, 5.832536292247394, 5.344665133738271, 0.9385201734413153, 4.080112827031108, 2.203072480148478, 4.128697186445816, 2.1544881207337703, 5.296080774323563, 0.9871045328560232, 2.5563563263094897, 3.7268289808700965, 0.5852363272803035, 5.697948979899283, 0.18146744064284118, 6.101717866536745, 2.960125212946952, 3.323060094232634, 4.833387172704859, 1.4497981344747266, 4.591390788064519, 1.6917945191150667, 2.7918615460197773, 3.491323761159809, 0.34973110757001596, 5.93345419960957, 5.482638688023586, 0.8005466191560002, 3.942139272745793, 2.341046034433793, 4.352790385465637, 1.9303949217139489, 5.0719875753037424, 1.2111977318758442, 3.717489897551027, 2.5656954096285594, 5.7072880632183525, 0.5758972439612341, 3.7910624104567034, 2.492122896722883, 5.633715550312676, 0.6494697568669106, 4.254298240434954, 2.0288870667446317, 5.170479720334425, 1.1127055868451614, 3.942688741814714, 2.340496565364872, 5.482089218954665, 0.8010960882249213, 3.7351591381391085, 2.5480261690404777, 5.689618822630271, 0.5935664845493153, 4.10417961184634, 2.179005695333246, 5.320598348923039, 0.9625869582565473, 3.003751473174286, 3.2794338340053004, 0.1378411804155074, 6.145344126764079, 5.3070021994447885, 0.976183107734798, 4.117775761324591, 2.1654095458549953, 4.5100654606574615, 1.7731198465221243, 4.914712500111917, 1.368472807067669, 3.070693289572808, 3.2124920176067784, 0.0708993640169854, 6.212285943162601, 1.0709319002825897, 5.212253406896997, 2.0706607533072034, 4.212524553872383, 4.841703525732235, 1.4414817814473504, 4.583074435037143, 1.700110872142443, 3.414808714892316, 2.8683765922872704, 6.0099692458770635, 0.27321606130252307, 3.2164313338205055, 3.0667539733590807, 6.208346626948874, 0.07483868023071243, 4.691381264211886, 1.5918040429677, 4.733396696557493, 1.549788610622093, 2.4243284953494575, 3.8588568118301287, 0.7172641582403358, 5.565921148939251, 5.63227740053863, 0.6509079066409567, 3.7925005602307498, 2.4906847469488365, 4.126005520072998, 2.157179787106588, 5.298772440696381, 0.9844128664832054, 2.171178120454246, 4.11200718672534, 0.9704145331355474, 5.312770774044039, 0.32015115313823256, 5.963034154041353, 2.8214415004515607, 3.4617438067280255, 5.1633484283055235, 1.1198368788740627, 4.261429532463856, 2.0217557747157304]

symmetries =  [1.         0.         0.         0.         0.         0.
 0.         0.         0.         0.         0.         0.
 0.         0.         0.         0.         0.         0.
 0.         0.         0.00217839 0.01504656 0.00589377]
symmetry error = 0.0
minimal relative intensity = 0.999956 rms = 0.00000907


216 points (theta,phi), l_max = 20 # from Graef
theta = [1.4670812889991416, 1.6807776913157724, 1.8764118508705887, 1.5547093937752132, 1.8561643540493502, 1.336266503287936, 2.03964246139458, 2.016200860016856, 1.6734080958274649, 1.9220953930372975, 2.197469951383588, 1.2906627184432984, 2.1518650475022127, 1.4648296130457186, 2.1932597155679527, 2.332786624278528, 1.0710433158674662, 2.047467972294827, 0.14751156544305982, 0.20645621335543038, 0.3091946187328603, 0.4013359095181712, 0.41217084031542817, 0.42874809816298526, 0.48486305623699183, 0.5220535869234858, 0.5799245141038418, 0.6263508730872094, 0.6299579243254362, 0.655537314752579, 0.6901008212075936, 0.734475469069472, 0.741543682344694, 0.7938063975827603, 0.8048693237624966, 0.84726362707589, 2.9940810881467335, 2.9351364402343627, 2.832398034856933, 2.740256744071622, 2.729421813274365, 2.712844555426808, 2.6567295973528013, 2.6195390666663076, 2.5616681394859513, 2.515241780502584, 2.5116347292643573, 2.486055338837214, 2.4514918323821995, 2.407117184520321, 2.4000489712450994, 2.347786256007033, 2.3367233298272967, 2.2943290265139034, 0.14751156544305982, 0.20645621335543038, 0.3091946187328603, 0.4013359095181712, 0.41217084031542817, 0.42874809816298526, 0.48486305623699183, 0.5220535869234858, 0.5799245141038418, 0.6263508730872094, 0.6299579243254362, 0.655537314752579, 0.6901008212075936, 0.734475469069472, 0.741543682344694, 0.7938063975827603, 0.8048693237624966, 0.84726362707589, 2.9940810881467335, 2.9351364402343627, 2.832398034856933, 2.740256744071622, 2.729421813274365, 2.712844555426808, 2.6567295973528013, 2.6195390666663076, 2.5616681394859513, 2.515241780502584, 2.5116347292643573, 2.486055338837214, 2.4514918323821995, 2.407117184520321, 2.4000489712450994, 2.347786256007033, 2.3367233298272967, 2.2943290265139034, 1.4662797502904452, 1.3967872407648523, 1.6162479702394423, 1.1698203334647546, 1.2817778528006152, 1.2188580816258208, 1.4562555210338757, 1.8246197611447998, 1.0023756573076994, 1.0763357318005426, 1.6267141330670243, 0.9964599010189659, 1.2424894298474791, 0.8475800120573038, 1.9187566125513245, 1.391491484967237, 1.0023134597455206, 0.9364997076854233, 1.6753129032993481, 1.7448054128249408, 1.525344683350351, 1.9717723201250386, 1.8598148007891782, 1.9227345719639723, 1.6853371325559174, 1.3169728924449933, 2.1392169962820935, 2.0652569217892505, 1.5148785205227688, 2.1451327525708273, 1.8991032237423142, 2.2940126415324893, 1.2228360410384689, 1.7501011686225563, 2.139279193844273, 2.20509294590437, 1.6753129032993481, 1.7448054128249408, 1.525344683350351, 1.9717723201250386, 1.8598148007891782, 1.9227345719639723, 1.6853371325559174, 1.3169728924449933, 2.1392169962820935, 2.0652569217892505, 1.5148785205227688, 2.1451327525708273, 1.8991032237423142, 2.2940126415324893, 1.2228360410384689, 1.7501011686225563, 2.139279193844273, 2.20509294590437, 1.4662797502904452, 1.3967872407648523, 1.6162479702394423, 1.1698203334647546, 1.2817778528006152, 1.2188580816258208, 1.4562555210338757, 1.8246197611447998, 1.0023756573076994, 1.0763357318005426, 1.6267141330670243, 0.9964599010189659, 1.2424894298474791, 0.8475800120573038, 1.9187566125513245, 1.391491484967237, 1.0023134597455206, 0.9364997076854233, 1.6745113645906518, 1.460814962274021, 1.2651808027192046, 1.5868832598145801, 1.2854282995404431, 1.8053261503018572, 1.101950192195213, 1.1253917935729374, 1.4681845577623285, 1.2194972605524956, 0.9441227022062053, 1.8509299351464947, 0.9897276060875806, 1.6767630405440748, 0.9483329380218403, 0.8088060293112654, 2.070549337722327, 1.0941246812949663, 1.6745113645906518, 1.460814962274021, 1.2651808027192046, 1.5868832598145801, 1.2854282995404431, 1.8053261503018572, 1.101950192195213, 1.1253917935729374, 1.4681845577623285, 1.2194972605524956, 0.9441227022062053, 1.8509299351464947, 0.9897276060875806, 1.6767630405440748, 0.9483329380218403, 0.8088060293112654, 2.070549337722327, 1.0941246812949663, 1.4670812889991416, 1.6807776913157724, 1.8764118508705887, 1.5547093937752132, 1.8561643540493502, 1.336266503287936, 2.03964246139458, 2.016200860016856, 1.6734080958274649, 1.9220953930372975, 2.197469951383588, 1.2906627184432984, 2.1518650475022127, 1.4648296130457186, 2.1932597155679527, 2.332786624278528, 1.0710433158674662, 2.047467972294827]
phi = [0.10508332145614331, 0.17507772178410805, 6.235523550963355, 0.40103085601433597, 0.30157421689965797, 0.36229439169219574, 0.12846850166749996, 6.0012092810988635, 0.5718018597785091, 0.5298744484971941, 6.214130610848407, 0.6008146099902621, 0.39602249785252763, 0.7282069371351721, 5.850073224393192, 0.24908407899724527, 0.6603186399582319, 0.7301199843937258, 0.7815628356572469, 5.7181653062803175, 4.562511171955505, 0.04118994665412522, 5.5039667569062845, 0.5931537997200833, 4.960128094730833, 4.184672560757499, 6.095146425611696, 5.655788644620704, 4.617376547078143, 0.47075681842393763, 5.243501245879731, 0.1584793745814857, 4.183195050945853, 4.965197853432517, 0.727338943348124, 5.624348988480797, 5.50162247152234, 0.565020000899269, 1.720674135224081, 6.241995360525461, 0.7792185502733021, 5.690031507459503, 1.323057212448753, 2.098512746422087, 0.18803888156789025, 0.6273966625588828, 1.6658087601014429, 5.812428488755649, 1.039684061299855, 6.124705932598101, 2.099990256233733, 1.317987453747069, 5.555846363831463, 0.6588363186987896, 3.9231554892470397, 2.5765726526905244, 1.420918518365712, 3.182782600243918, 2.3623741033164913, 3.734746453309876, 1.8185354411410404, 1.0430799071677062, 2.953553772021903, 2.5141959910309106, 1.4757838934883505, 3.6123494720137304, 2.1019085922899383, 3.3000720281712788, 1.0416023973560602, 1.8236051998427243, 3.8689315969379168, 2.4827563348910036, 2.3600298179325465, 3.706612654489062, 4.862266788813875, 3.100402706935668, 3.920811203863095, 2.54843885386971, 4.464649866038545, 5.24010540001188, 3.329631535157683, 3.7689893161486756, 4.8074014136912355, 2.670835835165856, 4.181276714889648, 2.9831132790083075, 5.241582909823526, 4.459580107336862, 2.4142537102416695, 3.8004289722885827, 1.4665101561310205, 1.6824710235096303, 1.8767380364334127, 1.5533233185242359, 1.8688829501926358, 1.3206357515220206, 2.042986196919583, 2.0320640837144888, 1.692641980509646, 1.9724523518508246, 2.1986038462330857, 1.2352099117296178, 2.189389615886955, 1.429239361644246, 2.2398031480993525, 2.3484521909287874, 0.9659372248758695, 2.17684264056662, 1.6750824974587726, 1.459121630080163, 1.2648546171563806, 1.5882693350655572, 1.2727097033971573, 1.8209569020677727, 1.0986064566702107, 1.1095285698753046, 1.4489506730801474, 1.1691403017389685, 0.9429888073567073, 1.9063827418601753, 0.9522030377028382, 1.7123532919455473, 0.9017895054904408, 0.7931404626610057, 2.175655428713924, 0.9647500130231732, 4.816675151048566, 4.600714283669956, 4.406447270746174, 4.72986198865535, 4.41430235698695, 4.962549555657565, 4.240199110260003, 4.2511212234650975, 4.59054332666994, 4.310732955328762, 4.0845814609465005, 5.047975395449969, 4.093795691292631, 4.85394594553534, 4.043382159080234, 3.934733116250799, 5.317248082303717, 4.106342666612966, 4.608102809720814, 4.824063677099423, 5.018330690023205, 4.694915972114029, 5.010475603782429, 4.462228405111814, 5.184578850509375, 5.173656737304282, 4.834234634099439, 5.114045005440618, 5.340196499822879, 4.3768025653194105, 5.330982269476748, 4.570832015234039, 5.381395801689146, 5.4900448445185805, 4.107529878465662, 5.318435294156413, 6.178101985723443, 6.108107585395478, 0.047661756216231484, 5.88215445116525, 5.981611090279928, 5.920890915487391, 6.154716805512086, 0.28197602608072253, 5.711383447401078, 5.753310858682392, 0.06905469633117904, 5.682370697189324, 5.887162809327059, 5.554978370044414, 0.43311208278639396, 6.034101228182341, 5.622866667221354, 5.55306532278586, 3.03650933213365, 2.966514931805685, 3.1892544098060247, 2.7405617975754573, 2.8400184366901353, 2.7792982618975977, 3.0131241519222933, 3.4235686796705154, 2.569790793811284, 2.611718205092599, 3.210647349920972, 2.540778043599531, 2.7455701557372656, 2.413385716454621, 3.574704736376187, 2.892508574592548, 2.481274013631561, 2.4114726691960673, 3.246675975045936, 3.316670375373901, 3.0939308973735615, 3.542623509604129, 3.443166870489451, 3.5038870452819886, 3.270061155257293, 2.859616627509071, 3.713394513368302, 3.671467102086987, 3.0725379572586142, 3.742407263580055, 3.5376151514423206, 3.8697995907249654, 2.7084805708033994, 3.390676732587038, 3.801911293548025, 3.871712637983519]

symmetries =  [1.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
 4.35525259e-25 2.11728665e-24 5.11671797e-24 4.35466260e-24
 1.27953361e-24 5.28856604e-24 4.24933871e-24 5.22615613e-25
 9.36703715e-24 1.21063154e-23 1.99402268e-23 2.27091137e-23
 6.78265041e-24 3.50288681e-24 3.85388952e-23 1.54785829e-23
 2.94343090e-23 8.95208447e-03 4.48179027e-03 4.26642112e-03]
symmetry error = 1.8122426377977265e-22
minimal relative intensity = 0.999965 rms = 0.00000573

Non t-Designs

In Shvydky2 we find the statement

"Future direct-drive ICF laser systems will look to improve uniformity, which means increasing t in t- design configurations using the least number of beams N ."

The strong t-designs from the previous section support that statement. But are t-designs really the best way to address uniformity?

The paper decomposes uniformity into contributions from different modes ⇐ l_max. For a typical t-design, you need to set l_max to about t+5 to cover the modes that matter most for uniformity.

The uniformity terms for each mode involve two components which are multiplied:

The first is laser positioning dependent (overall uniformity).
The second is beam shape dependent (single beam effect).

If a single beam covers a larger or smaller area, you need to adapt the positioning. t-designs have a useful property: the positioning-dependent component is zero for all modes ⇐ t. That gives good uniformity across many beam shapes. In practice, you cannot build a new system every time you adjust the laser parameters. A good design should cover a whole range of possible configurations.

So why not focus on t-designs and use the remaining freedom, when the number of beams N is large enough, to further improve uniformity? In practice, this means minimizing the roughly 5 uniformity components for modes l = t+1,…,t+5.

The main counterarguments are:

t-designs are a local minimum that no optimizer can escape.
Limiting the optimization to the last 5 components while keeping the t-design property is too restrictive in practice.
Non-t-designs can cover the same beam-shape area as t-designs.

The plots below show the beam-shape area covered by designs found by directly optimizing uniformity for a specific beam shape. The orange area marks the region where the uniformity rms stays below 0.1%. On the right you see the corresponding best t-designs. In all cases, the t-design covers a slightly smaller area.

The non-t-designs shown here do not only optimize uniformity for a specific beam shape, they also cover a large beam-shape area. For the optimization, we should choose beam-shape parameters in the center of the beam shape area we are interested in.

Note that this approach does not work well for designs with N < 132. In that case, the previous approach gives better results. t-designs cover a larger uniformity-rms area.

N = 132, l_max = 15

theta = [0.19058314910648566, 0.19058314910648566, 0.25326831366519414, 0.25326831366519414, 0.44381205087511444, 0.44381205087511444, 0.4663540479446312, 0.4663540479446312, 0.5269020092837822, 0.5269020092837822, 0.5535736063009062, 0.5535736063009062, 0.6065757477731419, 0.6065757477731419, 0.7305445803867652, 0.7305445803867652, 0.760678412119407, 0.760678412119407, 0.7947842116747352, 0.7947842116747352, 0.8153462038180218, 0.8153462038180218, 0.8322653495401049, 0.8322653495401049, 0.8550480900676015, 0.8550480900676015, 0.9311497677926796, 0.9311497677926796, 1.0172227204943038, 1.0172227204943038, 1.0524344483124528, 1.0524344483124528, 1.0627893228043859, 1.0627893228043859, 1.070840344657816, 1.070840344657816, 1.073386397057278, 1.073386397057278, 1.141000835180699, 1.141000835180699, 1.1756072515514133, 1.1756072515514133, 1.201149546307772, 1.201149546307772, 1.253585891690416, 1.253585891690416, 1.2998326929433284, 1.2998326929433284, 1.3283081136470907, 1.3283081136470907, 1.3314868702370424, 1.3314868702370424, 1.3794732151622056, 1.3794732151622056, 1.3910775197069516, 1.3910775197069516, 1.4429917922830295, 1.4429917922830295, 1.4493315558619464, 1.4493315558619464, 1.4991251744276173, 1.4991251744276173, 1.50805193795284, 1.50805193795284, 1.5707957972339772, 1.5707957972339772, 1.5707968563558161, 1.5707968563558161, 1.633540715636953, 1.633540715636953, 1.642467479162176, 1.642467479162176, 1.692261097727847, 1.692261097727847, 1.6986008613067636, 1.6986008613067636, 1.7505151338828417, 1.7505151338828417, 1.7621194384275878, 1.7621194384275878, 1.8101057833527507, 1.8101057833527507, 1.8132845399427027, 1.8132845399427027, 1.8417599606464647, 1.8417599606464647, 1.8880067618993772, 1.8880067618993772, 1.9404431072820214, 1.9404431072820214, 1.9659854020383798, 1.9659854020383798, 2.0005918184090943, 2.0005918184090943, 2.068206256532515, 2.068206256532515, 2.070752308931977, 2.070752308931977, 2.0788033307854072, 2.0788033307854072, 2.0891582052773403, 2.0891582052773403, 2.1243699330954895, 2.1243699330954895, 2.2104428857971135, 2.2104428857971135, 2.2865445635221917, 2.2865445635221917, 2.3093273040496882, 2.3093273040496882, 2.326246449771771, 2.326246449771771, 2.346808441915058, 2.346808441915058, 2.380914241470386, 2.380914241470386, 2.411048073203028, 2.411048073203028, 2.535016905816651, 2.535016905816651, 2.588019047288887, 2.588019047288887, 2.614690644306011, 2.614690644306011, 2.675238605645162, 2.675238605645162, 2.697780602714679, 2.697780602714679, 2.888324339924599, 2.888324339924599, 2.9510095044833076, 2.9510095044833076]
phi = [1.2334254061421728, 4.375018059731966, 5.993356278026351, 2.851763624436558, 5.171176172664568, 2.0295835190747757, 3.7792142002994904, 0.6376215467096975, 4.456128423206297, 1.3145357696165043, 3.1415916463038402, 6.283184299893633, 5.7042078165859245, 2.5626151629961313, 4.894985650705377, 1.7533929971155837, 0.8018823404069473, 3.94347499399674, 3.4801334610084678, 0.33854080741867476, 5.321816896861356, 2.180224243271563, 4.386081111009631, 1.244488457419838, 6.12191746124015, 2.9803248076503572, 5.737085127714493, 2.5954924741246996, 4.712389602920597, 1.5707969493308045, 3.7262183146820727, 0.5846256610922798, 0.14640155590468418, 3.287994209494477, 4.112491728877285, 0.9708990752874921, 5.075224909858596, 1.9336322562688029, 2.2871406841829196, 5.428733337772712, 2.934067575605919, 6.075660229195712, 4.421244491548798, 1.2796518379590054, 2.6154442700847946, 5.757036923674588, 3.52595423092626, 0.3843615773364669, 1.6446313963886712, 4.786224049978464, 0.7655728492267814, 3.907165502816574, 1.9737507503951244, 5.115343403984918, 3.205365529756412, 0.0637728761666193, 4.199798021777923, 1.0582053681881303, 2.2930172039646166, 5.43460985755441, 2.898467725381906, 6.040060378971699, 1.3907192981524739, 4.532311951742267, 2.588019047289114, 5.729611700878907, 0.5535736063006794, 3.695166259890472, 1.7508733554373195, 4.892466009027112, 0.24312492820788767, 3.3847175817976805, 0.8485754496251768, 3.9901681032149696, 5.224979938991456, 2.0833872854016633, 6.219412431012967, 3.077819777423174, 1.167841903194669, 4.309434556784462, 2.376019804363012, 5.517612457952805, 1.4969612572011222, 4.638553910790915, 5.898823729843119, 2.7572310762533263, 0.5261483835049988, 3.6677410370947916, 5.003533469220581, 1.861940815630788, 0.207525077983874, 3.349117731573667, 0.8544519694068737, 3.9960446229966666, 4.349553050910783, 1.2079603973209905, 5.312286231892094, 2.170693578302301, 2.995191097685109, 6.136783751274902, 5.698559646087307, 2.5569669924975136, 4.7123883578487815, 1.5707957042589888, 3.6876928330548866, 0.5461001794650937, 3.302860499529229, 0.16126784593943605, 5.038696849759749, 1.8971041961699553, 4.102961063908023, 0.9613684103182304, 5.944644499760911, 2.8030518461711185, 5.481302966772639, 2.339710313182846, 4.5297923100640025, 1.3881996564742094, 3.720570144183455, 0.5789774905936617, 1.007285953127783e-06, 3.141593660875746, 4.968649537563082, 1.827056883973289, 5.645563760469889, 2.503971106880096, 4.2536017881048105, 1.1120091345150178, 0.2898290291532351, 3.431421682743028, 5.0497599010374135, 1.9081672474476203]

symmetries =  [1.00000000e+00 0.00000000e+00 0.00000000e+00 3.72937003e-23
 1.31554197e-20 0.00000000e+00 2.89881472e-08 3.13699669e-19
 1.61212366e-19 5.48399621e-20 3.80130054e-12 1.75822578e-17
 8.17828061e-09 3.21015136e-15 1.48428925e-15 3.09467231e-06
 3.82774643e-04 4.71919194e-13 2.49589591e-03]
symmetry error = 3.131842547039528e-06
minimal relative intensity = 0.999818 rms = 0.00003867

N = 144, l_max = 16

theta = [0.21151854170026613, 0.21151854170026613, 0.21158151524808294, 0.21158151524808294, 0.4423761001010803, 0.4423761001010803, 0.4424362072009404, 0.4424362072009404, 0.4990932155155501, 0.4990932155155501, 0.49914593587246636, 0.49914593587246636, 0.662697056468213, 0.662697056468213, 0.662757857866129, 0.662757857866129, 0.7051649735590645, 0.7051649735590645, 0.705221921844034, 0.705221921844034, 0.8014052705698652, 0.8014052705698652, 0.8014105481600254, 0.8014105481600254, 0.8793299428973177, 0.8793299428973177, 0.8793707379950503, 0.8793707379950503, 0.9661752831826961, 0.9661752831826961, 0.9661830539126325, 0.9661830539126325, 0.9864229546167377, 0.9864229546167377, 0.9865086876996753, 0.9865086876996753, 1.1161146769171002, 1.1161146769171002, 1.116117203090332, 1.116117203090332, 1.1309726258317145, 1.1309726258317145, 1.1310314810888815, 1.1310314810888815, 1.1690811727152173, 1.1690811727152173, 1.1690977616635114, 1.1690977616635114, 1.2910240920957265, 1.2910240920957265, 1.2910850868157513, 1.2910850868157513, 1.2948020127107407, 1.2948020127107407, 1.2948399898095788, 1.2948399898095788, 1.4196393807581082, 1.4196393807581082, 1.4196982405555931, 1.4196982405555931, 1.423886217285154, 1.423886217285154, 1.4239150008380133, 1.4239150008380133, 1.4541397031131251, 1.4541397031131251, 1.4542095333492957, 1.4542095333492957, 1.525866688187864, 1.525866688187864, 1.5258796615193928, 1.5258796615193928, 1.6157129920704005, 1.6157129920704005, 1.6157259654019294, 1.6157259654019294, 1.6873831202404976, 1.6873831202404976, 1.687452950476668, 1.687452950476668, 1.71767765275178, 1.71767765275178, 1.7177064363046393, 1.7177064363046393, 1.7218944130342, 1.7218944130342, 1.7219532728316849, 1.7219532728316849, 1.8467526637802145, 1.8467526637802145, 1.8467906408790524, 1.8467906408790524, 1.850507566774042, 1.850507566774042, 1.8505685614940668, 1.8505685614940668, 1.972494891926282, 1.972494891926282, 1.972511480874576, 1.972511480874576, 2.010561172500912, 2.010561172500912, 2.010620027758079, 2.010620027758079, 2.0254754504994614, 2.0254754504994614, 2.025477976672693, 2.025477976672693, 2.155083965890118, 2.155083965890118, 2.1551696989730553, 2.1551696989730553, 2.175409599677161, 2.175409599677161, 2.1754173704070974, 2.1754173704070974, 2.262221915594743, 2.262221915594743, 2.2622627106924753, 2.2622627106924753, 2.3401821054297676, 2.3401821054297676, 2.340187383019928, 2.340187383019928, 2.4363707317457592, 2.4363707317457592, 2.4364276800307287, 2.4364276800307287, 2.4788347957236643, 2.4788347957236643, 2.4788955971215803, 2.4788955971215803, 2.642446717717327, 2.642446717717327, 2.642499438074243, 2.642499438074243, 2.699156446388853, 2.699156446388853, 2.699216553488713, 2.699216553488713, 2.9300111383417105, 2.9300111383417105, 2.9300741118895273, 2.9300741118895273]
phi = [3.941036229815513, 0.79944357622572, 2.3703358485890176, 5.511928502178811, 1.4657144250378764, 4.607307078627669, 3.036493694520311, 6.178086348110104, 3.7564103242755307, 0.6148176706857377, 2.1856956424252396, 5.327288296015032, 1.111944609041033, 4.253537262630826, 2.682845654718099, 5.824438308307892, 1.7512403596637995, 4.892833013253592, 0.18054059220337376, 3.3221332457931667, 2.2286014197243382, 5.370194073314131, 0.6577971515520897, 3.7993898051418826, 4.560710518436482, 1.419117864846689, 2.990010431312829, 6.131603084902622, 1.0076976445413943, 4.149290298131188, 5.720086752064338, 2.5784940984745455, 1.9035814385499685, 5.045174092139762, 3.4744045180781695, 0.33281186448837674, 2.255868131809863, 5.397460785399656, 3.8266726391312864, 0.6850799855414933, 1.620455545680741, 4.762048199270534, 0.04964350182599934, 3.191236155415792, 5.978470753049439, 2.8368780994596463, 4.407743411580516, 1.2661507579907234, 1.9897613098767066, 5.1313539634665, 0.4189397902626749, 3.560532443852468, 4.101858583628272, 0.9602659300384789, 2.530979072809317, 5.67257172639911, 4.861006062727723, 1.7194134091379298, 3.290179271215224, 0.14858661762543068, 2.988776868476079, 6.130369522065872, 1.4180407132019608, 4.559633366791754, 3.8387372284853805, 0.6971445748955876, 5.409485436214058, 2.267892782624265, 5.842886194729914, 2.7012935411401213, 1.1305564169727051, 4.272149070562499, 2.011036236617088, 5.152628890206881, 3.581891766039465, 0.4402991124496719, 4.015292524555321, 0.873699870965528, 5.5860407322839984, 2.4444480786942058, 1.7235519403878325, 4.865144593977625, 3.294408438703507, 0.152815785113714, 6.1345986895541555, 2.9930060359643624, 1.4221792444518635, 4.563771898041656, 0.6106135807804762, 3.752206234370269, 5.322919377141107, 2.1813267235513143, 2.7226528633271183, 5.864245516916911, 1.1518313437130865, 4.293423997302879, 5.017034549188863, 1.87544189559907, 3.44630720771994, 0.3047145541301472, 6.233541805353587, 3.091949151763794, 1.5211371079090523, 4.662729761498845, 5.5981053216380925, 2.4565126680483, 0.8857245217799304, 4.027317175369723, 5.95037344269121, 2.8087807891014167, 1.2380112150398248, 4.3796038686296175, 3.7046912087050408, 0.5630985551152476, 2.133895009048399, 5.2754876626381915, 0.1515822222769643, 3.2931748758667574, 4.864067442332898, 1.7224747887431044, 2.4837955020377036, 5.625388155627497, 0.912991233865455, 4.054583887455248, 2.9610520613864195, 6.102644714976212, 1.3903522939259938, 4.531944947515787, 3.600339652461487, 0.458746998871694, 5.171240698138553, 2.02964804454876, 4.097489664754347, 0.9558970111645535, 2.5267749829040556, 5.668367636493849, 3.246691612659275, 0.10509895906948208, 4.81747088214171, 1.6758782285519167, 0.7712568050007756, 3.9128494585905687, 2.3421490773640734, 5.4837417309538665]

symmetries =  [1.00000000e+00 0.00000000e+00 0.00000000e+00 2.28968246e-20
 1.93950864e-16 0.00000000e+00 1.45732725e-10 2.07704090e-17
 1.51368480e-13 5.79601693e-11 2.31436449e-10 1.68817667e-12
 6.40885121e-07 3.61739628e-08 2.42178318e-07 7.22552425e-08
 2.01267539e-04 4.76385211e-05 2.35827731e-05 5.69925155e-05]
symmetry error = 0.00020225946878413251
minimal relative intensity = 0.999889 rms = 0.00001849

N = 168, l_max = 17

theta = [0.1490408080889857, 0.1490408080889857, 0.25983512617146964, 0.25983512617146964, 0.3549578963500752, 0.3549578963500752, 0.40995546772604163, 0.40995546772604163, 0.46870803892220064, 0.46870803892220064, 0.5323254250661649, 0.5323254250661649, 0.5357276431431854, 0.5357276431431854, 0.6216804060646589, 0.6216804060646589, 0.6442177232987417, 0.6442177232987417, 0.699873460921509, 0.699873460921509, 0.7014943128249433, 0.7014943128249433, 0.7849096095733669, 0.7849096095733669, 0.7961319178340309, 0.7961319178340309, 0.7980093010418347, 0.7980093010418347, 0.8487919668951074, 0.8487919668951074, 0.8723350792626898, 0.8723350792626898, 0.9588409314963973, 0.9588409314963973, 0.9665232336236714, 0.9665232336236714, 0.9700383311560167, 0.9700383311560167, 0.989696673739025, 0.989696673739025, 1.0393197445924522, 1.0393197445924522, 1.0557173136801419, 1.0557173136801419, 1.069904448986343, 1.069904448986343, 1.0955067536866308, 1.0955067536866308, 1.1566184115963802, 1.1566184115963802, 1.1698135847279174, 1.1698135847279174, 1.2161116591555117, 1.2161116591555117, 1.2268378933390593, 1.2268378933390593, 1.2563857713275584, 1.2563857713275584, 1.2568541812610157, 1.2568541812610157, 1.2654947910274121, 1.2654947910274121, 1.2918118307141988, 1.2918118307141988, 1.3749242862231927, 1.3749242862231927, 1.3759330990451613, 1.3759330990451613, 1.3967817320267935, 1.3967817320267935, 1.4011194054898544, 1.4011194054898544, 1.4538430594136662, 1.4538430594136662, 1.4748508256270811, 1.4748508256270811, 1.4788334475916285, 1.4788334475916285, 1.4900578815602934, 1.4900578815602934, 1.516098168806205, 1.516098168806205, 1.557453306534981, 1.557453306534981, 1.5841393470548124, 1.5841393470548124, 1.6254944847835884, 1.6254944847835884, 1.6515347720295, 1.6515347720295, 1.6627592059981648, 1.6627592059981648, 1.666741827962712, 1.666741827962712, 1.687749594176127, 1.687749594176127, 1.740473248099939, 1.740473248099939, 1.7448109215629999, 1.7448109215629999, 1.7656595545446319, 1.7656595545446319, 1.7666683673666006, 1.7666683673666006, 1.8497808228755945, 1.8497808228755945, 1.8760978625623812, 1.8760978625623812, 1.8847384723287777, 1.8847384723287777, 1.885206882262235, 1.885206882262235, 1.914754760250734, 1.914754760250734, 1.9254809944342814, 1.9254809944342814, 1.971779068861876, 1.971779068861876, 1.984974241993413, 1.984974241993413, 2.0460858999031624, 2.0460858999031624, 2.0716882046034506, 2.0716882046034506, 2.0858753399096512, 2.0858753399096512, 2.102272908997341, 2.102272908997341, 2.151895979850768, 2.151895979850768, 2.1715543224337766, 2.1715543224337766, 2.175069419966122, 2.175069419966122, 2.182751722093396, 2.182751722093396, 2.2692575743271033, 2.2692575743271033, 2.2928006866946857, 2.2928006866946857, 2.3435833525479586, 2.3435833525479586, 2.345460735755762, 2.345460735755762, 2.3566830440164264, 2.3566830440164264, 2.44009834076485, 2.44009834076485, 2.441719192668284, 2.441719192668284, 2.4973749302910515, 2.4973749302910515, 2.5199122475251343, 2.5199122475251343, 2.6058650104466077, 2.6058650104466077, 2.6092672285236285, 2.6092672285236285, 2.6728846146675926, 2.6728846146675926, 2.7316371858637516, 2.7316371858637516, 2.786634757239718, 2.786634757239718, 2.8817575274183236, 2.8817575274183236, 2.9925518455008073, 2.9925518455008073]
phi = [2.474824877416353, 5.616417531006146, 3.8587694341808696, 0.7171767805910768, 4.750788836284852, 1.6091961826950592, 2.9378346021540596, 6.079427255743853, 2.2988168238734277, 5.440409477463221, 4.057179551328803, 0.9155868977390091, 3.4876326293236, 0.34603997573380707, 4.547149668283489, 1.4055570146936964, 1.9008146952049876, 5.0424073487947805, 5.783180911702366, 2.6415882581125727, 3.056777202707469, 6.198369856297262, 0.7996087689481939, 3.9412014225379868, 2.265688417937095, 5.407281071526888, 3.536394885693568, 0.39480223210377485, 4.336675177605305, 1.1950825240155125, 1.6422446186391815, 4.783837272228975, 3.2588991240472414, 0.11730647045744855, 6.044420104064823, 2.90282745047503, 1.9545004201085188, 5.096093073698311, 5.703864644869151, 2.5622719912793577, 3.749835528549191, 0.608242874959398, 4.090752037682531, 0.9491593840927384, 4.51369863939893, 1.372105985809137, 2.236195091898857, 5.37778774548865, 0.34459920012883594, 3.486191853718629, 1.6585080717478295, 4.800100725337622, 3.127363926847382, 6.268956580437175, 5.958213547475908, 2.816620893886115, 4.275434135578143, 1.1338414819883502, 2.5051778444978163, 5.6467704980876094, 5.073770122445372, 1.9321774688555786, 0.7580006898105435, 3.8995933434003365, 5.330185130142567, 2.1885924765527744, 1.397813458452324, 4.539406112042117, 3.6508987523918517, 0.5093060988020585, 3.3393323211181856, 0.1977396675283925, 1.663393555504899, 4.804986209094692, 4.097187066057945, 0.955594412468152, 6.165733454328628, 3.0241408007388353, 5.880816586758459, 2.739223933168666, 2.441873086625859, 5.583465740215652, 1.9255139646218828, 5.0671066182116755, 1.2160786889679105, 4.357671342557703, 0.6997195669639341, 3.841312220553727, 3.5439613740109204, 0.4023687204211276, 3.259044506440751, 0.11745185285095813, 5.3275908947114345, 2.1859982411216414, 1.478199098084894, 4.619791751674687, 6.085445639651193, 2.9438529860614007, 5.773879208377528, 2.6322865547877345, 1.743779195137469, 4.885371848727262, 4.094592830626812, 0.9530001770370188, 2.3835919637792498, 5.525184617369042, 4.351007838324008, 1.2094151847342147, 0.6364148090919766, 3.77800746268177, 5.149343825191236, 2.007751171601443, 3.4665644132934714, 0.32497175970367825, 0.014228726742411591, 3.1558213803322044, 1.4830845818419636, 4.624677235431757, 2.7969934534609573, 5.93858610705075, 0.9053975616909363, 4.046990215280729, 4.911079321370449, 1.7694866677806562, 5.334025923086848, 2.1924332694970547, 5.674942432220188, 2.533349778630395, 0.5793206623104354, 3.7209133159002286, 4.328684887071067, 1.1870922334812746, 0.23876520311476335, 3.3803578567045562, 6.165878836722138, 3.024286183132345, 4.640940688540405, 1.4993480349506119, 1.9465101295742808, 5.088102783164073, 2.7467904214860184, 5.8883830750758115, 4.017496889242491, 0.8759042356526983, 5.483576538231392, 2.3419838846415995, 3.2264081044721173, 0.08481545088232427, 0.5000043954772206, 3.6415970490670135, 4.382370611974599, 1.2407779583848058, 4.877628292485889, 1.736035638896097, 5.937145331445779, 2.7955526778559863, 2.226005755850784, 5.367598409440577, 3.9843684833061586, 0.8427758297163656, 3.3453507050255267, 0.20375805143573356, 1.5323964708947342, 4.673989124484527, 5.56600852658851, 2.4244158729987166, 3.8083604297632334, 0.6667677761734404]

symmetries =  [1.00000000e+00 0.00000000e+00 0.00000000e+00 3.76992577e-18
 4.56982427e-18 0.00000000e+00 1.35422678e-10 2.36382363e-15
 2.55265627e-15 1.30390646e-12 9.93092163e-11 3.62012367e-08
 2.59568068e-08 1.02630512e-08 4.45835015e-08 5.12751868e-07
 4.92802066e-05 1.36244989e-04 8.46190161e-05 2.92558507e-05
 1.68021906e-03]
symmetry error = 0.00018615518830865152
minimal relative intensity = 0.999916 rms = 0.00001406

N = 192, l_max = 18

theta = [0.173331207897696, 0.173331207897696, 0.18890561552715943, 0.18890561552715943, 0.40136830872921186, 0.40136830872921186, 0.40838567622839667, 0.40838567622839667, 0.4104481252288692, 0.4104481252288692, 0.41363802611286316, 0.41363802611286316, 0.5070684963768852, 0.5070684963768852, 0.6105634611654113, 0.6105634611654113, 0.6475065424729622, 0.6475065424729622, 0.6584479375394684, 0.6584479375394684, 0.6845577626483124, 0.6845577626483124, 0.6863296169369836, 0.6863296169369836, 0.6968243209089255, 0.6968243209089255, 0.7004245207132092, 0.7004245207132092, 0.8105127946287514, 0.8105127946287514, 0.8644556884668887, 0.8644556884668887, 0.9067197445263165, 0.9067197445263165, 0.9110599981249868, 0.9110599981249868, 0.9192017677958809, 0.9192017677958809, 0.9203931527012077, 0.9203931527012077, 0.9246468718425629, 0.9246468718425629, 0.9695796719351121, 0.9695796719351121, 0.9783274200305005, 0.9783274200305005, 0.9906591981183733, 0.9906591981183733, 1.0907457928030535, 1.0907457928030535, 1.1061112352307694, 1.1061112352307694, 1.108571332753095, 1.108571332753095, 1.1712036405906237, 1.1712036405906237, 1.1716511525696123, 1.1716511525696123, 1.1763215741479864, 1.1763215741479864, 1.1926980836816887, 1.1926980836816887, 1.2051026480002216, 1.2051026480002216, 1.2222072820551917, 1.2222072820551917, 1.230448050245938, 1.230448050245938, 1.2694054199419962, 1.2694054199419962, 1.2842495812908084, 1.2842495812908084, 1.3507425202339942, 1.3507425202339942, 1.3630388552835284, 1.3630388552835284, 1.4090908908951782, 1.4090908908951782, 1.4205401787821577, 1.4205401787821577, 1.4238563357160063, 1.4238563357160063, 1.430932274014307, 1.430932274014307, 1.4571705477870827, 1.4571705477870827, 1.4690862975711245, 1.4690862975711245, 1.4696100946002546, 1.4696100946002546, 1.482532706289407, 1.482532706289407, 1.5308410741624001, 1.5308410741624001, 1.534659483699689, 1.534659483699689, 1.6069331698901044, 1.6069331698901044, 1.6107515794273932, 1.6107515794273932, 1.6590599473003864, 1.6590599473003864, 1.6719825589895387, 1.6719825589895387, 1.6725063560186688, 1.6725063560186688, 1.6844221058027107, 1.6844221058027107, 1.7106603795754862, 1.7106603795754862, 1.717736317873787, 1.717736317873787, 1.7210524748076357, 1.7210524748076357, 1.732501762694615, 1.732501762694615, 1.778553798306265, 1.778553798306265, 1.7908501333557991, 1.7908501333557991, 1.8573430722989848, 1.8573430722989848, 1.8721872336477972, 1.8721872336477972, 1.9111446033438553, 1.9111446033438553, 1.9193853715346014, 1.9193853715346014, 1.9364900055895717, 1.9364900055895717, 1.9488945699081044, 1.9488945699081044, 1.965271079441807, 1.965271079441807, 1.969941501020181, 1.969941501020181, 1.9703890129991695, 1.9703890129991695, 2.0330213208366983, 2.0330213208366983, 2.0354814183590237, 2.0354814183590237, 2.0508468607867396, 2.0508468607867396, 2.15093345547142, 2.15093345547142, 2.163265233559293, 2.163265233559293, 2.172012981654681, 2.172012981654681, 2.2169457817472304, 2.2169457817472304, 2.2211995008885856, 2.2211995008885856, 2.2223908857939123, 2.2223908857939123, 2.230532655464806, 2.230532655464806, 2.2348729090634767, 2.2348729090634767, 2.2771369651229048, 2.2771369651229048, 2.3310798589610418, 2.3310798589610418, 2.441168132876584, 2.441168132876584, 2.4447683326808676, 2.4447683326808676, 2.4552630366528096, 2.4552630366528096, 2.4570348909414808, 2.4570348909414808, 2.483144716050325, 2.483144716050325, 2.494086111116831, 2.494086111116831, 2.531029192424382, 2.531029192424382, 2.634524157212908, 2.634524157212908, 2.72795462747693, 2.72795462747693, 2.731144528360924, 2.731144528360924, 2.7332069773613967, 2.7332069773613967, 2.7402243448605814, 2.7402243448605814, 2.952687038062634, 2.952687038062634, 2.9682614456920975, 2.9682614456920975]
phi = [0.629484480622239, 3.771077134212032, 2.2190385877502545, 5.360631241340048, 1.67321960169238, 4.814812255282173, 2.7640080336286235, 5.905600687218417, 4.169126605927731, 1.027533952337939, 3.395629470980574, 0.2540368173907809, 2.228475682947696, 5.370068336537489, 0.6734935557176218, 3.815086209307415, 4.652457541416726, 1.5108648878269326, 2.9970293420902645, 6.138621995680057, 1.107405639545406, 4.248998293135199, 0.25688101379976197, 3.398473667389555, 2.5945831275067177, 5.73617578109651, 5.057883715495376, 1.9162910619055835, 2.2618046275398727, 5.403397281129665, 3.76794288347954, 0.6263502298897471, 4.110398930973812, 0.9688062773840187, 4.507192173148715, 1.365599519558922, 2.863493794981481, 6.005086448571274, 4.8233790460434305, 1.6817863924536378, 3.186859373750794, 0.045266720161000966, 3.4914267332189124, 0.3498340796291194, 2.551176794254124, 5.692769447843917, 5.122941220439443, 1.9813485668496498, 2.251796164949759, 5.393388818539552, 1.1592647161521987, 4.300857369741992, 3.901137226373928, 0.7595445727841351, 1.4609232987739145, 4.602515952363707, 3.098226661183218, 6.239819314773011, 2.8142333120371807, 5.955825965626974, 4.870587045704628, 1.7289943921148352, 0.5005092963180321, 3.642101949907825, 3.3628594933635925, 0.22126683977379946, 2.5210287391271433, 5.662621392716936, 5.126596528782029, 1.9850038751922365, 0.9401208053153428, 4.081713458905136, 2.241376507736751, 5.382969161326544, 1.2142085872213237, 4.355801240811116, 3.8116234724444187, 0.6700308188546259, 3.026666338972188, 6.168258992561981, 5.900756261828765, 2.759163608238972, 1.4680797756247455, 4.609672429214538, 1.7220351643715528, 4.863627817961346, 3.282188092598901, 0.1405954390091083, 3.5433575587306985, 0.40176490514090546, 2.4882128013417093, 5.629805454931502, 1.9702784232918367, 5.11187107688163, 4.065746756738421, 0.9241541031486279, 2.217438550441165, 5.359031204030958, 1.1713142302979564, 4.31290688388775, 0.6533798522480839, 3.794972505837877, 5.881420402038681, 2.7398277484488878, 6.142589868170478, 3.000997214580685, 4.561150142808033, 1.4195574892182403, 1.6735128779650479, 4.815105531554841, 3.524021698940614, 0.3824290453508211, 0.11492631461760505, 3.256518968207398, 5.613154488324961, 2.4715618347351676, 5.068976719958263, 1.9273840663684696, 4.041808799442835, 0.900216145853042, 2.2014718482744504, 5.3430645018642435, 4.2981814319873495, 1.1565887783975568, 0.6205639144626499, 3.762156568052443, 2.9203258138159938, 6.061918467405786, 2.641083357271761, 5.782676010861554, 1.4125982614749582, 4.554190915064751, 3.4689519951424055, 0.32735934155261254, 3.1849586459963684, 0.04336599240657529, 4.822262008405672, 1.6806693548158789, 5.5236407343954514, 2.3820480808056583, 1.9823279374375944, 5.123920591027387, 4.031389142229827, 0.8897964886400342, 1.1602440867401436, 4.301836740329937, 3.732008512925462, 0.5904158593356691, 2.791758573960674, 5.9333512275504665, 6.237918587018585, 3.0963259334287923, 1.4598062611361555, 4.601398914725948, 0.278098858608312, 3.419691512198105, 1.7759931340308714, 4.917585787620665, 5.3143790297955675, 2.1727863762057744, 5.6568350772898395, 2.5152424237000464, 4.021380679639714, 0.8797880260499206, 4.366894245274002, 1.2253015916842098, 3.6886021796728685, 0.5470095260830755, 6.026304293379824, 2.8847116397900314, 5.17577966763418, 2.034187014044387, 3.2861559650893217, 0.14456331149952864, 4.772320419352654, 1.6307277657628607, 2.4680990978721713, 5.609691751461964, 4.05470962423189, 0.9131169706420971, 2.887555836199012, 6.029148489788805, 5.255651354841647, 2.1140587012518544, 3.5191772735509628, 0.3775846199611696, 4.609965705487206, 1.4683730518974132, 4.064146719429331, 0.9225540658395386, 5.653700826557348, 2.512108172967554]

symmetries =  [1.00000000e+00 0.00000000e+00 0.00000000e+00 4.30836344e-17
 7.48217823e-18 0.00000000e+00 3.41592489e-11 1.67379162e-15
 3.20042058e-15 1.23592326e-14 2.63949153e-13 5.07448579e-12
 1.79378073e-10 3.57732466e-11 1.99912925e-10 5.72015847e-09
 2.28261671e-06 3.96468809e-05 1.16882022e-05 1.16356828e-05
 1.96173339e-04 1.52072098e-03]
symmetry error = 5.362387458453542e-05
minimal relative intensity = 0.999959 rms = 0.00000690

N = 204, l_max = 19

theta = [0.12633947919054742, 0.12633947919054742, 0.2455084492819846, 0.2455084492819846, 0.34037275483429885, 0.34037275483429885, 0.3632672587861929, 0.3632672587861929, 0.38455378319007244, 0.38455378319007244, 0.48774140106084285, 0.48774140106084285, 0.5357163513278367, 0.5357163513278367, 0.5662436993150294, 0.5662436993150294, 0.5815572700425751, 0.5815572700425751, 0.5865967540463236, 0.5865967540463236, 0.6241172457803656, 0.6241172457803656, 0.6923712592380534, 0.6923712592380534, 0.7318255349419218, 0.7318255349419218, 0.7748863185149736, 0.7748863185149736, 0.7959180062589555, 0.7959180062589555, 0.7993836871212395, 0.7993836871212395, 0.8119040938108478, 0.8119040938108478, 0.8228503143098667, 0.8228503143098667, 0.8532609146212075, 0.8532609146212075, 0.8927923643982271, 0.8927923643982271, 0.9671924167957322, 0.9671924167957322, 0.9844097288552285, 0.9844097288552285, 0.9956511903488555, 0.9956511903488555, 1.000038611825488, 1.000038611825488, 1.010789741992626, 1.010789741992626, 1.0327465868053256, 1.0327465868053256, 1.0456215090888898, 1.0456215090888898, 1.0883683552453562, 1.0883683552453562, 1.089983867361501, 1.089983867361501, 1.1620576492161776, 1.1620576492161776, 1.1800720345153348, 1.1800720345153348, 1.183363351335183, 1.183363351335183, 1.199946527812733, 1.199946527812733, 1.2301504182083383, 1.2301504182083383, 1.2447400326838867, 1.2447400326838867, 1.249421548284146, 1.249421548284146, 1.2589298282324042, 1.2589298282324042, 1.27960291436567, 1.27960291436567, 1.3370820764199043, 1.3370820764199043, 1.3382719148447384, 1.3382719148447384, 1.3817908077354628, 1.3817908077354628, 1.3995962045656751, 1.3995962045656751, 1.4073275078092158, 1.4073275078092158, 1.4144987389584447, 1.4144987389584447, 1.4480885330185578, 1.4480885330185578, 1.471297251089027, 1.471297251089027, 1.4737165250159938, 1.4737165250159938, 1.495160557342687, 1.495160557342687, 1.4969861496183732, 1.4969861496183732, 1.5408729955603786, 1.5408729955603786, 1.5679685659262228, 1.5679685659262228, 1.5736240876635703, 1.5736240876635703, 1.6007196580294147, 1.6007196580294147, 1.6446065039714202, 1.6446065039714202, 1.6464320962471064, 1.6464320962471064, 1.6678761285737995, 1.6678761285737995, 1.6702954025007661, 1.6702954025007661, 1.6935041205712356, 1.6935041205712356, 1.7270939146313486, 1.7270939146313486, 1.7342651457805776, 1.7342651457805776, 1.7419964490241182, 1.7419964490241182, 1.7598018458543305, 1.7598018458543305, 1.8033207387450547, 1.8033207387450547, 1.804510577169889, 1.804510577169889, 1.8619897392241231, 1.8619897392241231, 1.8826628253573892, 1.8826628253573892, 1.8921711053056471, 1.8921711053056471, 1.8968526209059067, 1.8968526209059067, 1.911442235381455, 1.911442235381455, 1.9416461257770603, 1.9416461257770603, 1.9582293022546104, 1.9582293022546104, 1.9615206190744585, 1.9615206190744585, 1.9795350043736157, 1.9795350043736157, 2.051608786228292, 2.051608786228292, 2.053224298344437, 2.053224298344437, 2.0959711445009033, 2.0959711445009033, 2.1088460667844675, 2.1088460667844675, 2.1308029115971676, 2.1308029115971676, 2.141554041764305, 2.141554041764305, 2.1459414632409377, 2.1459414632409377, 2.1571829247345646, 2.1571829247345646, 2.174400236794061, 2.174400236794061, 2.248800289191566, 2.248800289191566, 2.2883317389685858, 2.2883317389685858, 2.3187423392799267, 2.3187423392799267, 2.3296885597789454, 2.3296885597789454, 2.3422089664685535, 2.3422089664685535, 2.3456746473308376, 2.3456746473308376, 2.3667063350748196, 2.3667063350748196, 2.4097671186478715, 2.4097671186478715, 2.44922139435174, 2.44922139435174, 2.5174754078094277, 2.5174754078094277, 2.55499589954347, 2.55499589954347, 2.560035383547218, 2.560035383547218, 2.575348954274764, 2.575348954274764, 2.6058763022619567, 2.6058763022619567, 2.65385125252895, 2.65385125252895, 2.757038870399721, 2.757038870399721, 2.7783253948036, 2.7783253948036, 2.8012198987554946, 2.8012198987554946, 2.8960842043078086, 2.8960842043078086, 3.0152531743992457, 3.0152531743992457]
phi = [4.472654777426582, 1.331062123836789, 2.833325445808794, 5.974918099398587, 0.5355589154750018, 3.6771515690647947, 2.0465420831272096, 5.188134736717003, 1.3094589511470875, 4.451051604736881, 2.9796425014734824, 6.1212351550632755, 5.604852786678192, 2.463260133088399, 3.92297682124582, 0.7813841676560269, 1.7526273382966133, 4.894219991886406, 0.28506201418977556, 3.4266546677795686, 1.2434765262778766, 4.38506917986767, 5.263181375156419, 2.121588721566626, 5.931112697780932, 2.789520044191139, 3.1456344259249103, 0.004041772335117278, 4.7084313315898765, 1.5668386780000836, 0.7963443612489938, 3.937937014838787, 3.578205329692207, 0.4366126761024139, 5.59796194224449, 2.456369288654697, 4.292945256163722, 1.1513526025739291, 5.012759722101962, 1.871167068512169, 2.169419325738844, 5.311011979328637, 3.3691141696640066, 0.22752151607421367, 2.732089577246974, 5.873682230836767, 3.023270478672824, 6.164863132262617, 1.386020863202451, 4.527613516792244, 0.6234722820302488, 3.7650649356200416, 0.9369105517484306, 4.078503205338224, 5.587723330084959, 2.446130676495166, 4.797719221702059, 1.6561265681122659, 1.92736899456307, 5.068961648152863, 4.291460241033725, 1.1498675874439324, 3.5655233920206464, 0.42393073843085344, 0.10418552187482494, 3.245778175464618, 5.327584664499425, 2.1859920109096325, 5.850227536069455, 2.708634882479662, 6.110809530898661, 2.9692168773088676, 3.904263454633937, 0.7626708010441442, 1.3919928168378681, 4.533585470427662, 1.6466731815679878, 4.788265835157781, 2.441109730748976, 5.582702384338769, 0.9723155765278307, 4.113908230117624, 3.4372355489763837, 0.2956428953865905, 1.8966726615467067, 5.0382653151365, 0.5677622156948525, 3.7093548692846454, 0.03015010260753213, 3.1717427561973253, 2.1447485957703964, 5.28634124936019, 1.1981063312191147, 4.339698984808908, 2.659283942308, 5.800876595897793, 6.0488210498797335, 2.9072283962899403, 1.448033298189593, 4.589625951779386, 0.7748822354258226, 3.916474889015616, 2.3667104181639704, 5.5083030717537635, 1.6935593554002004, 4.8351520089899935, 3.375956910889646, 0.2343642572998531, 3.6239013648715863, 0.4823087112817935, 1.9434863223706784, 5.085078975960472, 4.13843671140919, 0.996844057819397, 3.111442550982261, 6.253035204572054, 2.573830437894941, 5.715423091484734, 4.386512645632879, 1.2449199920430865, 5.987542411792996, 2.8459497582032025, 5.310869730651755, 2.1692770770619623, 0.7004829228408171, 3.8420755764306103, 1.4949194720218053, 4.636512125611598, 4.891192490341718, 1.749599836751925, 2.3789218525456493, 5.520514506135442, 3.3139684298707186, 0.17237577628092549, 3.574550424699924, 0.43295777111013106, 4.097193296269953, 0.9556006426801609, 6.178999785304761, 3.0374071317149682, 2.71766191515894, 5.8592545687487325, 1.991725066145861, 5.133317719735654, 1.214223659026723, 4.355816312616517, 4.62705873906732, 1.4854660854775272, 0.6954619770946275, 3.8370546306844204, 5.346274755431155, 2.2046821018413625, 5.659713025149338, 2.5181203715595446, 1.7555717903873422, 4.897164443977135, 0.11832217491696906, 3.259914828506762, 0.409503076342819, 3.5510957299326122, 6.055663791105372, 2.9140711375155797, 4.113765981440743, 0.9721733278509493, 1.270425585077624, 4.412018238667417, 1.9902400510158642, 5.131832704605657, 0.6852233649350961, 3.826816018524889, 2.7049799774873793, 5.846572631077172, 5.4868409459305925, 2.3452482923407993, 1.5747539755897098, 4.716346629179503, 3.137550881254676, 6.279143534844469, 0.3520726093986541, 3.493665262988447, 1.0200039320231669, 4.16159658561296, 5.03970878090171, 1.8981161273119165, 5.998123292989811, 2.8565306394000176, 4.530557968882973, 1.3889653152931798, 2.360208485933766, 5.501801139523559, 0.6783325205013945, 3.8199251740911873, 3.303542805706104, 0.16195015211631075, 4.973726356032499, 1.8321337024427058, 4.236643224052377, 1.0950505704625833, 5.747626391704585, 2.6060337381147916, 3.449859861370792, 0.3082672077809991, 1.810530529753004, 4.952123183342797]

symmetries =  [1.00000000e+00 0.00000000e+00 0.00000000e+00 6.36295624e-18
 7.04722094e-18 0.00000000e+00 2.62667417e-13 1.61304800e-17
 3.49732219e-17 1.29725539e-15 7.37004751e-15 2.80331627e-10
 2.32956954e-10 2.39722281e-11 1.18187530e-10 1.96131340e-09
 3.71007858e-07 6.99759598e-07 8.49790608e-07 8.52382784e-07
 5.90642299e-05 7.24709303e-04 1.28446790e-04]
symmetry error = 2.7755578809176694e-06
minimal relative intensity = 0.9999742764 rms = 4.5533348123e-06

N = 216, l_max = 20

theta = [0.11953761381308146, 0.11953761381308146, 0.24457976837526674, 0.24457976837526674, 0.3201973907465535, 0.3201973907465535, 0.36001118897928336, 0.36001118897928336, 0.37540746263252284, 0.37540746263252284, 0.47408332471653525, 0.47408332471653525, 0.5159914464135475, 0.5159914464135475, 0.522945404956535, 0.522945404956535, 0.5817421698346954, 0.5817421698346954, 0.6138213162151603, 0.6138213162151603, 0.6147083556490875, 0.6147083556490875, 0.6181213350558544, 0.6181213350558544, 0.7238281555321838, 0.7238281555321838, 0.749426761047236, 0.749426761047236, 0.7524836709791594, 0.7524836709791594, 0.7657978052416324, 0.7657978052416324, 0.8081430664155357, 0.8081430664155357, 0.8292035035759499, 0.8292035035759499, 0.8375634978604874, 0.8375634978604874, 0.8555829311546385, 0.8555829311546385, 0.9163119832087416, 0.9163119832087416, 0.9267405574525506, 0.9267405574525506, 0.9505567091220066, 0.9505567091220066, 0.9569270050208433, 0.9569270050208433, 1.0048957257950584, 1.0048957257950584, 1.0172098495013833, 1.0172098495013833, 1.0644012088938197, 1.0644012088938197, 1.0669845203573667, 1.0669845203573667, 1.0810955593735474, 1.0810955593735474, 1.089573921184617, 1.089573921184617, 1.092377953063416, 1.092377953063416, 1.1178030731282758, 1.1178030731282758, 1.1593156742974293, 1.1593156742974293, 1.1844900886329754, 1.1844900886329754, 1.2123998512047507, 1.2123998512047507, 1.2318843170447087, 1.2318843170447087, 1.2432125754869987, 1.2432125754869987, 1.254870402443791, 1.254870402443791, 1.305363267591162, 1.305363267591162, 1.311748912629713, 1.311748912629713, 1.3245486011517007, 1.3245486011517007, 1.3269903912116425, 1.3269903912116425, 1.3360616232490756, 1.3360616232490756, 1.357102276732546, 1.357102276732546, 1.3759459937979721, 1.3759459937979721, 1.4147482553414827, 1.4147482553414827, 1.4405513828421752, 1.4405513828421752, 1.4556023288269926, 1.4556023288269926, 1.4687524584894123, 1.4687524584894123, 1.4782522238851186, 1.4782522238851186, 1.538190205491026, 1.538190205491026, 1.539006644259338, 1.539006644259338, 1.5426789195689898, 1.5426789195689898, 1.5517404358323337, 1.5517404358323337, 1.5898522177574597, 1.5898522177574597, 1.5989137340208035, 1.5989137340208035, 1.6025860093304551, 1.6025860093304551, 1.6034024480987674, 1.6034024480987674, 1.6633404297046746, 1.6633404297046746, 1.672840195100381, 1.672840195100381, 1.6859903247628005, 1.6859903247628005, 1.7010412707476181, 1.7010412707476181, 1.7268443982483106, 1.7268443982483106, 1.7656466597918212, 1.7656466597918212, 1.7844903768572473, 1.7844903768572473, 1.8055310303407177, 1.8055310303407177, 1.8146022623781506, 1.8146022623781506, 1.8170440524380926, 1.8170440524380926, 1.8298437409600803, 1.8298437409600803, 1.8362293859986314, 1.8362293859986314, 1.886722251146002, 1.886722251146002, 1.8983800781027944, 1.8983800781027944, 1.9097083365450844, 1.9097083365450844, 1.9291928023850424, 1.9291928023850424, 1.9571025649568177, 1.9571025649568177, 1.982276979292364, 1.982276979292364, 2.0237895804615174, 2.0237895804615174, 2.0492147005263774, 2.0492147005263774, 2.0520187324051764, 2.0520187324051764, 2.0604970942162457, 2.0604970942162457, 2.0746081332324264, 2.0746081332324264, 2.0771914446959734, 2.0771914446959734, 2.12438280408841, 2.12438280408841, 2.136696927794735, 2.136696927794735, 2.18466564856895, 2.18466564856895, 2.1910359444677865, 2.1910359444677865, 2.2148520961372427, 2.2148520961372427, 2.2252806703810517, 2.2252806703810517, 2.2860097224351548, 2.2860097224351548, 2.3040291557293058, 2.3040291557293058, 2.3123891500138436, 2.3123891500138436, 2.3334495871742575, 2.3334495871742575, 2.3757948483481606, 2.3757948483481606, 2.389108982610634, 2.389108982610634, 2.392165892542557, 2.392165892542557, 2.4177644980576094, 2.4177644980576094, 2.523471318533939, 2.523471318533939, 2.526884297940706, 2.526884297940706, 2.5277713373746327, 2.5277713373746327, 2.5598504837550977, 2.5598504837550977, 2.618647248633258, 2.618647248633258, 2.6256012071762456, 2.6256012071762456, 2.667509328873258, 2.667509328873258, 2.7661851909572706, 2.7661851909572706, 2.78158146461051, 2.78158146461051, 2.8213952628432395, 2.8213952628432395, 2.8970128852145267, 2.8970128852145267, 3.0220550397767116, 3.0220550397767116]
phi = [1.3010071499519775, 4.44259980354177, 0.07877169651550778, 3.2203643501053008, 5.451686207008141, 2.310093553418348, 4.805061336640245, 1.6634686830504515, 3.938885094613049, 0.7972924410232561, 5.99470914421585, 2.853116490626057, 3.349548327750985, 0.20795567416119215, 1.1726848415683544, 4.314277495158148, 2.3860508265851834, 5.5276434801749765, 3.756881188997621, 0.6152885354078279, 4.761155859970208, 1.6195632063804153, 2.004953753626024, 5.146546407215817, 3.0015948845376, 6.143187538127393, 1.078870722278551, 4.220463375868344, 5.811264618305538, 2.669671964715745, 3.3677439436661176, 0.2261512900763247, 0.7353537255726454, 3.8769463791624386, 1.3584405757648468, 4.5000332293546395, 2.243308174473941, 5.384900828063734, 1.693511742120595, 4.835104395710388, 1.9731883000873198, 5.114780953677113, 3.555688878857024, 0.4140962252672311, 2.536534237961856, 5.678126891551649, 3.1071928758078284, 6.2487855293976216, 4.100266999385811, 0.9586743457960182, 5.992543710506219, 2.8509510569164256, 3.8016018799704554, 0.6600092263806626, 1.4542126684775096, 4.595805322067303, 4.326116885392563, 1.1845242318027696, 5.427495601075375, 2.2859029474855816, 3.361467979871221, 0.21987532628142784, 4.857341697961084, 1.7157490443712913, 1.9944377256426722, 5.136030379232466, 2.6951202347030137, 5.836712888292807, 6.2483659717042155, 3.1067733181144224, 3.663726966237846, 0.5221343126480533, 4.025458255651474, 0.8838656020616803, 2.446513160634025, 5.588105814223818, 3.410269947768184, 0.2686772941783909, 1.2959676983625448, 4.437560351952338, 5.285316335990684, 2.143723682400891, 1.551159634034363, 4.692752287624156, 6.063368150847205, 2.9217754972574115, 4.952693536113321, 1.811100882523528, 1.082348727397881, 4.223941380987674, 0.7529190477791333, 3.894511701368926, 2.6844372177535263, 5.826029871343319, 3.1735944999151804, 0.032001846325387384, 3.648289397268018, 0.5066967436782251, 5.564232839095837, 2.422640185506044, 5.070984667136784, 1.9293920135469915, 1.4555438385652302, 4.597136492155023, 5.326536982814002, 2.1849443292242086, 0.24385110737808444, 3.3854437609678776, 6.039334199801502, 2.8977415462117087, 4.098240977955378, 0.9566483243655848, 4.827641468614356, 1.686048815024563, 4.353793293632595, 1.2122006400428016, 3.860545121673542, 0.7189524680837491, 5.776488563501361, 2.634895909911568, 3.109590807264406, 6.2511834608541985, 0.4571554358362671, 3.59874808942606, 2.38867360581066, 5.530266259400453, 5.200836579781705, 2.059243926191912, 1.3304917710662654, 4.4720844246560585, 0.21981715633238189, 3.3614098099221748, 1.5904330195554301, 4.732025673145223, 0.9978689711889023, 4.139461624778695, 1.8456249552272483, 4.9872176088170415, 6.014508013001195, 2.8729153594114023, 0.6950794929557682, 3.836672146545561, 2.257727051528113, 5.399319705117906, 2.61945834094174, 5.761050994531533, 0.034819335475371, 3.176411989065164, 0.4464724188867795, 3.5880650724765726, 1.147154927947121, 4.288747581536914, 1.425843609218502, 4.567436262808295, 2.9217173273083654, 6.0633099808981585, 0.8556897061042117, 3.9972823596940046, 1.9570684217870236, 5.098661075376817, 4.828972638702076, 1.6873799851122837, 2.481583427209131, 5.623176080798924, 0.29064159667336786, 3.4322342502631606, 2.1829183077937753, 5.324510961383568, 3.175992431371758, 0.03439977778196499, 3.74665106921773, 0.6050584156279369, 2.7274964283225622, 5.869089081912355, 1.1684043535024735, 4.309997007092266, 4.589673565058991, 1.4480809114691984, 4.039877132705645, 0.8982844791158524, 4.924744731414739, 1.7831520778249466, 5.547831581606941, 2.4062389280171477, 2.9154413635134686, 6.057034017103262, 0.47192068887404826, 3.613513342463841, 5.204314584901035, 2.062721931311242, 3.2815904226419863, 0.1399977690521933, 4.278231553553562, 1.1366388999637693, 1.522029447209378, 4.663622100799171, 2.526304118181965, 5.667896771771758, 3.897134480594403, 0.7555418270046099, 5.110500465611231, 1.9689078120214387, 2.9336369794286012, 6.075229633018394, 0.28847616296373596, 3.430068816553529, 5.48589286615633, 2.3443002125665373, 1.4781239705393419, 4.6197166241291345, 0.8314991001714451, 3.973091753761238, 6.204413610664078, 3.0628209570742855, 4.982178157227609, 1.8405855036378158]

symmetries =  [1.00000000e+00 0.00000000e+00 0.00000000e+00 1.74532536e-20
 7.43486149e-19 0.00000000e+00 1.35950916e-14 8.87957255e-18
 2.01592378e-17 5.61736880e-16 9.12938387e-15 1.03327231e-12
 1.15174805e-11 1.44043285e-12 1.98132437e-12 2.38495436e-10
 1.14696252e-08 2.45721141e-07 2.83481050e-07 2.63100672e-06
 1.55621516e-05 9.73102049e-03 3.82873019e-04 5.72101450e-04]
symmetry error = 1.873408463883804e-05
minimal relative intensity = 0.999974 rms = 0.00000373

N = 228, l_max = 20

N = 228 l_max = 20:
theta = [0.12562486925147748, 0.12562486925147748, 0.22410231641197573, 0.22410231641197573, 0.30709021204298004, 0.30709021204298004, 0.34633042221477855, 0.34633042221477855, 0.4205309294310804, 0.4205309294310804, 0.44023365476901877, 0.44023365476901877, 0.44904357256663874, 0.44904357256663874, 0.521868116821457, 0.521868116821457, 0.5921961916089036, 0.5921961916089036, 0.5984103852004983, 0.5984103852004983, 0.600204282975355, 0.600204282975355, 0.6138240611211366, 0.6138240611211366, 0.6922803932881509, 0.6922803932881509, 0.6929001756362099, 0.6929001756362099, 0.7307286985059968, 0.7307286985059968, 0.7502301952377657, 0.7502301952377657, 0.7915405568513033, 0.7915405568513033, 0.8208594725768449, 0.8208594725768449, 0.8279193142095361, 0.8279193142095361, 0.8435291079583914, 0.8435291079583914, 0.8460660132036933, 0.8460660132036933, 0.9285694544837461, 0.9285694544837461, 0.9364561726570964, 0.9364561726570964, 0.9391735810424884, 0.9391735810424884, 0.9472562201597544, 0.9472562201597544, 0.9818751744552506, 0.9818751744552506, 0.9980330108737115, 0.9980330108737115, 1.0305058867933872, 1.0305058867933872, 1.0526128381143838, 1.0526128381143838, 1.0648269433676167, 1.0648269433676167, 1.0861652414893357, 1.0861652414893357, 1.0956350889691655, 1.0956350889691655, 1.1390976459137834, 1.1390976459137834, 1.1534344002967525, 1.1534344002967525, 1.1637953644030885, 1.1637953644030885, 1.1823722305920863, 1.1823722305920863, 1.2120597164849694, 1.2120597164849694, 1.2251760506323048, 1.2251760506323048, 1.2263293640976531, 1.2263293640976531, 1.2544142188421914, 1.2544142188421914, 1.2795565008606808, 1.2795565008606808, 1.2857520275689889, 1.2857520275689889, 1.307840578509683, 1.307840578509683, 1.3484178127765838, 1.3484178127765838, 1.3484880893990736, 1.3484880893990736, 1.360935368998428, 1.360935368998428, 1.3806930846209573, 1.3806930846209573, 1.4148017161050448, 1.4148017161050448, 1.4367794063510457, 1.4367794063510457, 1.4535633401827648, 1.4535633401827648, 1.4552461850893197, 1.4552461850893197, 1.4596461729319634, 1.4596461729319634, 1.468288924414246, 1.468288924414246, 1.5143950887985875, 1.5143950887985875, 1.521723137177635, 1.521723137177635, 1.5495085295677342, 1.5495085295677342, 1.5536896556067588, 1.5536896556067588, 1.5879029979830344, 1.5879029979830344, 1.5920841240220591, 1.5920841240220591, 1.6198695164121584, 1.6198695164121584, 1.6271975647912056, 1.6271975647912056, 1.673303729175547, 1.673303729175547, 1.68194648065783, 1.68194648065783, 1.6863464685004736, 1.6863464685004736, 1.6880293134070286, 1.6880293134070286, 1.7048132472387474, 1.7048132472387474, 1.7267909374847485, 1.7267909374847485, 1.760899568968836, 1.760899568968836, 1.7806572845913653, 1.7806572845913653, 1.7931045641907195, 1.7931045641907195, 1.7931748408132095, 1.7931748408132095, 1.8337520750801102, 1.8337520750801102, 1.8558406260208045, 1.8558406260208045, 1.8620361527291125, 1.8620361527291125, 1.887178434747602, 1.887178434747602, 1.91526328949214, 1.91526328949214, 1.9164166029574885, 1.9164166029574885, 1.929532937104824, 1.929532937104824, 1.959220422997707, 1.959220422997707, 1.9777972891867048, 1.9777972891867048, 1.9881582532930406, 1.9881582532930406, 2.0024950076760097, 2.0024950076760097, 2.045957564620628, 2.045957564620628, 2.0554274121004577, 2.0554274121004577, 2.0767657102221766, 2.0767657102221766, 2.0889798154754096, 2.0889798154754096, 2.111086766796406, 2.111086766796406, 2.143559642716082, 2.143559642716082, 2.1597174791345424, 2.1597174791345424, 2.194336433430039, 2.194336433430039, 2.202419072547305, 2.202419072547305, 2.2051364809326968, 2.2051364809326968, 2.213023199106047, 2.213023199106047, 2.2955266403861, 2.2955266403861, 2.298063545631402, 2.298063545631402, 2.313673339380257, 2.313673339380257, 2.3207331810129483, 2.3207331810129483, 2.35005209673849, 2.35005209673849, 2.3913624583520274, 2.3913624583520274, 2.4108639550837965, 2.4108639550837965, 2.4486924779535832, 2.4486924779535832, 2.4493122603016424, 2.4493122603016424, 2.5277685924686564, 2.5277685924686564, 2.5413883706144382, 2.5413883706144382, 2.543182268389295, 2.543182268389295, 2.5493964619808898, 2.5493964619808898, 2.619724536768336, 2.619724536768336, 2.6925490810231545, 2.6925490810231545, 2.7013589988207745, 2.7013589988207745, 2.7210617241587127, 2.7210617241587127, 2.795262231375015, 2.795262231375015, 2.8345024415468134, 2.8345024415468134, 2.9174903371778176, 2.9174903371778176, 3.015967784338316, 3.015967784338316]
phi = [1.9730628492763607, 5.114655502866153, 3.695858933410258, 0.5542662798204651, 5.907466072617035, 2.7658734190272423, 4.775138324650008, 1.633545671060215, 1.035273004217482, 4.176865657807275, 2.22680897402463, 5.368401627614423, 3.5076028169705173, 0.36601016338072456, 3.0282684289876634, 6.169861082577457, 4.954112313648563, 1.8125196600587703, 2.6067341775783976, 5.748326831168191, 1.3886182379630647, 4.530210891552858, 0.7577029521939567, 3.89929560578375, 0.35280413110658293, 3.494396784696376, 2.205457303041397, 5.3470499566311895, 1.0858352673562843, 4.227427920946077, 3.1666842522091887, 0.025091598619395743, 2.8264938665865094, 5.9680865201763025, 4.689009659294763, 1.5474170057049703, 5.016404183047266, 1.8748115294574728, 0.673436111136829, 3.815028764726622, 5.62579558818235, 2.484202934592557, 1.2917648202733207, 4.433357473863114, 2.174838240033024, 5.316430893622817, 0.39575586738173263, 3.5373485209715256, 4.101158010842463, 0.9595653572526696, 3.264963705408144, 0.12337105181835133, 2.9819252570487507, 6.123517910638544, 2.684169634108929, 5.825762287698722, 4.777324694759285, 1.6357320411694916, 1.9052685833511558, 5.0468612369409485, 0.7206890804307147, 3.862281734020508, 5.553708812169681, 2.412116158579888, 1.1198217346944546, 4.261414388284248, 1.4000163205525438, 4.541608974142337, 3.614676974672687, 0.47308432108289383, 5.301698712822507, 2.1601060592327133, 3.3659575193919786, 0.2243648658021855, 3.1189666716453512, 6.260559325235144, 2.861797676413095, 6.003390330002888, 4.041999015756334, 0.9004063621665408, 2.6110953308642455, 5.752687984454038, 1.68664065025959, 4.828233303849383, 1.9280644665219433, 5.069657120111736, 0.6612476402578492, 3.802840293847642, 5.516883724904434, 2.3752910713146416, 4.345227731893792, 1.2036350783039995, 4.592995080063333, 1.4514024264735401, 0.42281886996014995, 3.5644115235499427, 5.290998147433357, 2.149405493843564, 3.333025997755184, 0.19143334416539093, 3.0921897554965803, 6.233782409086373, 2.8547284438136713, 5.996321097403464, 0.9783460112895883, 4.119938664879381, 5.764093523957076, 2.622500870367283, 1.6864863652071544, 4.828079018796948, 1.9164982067798908, 5.058090860369684, 0.7500731725918305, 3.8916658261816233, 2.391519480997963, 5.533112134587756, 4.366687100399695, 1.2250944468099025, 4.596698941972432, 1.4551062883826387, 3.660684436812303, 0.51909178322251, 2.163246642300205, 5.304839295889998, 0.28686420977612187, 3.428456863365915, 3.190995551683006, 0.04940289809321296, 2.9501593094244023, 6.091751963014195, 0.9921871597462291, 4.133779813336022, 5.860366437219437, 2.7187737836296435, 1.6901902271162532, 4.831782880706046, 5.079550228875586, 1.9379575752857938, 0.7663015822751517, 3.9078942358649447, 5.621937666921737, 2.4803450133319442, 4.3551208406576425, 1.21352818706785, 4.596544656919996, 1.454952003330203, 3.6720899763153407, 0.5304973227255477, 2.2411862914232525, 5.382778945013046, 3.421387630766491, 0.27979497717669827, 3.164218635534235, 0.02262598194444204, 2.9172277877876076, 6.058820441377401, 0.9814865943570799, 4.1230792479468725, 2.6685083325068994, 5.810100986096693, 4.883168986627043, 1.7415763330372493, 5.163363572485132, 2.0217709188953386, 0.7294764950099053, 3.871069148599698, 5.562496226748872, 2.4209035731590784, 4.377916723828431, 1.2363240702386373, 1.5058606124203016, 4.6474532660100945, 3.599015673070657, 0.4574230194808641, 0.1596673965410426, 3.3012600501308356, 6.159814255361235, 3.018221601771442, 2.1820272963371234, 5.323619949926917, 5.887429439797853, 2.7458367862080606, 4.108347067146562, 0.966754413556769, 1.8498278333164724, 4.991420486906265, 0.6573897189972361, 3.7989823725870293, 5.609749196042757, 2.4681565424529643, 1.2667811241323204, 4.4083737777221135, 1.5941756478848228, 4.735768301474616, 3.456691440593077, 0.31509878700328375, 3.1165010549703975, 6.258093708560191, 5.197350039823302, 2.055757386233509, 4.077728004138189, 0.9361353505483965, 2.7887885224832103, 5.930381176073003, 5.52548235498563, 2.3838897013958364, 4.894567069216522, 1.7529744156267284, 3.6764511296011886, 0.5348584760113956, 4.470665647120816, 1.3290729935310228, 3.254916878191923, 0.11332422460212974, 2.775582490209069, 5.917175143798862, 4.056376333154956, 0.9147836795651635, 5.247912302962105, 2.1063196493723115, 1.5080469825295781, 4.649639636119371, 0.3757192345625512, 3.517311888152344, 2.5873263737693284, 5.7289190273591215, 4.310122457903225, 1.1685298043134327]

symmetries =  [1.00000000e+00 0.00000000e+00 0.00000000e+00 3.61190098e-16
 1.65331415e-16 0.00000000e+00 2.11212843e-11 6.77105928e-14
 7.96816830e-14 1.79246379e-13 4.62984617e-12 1.82060907e-10
 3.44486576e-10 1.30711098e-10 6.49999281e-10 4.81630012e-09
 3.32574229e-07 5.57032129e-07 5.77490249e-07 1.21395306e-06
 2.94755696e-06 1.96542267e-04 9.55850631e-06 9.58256282e-05]
symmetry error = 5.6347562549844545e-06
minimal relative intensity = 0.999978 rms = 0.00000331

N = 240, l_max = 21

theta = [0.1159268231911234, 0.1159268231911234, 0.24293764136119148, 0.24293764136119148, 0.2700369312525688, 0.2700369312525688, 0.3431328562315875, 0.3431328562315875, 0.40426837922782455, 0.40426837922782455, 0.43706075059724214, 0.43706075059724214, 0.471085526959959, 0.471085526959959, 0.47137081458576924, 0.47137081458576924, 0.5490485585443254, 0.5490485585443254, 0.5902359353108487, 0.5902359353108487, 0.6116708836409961, 0.6116708836409961, 0.625489795846225, 0.625489795846225, 0.6350850731507843, 0.6350850731507843, 0.6748201782663785, 0.6748201782663785, 0.7300326084184376, 0.7300326084184376, 0.7324927367993346, 0.7324927367993346, 0.7591346667234564, 0.7591346667234564, 0.7869492038272373, 0.7869492038272373, 0.7929938223129239, 0.7929938223129239, 0.8254206484229261, 0.8254206484229261, 0.8494626590698545, 0.8494626590698545, 0.899165069667053, 0.899165069667053, 0.909644376679177, 0.909644376679177, 0.9148553493534006, 0.9148553493534006, 0.9382112829325204, 0.9382112829325204, 0.9589634540980594, 0.9589634540980594, 0.9598059939525113, 0.9598059939525113, 0.9838561102626749, 0.9838561102626749, 0.9876737260853569, 0.9876737260853569, 1.0667438996714789, 1.0667438996714789, 1.0719985093250615, 1.0719985093250615, 1.089461417367628, 1.089461417367628, 1.0997108363422818, 1.0997108363422818, 1.111274856603816, 1.111274856603816, 1.123502477348684, 1.123502477348684, 1.1552999614134059, 1.1552999614134059, 1.1636842088644872, 1.1636842088644872, 1.1658229823353372, 1.1658229823353372, 1.2129569337260622, 1.2129569337260622, 1.2212392583383769, 1.2212392583383769, 1.2413852017692848, 1.2413852017692848, 1.2461168452465514, 1.2461168452465514, 1.2491554524216775, 1.2491554524216775, 1.3285693341118048, 1.3285693341118048, 1.330785358164266, 1.330785358164266, 1.3345382160198374, 1.3345382160198374, 1.3465930531835146, 1.3465930531835146, 1.360647118056854, 1.360647118056854, 1.3704440955399928, 1.3704440955399928, 1.3783303797370396, 1.3783303797370396, 1.3837597428465933, 1.3837597428465933, 1.4430284912898992, 1.4430284912898992, 1.4552454158774557, 1.4552454158774557, 1.4570990964219974, 1.4570990964219974, 1.475013096958894, 1.475013096958894, 1.478208332982087, 1.478208332982087, 1.5339216836719112, 1.5339216836719112, 1.5455008461579907, 1.5455008461579907, 1.5482670940716208, 1.5482670940716208, 1.5706244882723062, 1.5706244882723062, 1.5709681653174872, 1.5709681653174872, 1.5933255595181726, 1.5933255595181726, 1.5960918074318027, 1.5960918074318027, 1.6076709699178822, 1.6076709699178822, 1.6633843206077064, 1.6633843206077064, 1.6665795566308994, 1.6665795566308994, 1.684493557167796, 1.684493557167796, 1.6863472377123376, 1.6863472377123376, 1.6985641622998942, 1.6985641622998942, 1.7578329107431998, 1.7578329107431998, 1.7632622738527537, 1.7632622738527537, 1.7711485580498005, 1.7711485580498005, 1.7809455355329393, 1.7809455355329393, 1.7949996004062787, 1.7949996004062787, 1.807054437569956, 1.807054437569956, 1.8108072954255272, 1.8108072954255272, 1.8130233194779883, 1.8130233194779883, 1.8924372011681159, 1.8924372011681159, 1.895475808343242, 1.895475808343242, 1.9002074518205085, 1.9002074518205085, 1.9203533952514165, 1.9203533952514165, 1.9286357198637312, 1.9286357198637312, 1.9757696712544561, 1.9757696712544561, 1.977908444725306, 1.977908444725306, 1.9862926921763875, 1.9862926921763875, 2.018090176241109, 2.018090176241109, 2.0303177969859774, 2.0303177969859774, 2.0418818172475115, 2.0418818172475115, 2.052131236222165, 2.052131236222165, 2.0695941442647316, 2.0695941442647316, 2.0748487539183142, 2.0748487539183142, 2.1539189275044364, 2.1539189275044364, 2.1577365433271183, 2.1577365433271183, 2.181786659637282, 2.181786659637282, 2.1826291994917337, 2.1826291994917337, 2.2033813706572727, 2.2033813706572727, 2.2267373042363925, 2.2267373042363925, 2.2319482769106163, 2.2319482769106163, 2.24242758392274, 2.24242758392274, 2.2921299945199385, 2.2921299945199385, 2.3161720051668673, 2.3161720051668673, 2.3485988312768695, 2.3485988312768695, 2.354643449762556, 2.354643449762556, 2.3824579868663367, 2.3824579868663367, 2.4090999167904585, 2.4090999167904585, 2.4115600451713557, 2.4115600451713557, 2.466772475323415, 2.466772475323415, 2.506507580439009, 2.506507580439009, 2.5161028577435682, 2.5161028577435682, 2.529921769948797, 2.529921769948797, 2.5513567182789445, 2.5513567182789445, 2.5925440950454677, 2.5925440950454677, 2.670221839004024, 2.670221839004024, 2.6705071266298344, 2.6705071266298344, 2.704531902992551, 2.704531902992551, 2.737324274361969, 2.737324274361969, 2.798459797358206, 2.798459797358206, 2.8715557223372246, 2.8715557223372246, 2.898655012228602, 2.898655012228602, 3.02566583039867, 3.02566583039867]
phi = [4.516376611895616, 1.3747839583058228, 0.15386145292166128, 3.295454106511454, 5.483653641801431, 2.342060988211638, 4.434000067470425, 1.2924074138806319, 3.7789384281764185, 0.6373457745866253, 5.018164063715621, 1.8765714101258277, 0.00037862084837239873, 3.1419712744381654, 5.77155851470513, 2.6299658611153367, 1.1595816052466061, 4.301174258836399, 2.251471506414814, 5.393064160004607, 4.756449192248552, 1.6148565386587594, 0.3468196068173647, 3.4884122604071575, 0.7263335624086035, 3.8679262159983967, 6.097579384808465, 2.955986731218672, 5.08030873708546, 1.9387160834956667, 2.6039167086495123, 5.745509362239305, 4.230479453370112, 1.0888867997803187, 3.2770481210504268, 0.13545546746063375, 1.436154699695427, 4.57774735328522, 5.429569077647325, 2.2879764240575318, 3.805911325273036, 0.6643186716832433, 5.9717591239100525, 2.8301664703202594, 1.7173946826236646, 4.858987336213458, 0.414293198038173, 3.555885851627966, 4.100959990981628, 0.9593673373918351, 5.1440462344087745, 2.0024535808189823, 3.1107082520349882, 6.252300905624781, 2.5227943209461134, 5.664386974535907, 1.3300650285468403, 4.471657682136633, 5.397133675173597, 2.2555410215838045, 3.3814347384670005, 0.2398420848772077, 0.7302016831557403, 3.871794336745533, 1.5706034829834914, 4.712196136573285, 4.257240366273048, 1.1156477126832551, 2.7426236839546183, 5.884216337544411, 3.001868317911936, 6.143460971501729, 1.8153449876315555, 4.956937641221348, 3.6451580979077334, 0.5035654443179405, 2.0507952100144937, 5.192387863604287, 2.4838400333556643, 5.625432686945457, 3.2394580996739495, 0.0978654460841563, 4.013290918113562, 0.8716982645237692, 4.463096308237324, 1.321503654647531, 2.2665658774194846, 5.408158531009278, 4.6744256310175825, 1.5328329774277891, 3.4727697125400527, 0.3311770589502598, 2.7233843946375815, 5.864977048227375, 1.0597029867072907, 4.201295640297084, 3.7382404292716758, 0.5966477756818828, 2.950994752935744, 6.092587406525537, 4.908315184782408, 1.7667225311926145, 1.989921323186222, 5.131513976776015, 2.4752073596417614, 5.616800013231554, 3.164268318189999, 0.022675664600206105, 0.7823480829703473, 3.9239407365601404, 1.2399142452251382, 4.381506898814931, 3.381770103294155, 0.24017744970436214, 5.323603468097476, 2.1820108145076826, 4.598662764410333, 1.4570701108205406, 3.6126781515623105, 0.47108549797251764, 5.812099809207068, 2.6705071556172757, 4.826115196359046, 1.6845225427692525, 0.9595818390821107, 4.101174492671904, 6.043007857475224, 2.9014152038854313, 5.043271061954448, 1.9016784083646552, 2.359244570619446, 5.5008372242092385, 3.118916988989587, 6.26050964257938, 0.6663852939480318, 3.807977947537825, 4.293263983993365, 1.1516713304035713, 1.3748701223971787, 4.516462775986971, 3.332190554243842, 0.19059790065404936, 2.5449448779079105, 5.686537531497703, 5.2234823204722955, 2.0818896668825024, 3.5598009125420047, 0.4182082589522117, 5.952008248229326, 2.8104155946395335, 4.750352329751797, 1.608759676162004, 4.016619429760102, 0.8750267761703089, 1.8200889989422624, 4.9616816525320555, 2.269894389066024, 5.411487042655817, 3.0437272075056367, 6.18531986109543, 3.799345273823922, 0.657752620234129, 4.2323900971650925, 1.0907974435752994, 2.638027209271853, 5.7796198628616455, 4.4678403195480305, 1.3262476659582378, 3.2813169892676504, 0.13972433567785733, 3.540561623224968, 0.3989689696351749, 2.0259449409065384, 5.167537594496331, 4.712581824196095, 1.570989170606302, 5.552983624023846, 2.411390970434053, 2.9017505687125857, 6.043343222302378, 0.8860516320059887, 4.027644285595782, 4.9531202786327455, 1.811527625042953, 3.760390986233473, 0.6187983326436796, 3.172477055144598, 0.030884401554805197, 1.1391390727708113, 4.280731726360604, 2.1822253161979583, 5.3238179697877515, 5.868892109141413, 2.7272994555516203, 1.4241979709661288, 4.565790624555921, 0.31142618326953403, 3.453018836859327, 2.47727398190655, 5.618866635496343, 0.8536162295322614, 3.9952088831220545, 4.847030607484159, 1.705437953894366, 3.0061371861291595, 6.147729839718952, 5.194298507399267, 2.0527058538094747, 3.679268598530074, 0.5376759449402807, 1.2028765700941266, 4.3444692236839195, 3.327198575960914, 0.18560592237112114, 5.556851744770983, 2.4152590911811895, 5.936365700362222, 2.7947730467724288, 1.526736114931034, 4.668328768520826, 4.031713800764772, 0.8901211471749793, 5.12360370193298, 1.9820110483431872, 0.5116267924744563, 3.6532194460642495, 6.282806686331214, 3.141214032741421, 1.2650212434639656, 4.4066138970537585, 2.504246879003168, 5.6458395325929605, 1.8491852397091615, 4.990777893298954, 0.7995316653781555, 3.9411243189679483, 6.129323854257925, 2.987731200668132, 1.7668086952839706, 4.908401348873763]

symmetries =  [1.00000000e+00 0.00000000e+00 0.00000000e+00 6.81057849e-18
 2.70401154e-17 0.00000000e+00 2.70416714e-11 5.93953530e-14
 5.45794664e-14 1.51207850e-13 1.46199291e-12 1.88425463e-10
 2.48341801e-10 3.42980195e-11 9.51446272e-11 1.82819016e-09
 9.90362821e-08 4.89390627e-07 9.96403594e-08 5.57891033e-07
 8.78189368e-06 1.61416085e-03 1.30031978e-04 2.14570816e-04
 3.77788004e-03]
symmetry error = 0.0016241911277306083
minimal relative intensity = 0.999983 rms = 0.00000244

Table of Minimal Weighted Spherical t-Designs

Note that Lebedev weighted designs use negative or highly uneven weights compared to our approach. Also, except for the l_max = 27 design, they require more points. It seems this optimization approach reaches its limits at l_max = 27.

N = 2, l_max = 1

best uniform solution: 2.1
no better weighted solution

N = 4, l_max = 2

best uniform solution: 4.2
no better weighted solution

N = 6, l_max = 3

best uniform solution: 6.3
no better weighted solution

N = 10, l_max = 4

best uniform solution: 12.5

10 points (theta,phi) 10 different weights:
theta: [1.4516996489338678, 1.6562633379105325, 1.6129609604452073, 0.8897714308536847, 2.452422956704758, 0.36835868558281637, 2.5336417905986237, 0.9464589242062572, 2.2319818152820456, 1.4713284103944169]
phi: [3.921433007771677, 1.1950263894901938, 2.878850890489476, 1.9966581706432622, 2.0523464207625484, 4.138712521668158, 4.29207449038794, 0.1687053786068844, 6.282943053143298, 5.181919450794649]
weights: [0.8978324791976032, 0.913796613393476, 0.9475650144667062, 0.9510414672921037, 0.9577588400322337, 0.9993755249586004, 1.0352786372275602, 1.0867099249106358, 1.1013477275085026, 1.1092937710125774]

symmetries =  [1.         0.         0.         0.         0.         0.29831245
 0.14476503 0.03427155]
symmetry error = 0.0
minimal relative intensity = 0.417921 rms = 0.14275317

N = 12, l_max = 5

best uniform solution: 12.5
no better weighted solution

N = 18, l_max = 6

best uniform solution: 24.7

18 points (theta,phi) 18 different weights:
theta: [1.3794825249625597, 1.5442876424317489, 0.903718118606218, 0.798924645429727, 2.4329690291933557, 2.863852374133951, 1.613434080498815, 1.6547029190286324, 2.323744026619822, 0.9252547711830056, 1.507849193449719, 1.0400973033048473, 0.17371176290042326, 0.7867145723183455, 2.0009353553459466, 1.689906615770816, 2.406587003412421, 2.0413480583771304]
phi: [0.7053989299824818, 3.727749881354547, 1.3605165458742978, 3.9085935277068713, 5.160497538801281, 0.8203146262171511, 2.95286824295287, 5.556708316145729, 2.3999972471440367, 5.043600842243375, 2.0069487784257585, 6.213641982317595, 0.17951085450289325, 2.663325990140533, 1.2546403769327992, 4.559280126288992, 3.7261051975949595, 0.12329341373268046]
weights: [0.8437542170903141, 0.8481698493367164, 0.960777771257228, 0.9721599004335852, 0.9928861097001147, 0.993832216798389, 0.9955750660289733, 0.9959660937509808, 0.9984926029138201, 1.0044177028152932, 1.013311974960087, 1.0199432721531014, 1.0275314303822074, 1.0354089608560364, 1.0520962113996648, 1.0718586787258402, 1.0850959179032045, 1.088722023494444]

symmetries =  [1.         0.         0.         0.         0.         0.
 0.         0.2183526  0.10240868 0.02657944]
symmetry error = 0.0
minimal relative intensity = 0.726816 rms = 0.07977770

N = 22, l_max = 7

best uniform solution: 24.7

22 points (theta,phi) 11 different weights:
theta: [0.7049259702640659, 2.4366666833146704, 1.8639989713740825, 1.2775936822121738, 2.9294471666889677, 0.21214548689940563, 1.9047883620774533, 1.236804291507289, 0.5755239253907631, 2.566068728193087, 1.5854801186860177, 1.5561125349091527, 0.98826095596607, 2.1533316976233796, 1.7633943826310121, 1.3781982709591631, 2.396225468209055, 0.7453671853772167, 1.294300951877279, 1.8472917017129107, 2.222127386077185, 0.9194652675061288]
phi: [0.3518468842047877, 3.493439537798776, 3.9901654251934264, 0.848572771602036, 4.927927795810437, 1.7863351422523912, 0.4781188222093335, 3.619711475794406, 4.064810460466948, 0.923217806872719, 2.2235407034585357, 5.365133357048264, 1.6529769773526286, 4.794569630948245, 3.0184538510580032, 6.160046504648294, 2.2681338808099034, 5.409726534393158, 4.571805694725872, 1.4302130411287886, 5.894696922862803, 2.7531042692773657]
weights: [0.8939042691561827, 0.8939042691561827, 0.9221634486244367, 0.9221634486244367, 0.9298803847556395, 0.9298803847556395, 0.9535499480435141, 0.9535499480435141, 0.9557541349933787, 0.9557541349933787, 1.0163846032882542, 1.0163846032882542, 1.0453441768326468, 1.0453441768326468, 1.0573220743394074, 1.0573220743394074, 1.0618048673867042, 1.0618048673867042, 1.0808310528576806, 1.0808310528576806, 1.0830610397221547, 1.0830610397221547]

symmetries =  [1.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
 0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
 2.62710780e-01 4.21578291e-23 7.07112770e-02]
symmetry error = 0.0
minimal relative intensity = 0.724756 rms = 0.06836737

N = 28, l_max = 8

best uniform solution: 36.8

28 points (theta,phi) 3 different weights:
theta: [2.049876390339078, 1.3007112093002084, 1.4547202926025566, 2.1925187275294853, 0.7362797334446627, 2.5753643190171465, 2.593644294723892, 1.186025165225192, 1.4923820091472486, 0.8022940272381623, 0.6480426871480323, 1.9057056772327552, 1.5457718333104122, 2.7005600639742435, 0.8051879303993821, 1.3833874989725652, 1.3380356542015006, 0.7240644353600194, 2.1364459491264776, 1.8489091016817272, 1.3953106919313747, 2.052146372313022, 0.7780643073078928, 1.3682510756225863, 2.752734472070327, 2.0306530950713326, 2.0085252797811823, 0.050586173274131806]
phi: [3.07007712412391, 6.05300623953674, 3.408147717572203, 5.36448956195994, 2.414093425280563, 1.6267466909798938, 4.51930406157485, 4.8840692708724545, 0.4327806293826175, 1.41973831299009, 4.340722295429118, 1.4744311210984817, 5.4416769308172555, 3.055346542855945, 3.3627454208365664, 1.080596431313999, 4.10814142008108, 5.572248759431854, 0.7168671733468445, 4.659935101031482, 2.6880518421611543, 3.8806497900171575, 0.41550525406244987, 1.9268788933797272, 6.265330983937306, 2.282955115894484, 6.152870375913103, 1.3144099075195315]
weights: [0.951554691788582, 0.951554691788582, 0.951554691788582, 0.951554691788582, 0.951554691788582, 0.951554691788582, 0.951554691788582, 0.951554691788582, 0.951554691788582, 0.951554691788582, 0.951554691788582, 0.951554691788582, 0.9692307692264744, 0.9692307692264744, 0.9692307692264744, 0.9692307692264744, 1.0587017184692598, 1.0587017184692598, 1.0587017184692598, 1.0587017184692598, 1.0587017184692598, 1.0587017184692598, 1.0587017184692598, 1.0587017184692598, 1.0587017184692598, 1.0587017184692598, 1.0587017184692598, 1.0587017184692598]

symmetries =  [1.         0.         0.         0.         0.         0.
 0.         0.         0.         0.17312411 0.09176089 0.01504159]
symmetry error = 0.0
minimal relative intensity = 0.827053 rms = 0.04465091

N = 34, l_max = 9

best uniform solution: 48.9

34 points (theta,phi) 17 different weights:
theta: [0.6160919136355196, 2.525500739967017, 2.0389059510818512, 1.1026867025178098, 2.9477064973420677, 0.19388615625617087, 0.38054796061000323, 2.76104469297918, 2.1984755908140685, 0.9431170627712412, 1.3222763581034396, 1.819316295488534, 2.420607202486368, 0.7209854511065524, 1.8540946405307839, 1.2874980130594962, 1.6695061537302764, 1.472086499854895, 1.305505678616015, 1.8360869749746889, 1.4851803513200668, 1.6564123022670856, 2.1745261184883597, 0.967066535101684, 2.4497041144795775, 0.6918885391140858, 1.8974102721351327, 1.2441823814551505, 1.0081228487051588, 2.133469804884713, 1.3500163630850335, 1.7915762905077361, 2.4788512035211747, 0.662741450063009]
phi: [2.82219674025365, 5.963789393851405, 5.758038832011148, 2.6164461784143844, 0.6162162684980208, 3.7578089220285316, 0.3983111426043742, 3.5399037962011324, 3.9821265298707638, 0.8405338762782075, 4.201215708286491, 1.059623054694641, 1.2526647897569014, 4.3942574433473345, 5.169981791119339, 2.028389137527214, 6.227784162607241, 3.0861915090162904, 1.3797985366282635, 4.521391190222528, 0.5496168969531845, 3.691209550542608, 0.4038966932277184, 3.545489346825146, 2.2967702513972945, 5.43836290499337, 1.77355712078464, 4.915149774373809, 6.282014717374506, 3.140422063788476, 5.626438823358258, 2.4848461697659494, 4.888175544986312, 1.7465828913994825]
weights: [0.7973318435404521, 0.7973318435404521, 0.8748787072549777, 0.8748787072549777, 0.8882539021944679, 0.8882539021944679, 0.9712608232819234, 0.9712608232819234, 0.9752837593522342, 0.9752837593522342, 0.9794969056925997, 0.9794969056925997, 0.9805981137194081, 0.9805981137194081, 0.9822535980104351, 0.9822535980104351, 0.9939401664896705, 0.9939401664896705, 0.9990394311743611, 0.9990394311743611, 1.0053037961180802, 1.0053037961180802, 1.0858347226892673, 1.0858347226892673, 1.0867072392137682, 1.0867072392137682, 1.0921328710052833, 1.0921328710052833, 1.0933775087834647, 1.0933775087834647, 1.0949678055152163, 1.0949678055152163, 1.0993388059643907, 1.0993388059643907]

symmetries =  [1.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
 0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
 0.00000000e+00 0.00000000e+00 2.02862068e-01 7.06412102e-23
 4.91047165e-02]
symmetry error = 0.0
minimal relative intensity = 0.897265 rms = 0.01825170

N = 42, l_max = 10

best uniform solution: 60.10

42 points (theta,phi) 42 different weights:
theta: [0.912090306934545, 2.746761190754236, 2.2986410362170338, 1.745862929099355, 0.6087172979572448, 0.8750179958146024, 2.003199964794502, 1.914019919169261, 1.3719018453624228, 2.0783535824454447, 3.0416516361988566, 1.9030536604204784, 2.5555780724806496, 0.6356259587455823, 1.0585497682737501, 1.4679930768047569, 0.38392536310433395, 2.42688314878376, 1.9802333048552152, 2.123718951551832, 1.5987738995397742, 1.377057139594396, 2.061148509982718, 1.4517070680861281, 1.4353585379167855, 1.1643720284459766, 0.8907621209089062, 1.795193579750673, 1.4049928360005761, 0.9936439341556538, 1.5103957757421282, 1.1852333762660772, 0.27975686369694425, 2.447217284242543, 1.4814488883229116, 0.4161660936306942, 2.53426390482221, 2.5343394636596, 0.9022157881722463, 2.049939403516931, 0.8567644078993933, 1.8001113473988923]
phi: [1.6189186869012622, 4.417890964353101, 3.925717844888819, 4.165114459914451, 1.024244687866028, 3.571479923677028, 5.6761304055135895, 6.205336584239582, 3.8328097495962488, 0.4743672484760749, 0.986147115054497, 3.5546001474686792, 2.0704647095249578, 4.915642480324595, 2.1949792038737597, 4.610023292590779, 3.8262559858367093, 5.258320077680217, 2.977183013149219, 4.598896945880547, 2.0519720817232328, 6.162612488037494, 2.370264024699624, 5.604941266160562, 1.5141547998093383, 5.098644649280042, 2.8593015359409786, 5.096594585219709, 3.238975312951073, 4.308797093290831, 0.4550427435469594, 0.9465296848185534, 6.155149089475483, 1.0303914048801714, 2.6508963308118045, 2.2568415894699476, 3.1129991266539556, 6.209840414334106, 0.2816873934360353, 1.6296533929828394, 5.769891533320035, 1.03323260646458]
weights: [0.8702450096407891, 0.891068953969204, 0.8955518026815953, 0.8990632826491868, 0.9112149039905258, 0.919575366373204, 0.9247050688209141, 0.9249481421631185, 0.9374736698059064, 0.9441339800127497, 0.9460695594614713, 0.9624139527428935, 0.9626781153524502, 0.9694810363793928, 0.9706018847174712, 0.9731112520873746, 0.9777700998281489, 0.9796868643955328, 0.9803515510110513, 0.9886895472366434, 0.9945191426107445, 1.0026057961079573, 1.006543582504591, 1.006936853157852, 1.02102322906593, 1.0237625319134556, 1.0366919458662687, 1.0463681488559085, 1.0481872408048207, 1.0484653593209248, 1.0569115307744548, 1.0613689554089596, 1.0660550338853683, 1.0686923989463608, 1.0696719846263136, 1.0700103666558665, 1.07433486681468, 1.0891862263724188, 1.0909117206860226, 1.091665763486926, 1.0964660660903478, 1.1007872127242062]

symmetries =  [1.         0.         0.         0.         0.         0.
 0.         0.         0.         0.         0.         0.1241871
 0.06575862 0.02474867]
symmetry error = 0.0
minimal relative intensity = 0.976213 rms = 0.00393389

N = 48, l_max = 11

best uniform solution: 70.11

48 points (theta,phi) 2 different weights:
theta: [1.2336997948453468, 1.9078928587444466, 1.9078928587444466, 1.2336997948453468, 1.4598542464697268, 1.6817384071200663, 1.6817384071200663, 1.4598542464697268, 2.7853163157342573, 0.3562763378555358, 0.3562763378555358, 2.7853163157342573, 2.7853163157330116, 0.35627633785678164, 0.35627633785678164, 2.7853163157330116, 1.45985424646999, 1.6817384071198034, 1.6817384071198034, 1.45985424646999, 1.907892858745844, 1.2336997948439494, 1.2336997948439494, 1.907892858745844, 0.8824843417477756, 2.2591083118420174, 2.2591083118420174, 0.8824843417477756, 1.8616874665689487, 1.2799051870208444, 1.2799051870208444, 1.8616874665689487, 2.259108311842379, 0.8824843417474144, 0.8824843417474144, 2.259108311842379, 1.2799051870219895, 1.8616874665678038, 1.8616874665678038, 1.2799051870219895, 2.370415958253471, 0.7711766953363222, 0.7711766953363222, 2.370415958253471, 0.771176695336047, 2.370415958253746, 2.370415958253746, 0.771176695336047]
phi: [4.8299771574945405, 1.453208149685046, 4.594800803274839, 1.6883845039047471, 5.943919993516647, 0.33926531366293894, 3.480857967252732, 2.8023273399268542, 3.4646084238352346, 2.8185768833443516, 5.960169536934145, 0.3230157702454418, 5.035404750628221, 1.2477805565513658, 4.389373210141159, 1.8938120970384273, 4.373123666720354, 1.9100616404592323, 5.051654294049025, 1.231531013130561, 6.165597130069957, 0.11758817710962952, 3.2591808306994223, 3.024004476480164, 4.331919117226351, 1.9512661899532349, 5.092858843543028, 1.1903264636365583, 2.416651002899945, 3.866534304279641, 0.7249416506898482, 5.558243656489738, 3.5220625167467206, 2.7611227904328657, 5.902715444022658, 0.3804698631569275, 5.437330631074623, 0.8458546761049632, 3.9874473296947563, 2.29573797748483, 2.7174950230862995, 3.5656902840932867, 0.42409763050349364, 5.859087676676093, 5.1364866108865845, 1.146698696293002, 4.288291349882795, 1.9948939572967912]
weights: [0.9495675564588315, 0.9495675564588315, 0.9495675564588315, 0.9495675564588315, 0.9495675564588315, 0.9495675564588315, 0.9495675564588315, 0.9495675564588315, 0.9495675564588315, 0.9495675564588315, 0.9495675564588315, 0.9495675564588315, 0.9495675564588315, 0.9495675564588315, 0.9495675564588315, 0.9495675564588315, 0.9495675564588315, 0.9495675564588315, 0.9495675564588315, 0.9495675564588315, 0.9495675564588315, 0.9495675564588315, 0.9495675564588315, 0.9495675564588315, 1.0504324435411683, 1.0504324435411683, 1.0504324435411683, 1.0504324435411683, 1.0504324435411683, 1.0504324435411683, 1.0504324435411683, 1.0504324435411683, 1.0504324435411683, 1.0504324435411683, 1.0504324435411683, 1.0504324435411683, 1.0504324435411683, 1.0504324435411683, 1.0504324435411683, 1.0504324435411683, 1.0504324435411683, 1.0504324435411683, 1.0504324435411683, 1.0504324435411683, 1.0504324435411683, 1.0504324435411683, 1.0504324435411683, 1.0504324435411683]

symmetries =  [1.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
 0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
 0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
 1.14621038e-01 9.45072435e-02 7.03566689e-05]
symmetry error = 0.0
minimal relative intensity = 0.982725 rms = 0.00432292

N = 58/60, l_max = 12

best uniform solution: 84.12

58 points (theta,phi) 58 different weights:
theta: [1.4625732945102392, 1.5124900499321432, 0.749948790760706, 1.8055944536325494, 2.061310567834104, 2.23226434495121, 1.6456869322125696, 1.564732590741013, 1.4957407024337483, 0.8041339533280961, 0.9749327633824536, 2.2530865495040144, 2.4196140732903504, 1.1234157722963025, 2.9120625356827894, 0.538112422091656, 2.579071228464289, 1.1673808518901907, 1.1958027838302598, 1.6991198381568426, 2.6515073694563775, 2.198845072848418, 0.3573606091636152, 0.5486085347557721, 0.8883386793240915, 0.6789416862870981, 1.6670247565036371, 1.9595090731976386, 1.0824457996934733, 1.7217704594909287, 2.895787806997049, 1.117677748388637, 1.8673844360870282, 1.2191549430750221, 1.927285154465101, 0.7154107707199917, 1.3440884565485327, 2.0172009133418474, 1.3940910225834287, 2.1886125856021015, 1.554395366080808, 0.9025077721183872, 2.4776210135817545, 2.681104400095342, 1.0223476998510461, 1.7910367280457309, 2.4510517150186244, 1.657081274078465, 0.36673989020787107, 1.2852150762492365, 2.0607184906845792, 1.2815538491964542, 2.15571242979144, 0.19737192872794945, 1.9462168306606145, 1.4615780340408329, 2.414340849387238, 0.7441936799638402]
phi: [6.093046430729653, 2.497984534320207, 4.661672382384337, 6.062793086597214, 0.5029541743695377, 0.9998465280485642, 4.763853560130005, 2.0417176855383357, 1.6291640801891727, 0.4138427514745532, 3.645646402536016, 3.5181797115811313, 1.5350347118450354, 2.17878262234884, 4.66427341850738, 1.5386371524678506, 0.52761351708754, 0.06308351359734513, 4.730861138756998, 5.594023873585617, 5.872128820146829, 3.383892846989375e-08, 0.4911347637466857, 3.993892843564511, 1.0227092186157694, 2.328329180993931, 2.913667422738485, 2.397175382603447, 4.18228214464703, 0.8458413657665363, 1.517021562991158, 2.749032932035534, 1.3415982723199287, 5.744060604676913, 5.128239532310928, 3.1093108232963274, 1.1678137651302691, 1.8546938350783773, 5.215498358161127, 5.64678437287598, 4.295364236137228, 5.279284759766376, 4.140574036565011, 3.1297111908066637, 1.6539868090786796, 3.4021765877688606, 2.3041965870967176, 0.27416250207874604, 5.263709270011244, 3.2452205366176208, 4.518646321073305, 0.6095694593202984, 2.9201672205986777, 2.905130104600923, 3.966649614743391, 3.7743774240100443, 4.997897634883493, 5.966385688898415]
weights: [0.5950876792889014, 0.8563462352677825, 0.8737840492391372, 0.8763365519399436, 0.8765735986256921, 0.8871945100879534, 0.8901570382398424, 0.9032691525738665, 0.9125744573276605, 0.9148515573746437, 0.9367675355371524, 0.9496026529334867, 0.9510053307449832, 0.9553472437314717, 0.963340169956153, 0.967971978400785, 0.9821983006942473, 0.9854613463006211, 0.9873931636061192, 0.9915489714651599, 0.9918567334238754, 0.9932899851366229, 0.9963303152777864, 1.0046071663622314, 1.007529744120311, 1.0095982144815758, 1.0144689262853812, 1.0161286682563666, 1.0265946992516373, 1.0303076649347143, 1.031102598230703, 1.033585586061885, 1.03594510604714, 1.0370671188671112, 1.039242114356098, 1.0426135657875877, 1.044301868705987, 1.0464884771017846, 1.0495777954528496, 1.049865102651683, 1.0534229170116134, 1.0545234695792765, 1.055638455125709, 1.0568405650969255, 1.0568586778257856, 1.060560981064404, 1.0632468849300942, 1.0633170371517284, 1.063811351416183, 1.0691801255766094, 1.0698326759861898, 1.075775796958918, 1.0760014680058143, 1.0762435812041282, 1.0840576179023762, 1.085668298658389, 1.088599071254965, 1.089108051121954]

symmetries =  [1.         0.         0.         0.         0.         0.
 0.         0.         0.         0.         0.         0.
 0.         0.0934051  0.06649197 0.01707206]
symmetry error = 0.0
minimal relative intensity = 0.981192 rms = 0.00371910

60 points (theta,phi) 5 different weights:
theta: [1.6274940267722833, 1.51409862681751, 1.51409862681751, 1.6274940267722833, 0.8638557178204358, 2.2777369357693575, 2.2777369357693575, 0.8638557178204358, 0.7101900904428751, 2.4314025631469183, 2.4314025631469183, 0.7101900904428751, 2.4226326692470517, 0.7189599843427416, 0.7189599843427416, 2.4226326692470517, 1.9389988215692207, 1.2025938320205727, 1.2025938320205727, 1.9389988215692207, 0.9865818998776599, 2.1550107537121335, 2.1550107537121335, 0.9865818998776599, 1.0615792547795029, 2.0800133988102907, 2.0800133988102907, 1.0615792547795029, 0.674609763729876, 2.466982889859917, 2.466982889859917, 0.674609763729876, 1.9719368622435383, 1.1696557913462549, 1.1696557913462549, 1.9719368622435383, 2.8875499597023726, 0.2540426938874208, 0.2540426938874208, 2.8875499597023726, 1.816324864404675, 1.3252677891851181, 1.3252677891851181, 1.816324864404675, 1.6347041340303299, 1.5068885195594635, 1.5068885195594635, 1.6347041340303299, 0.4325408030670978, 2.7090518505226955, 2.7090518505226955, 0.4325408030670978, 1.4551037073027548, 1.6864889462870385, 1.6864889462870385, 1.4551037073027548, 1.1560352805339638, 1.9855573730558294, 1.9855573730558294, 1.1560352805339638]
phi: [0.8624800626257317, 5.420705244553854, 2.2791125909640613, 4.004072716215525, 3.0669957854499774, 3.216189521729609, 0.0745968681398157, 6.2085884390397705, 4.799414810465431, 1.4837704967141547, 4.625363150303947, 1.6578221568756386, 2.14899866351929, 4.134186643660296, 0.9925939900705033, 5.290591317109083, 5.344893613415652, 0.9382916937639341, 4.079884347353727, 2.2033009598258593, 0.4461617235020494, 5.837023583677537, 2.6954309300877437, 3.5877543770918425, 2.034436987188264, 4.248748319991322, 1.1071556664015294, 5.176029640778057, 2.466259676091016, 3.81692563108857, 0.6753329774987773, 5.607852329680809, 0.5580462589752347, 5.725139048204351, 2.5835463946145585, 3.6996389125650277, 4.455454775662978, 1.8277305315166086, 4.9693231851064015, 1.3138621220731848, 3.2074792013766307, 3.0757061058029556, 6.217298759392748, 0.06588654778683765, 4.466347818334601, 1.816837488844985, 4.9584301424347785, 1.3247551647448081, 3.4205806052235346, 2.8626047019560517, 6.004197355545845, 0.2789879516337414, 0.4177261924096632, 5.865459114769923, 2.72386646118013, 3.559318845999456, 4.585923306932992, 1.6972620002465937, 4.838854653836387, 1.4443306533431994]
weights: [0.8935966646215763, 0.8935966646215763, 0.8935966646215763, 0.8935966646215763, 0.8935966646215763, 0.8935966646215763, 0.8935966646215763, 0.8935966646215763, 0.8935966646215763, 0.8935966646215763, 0.8935966646215763, 0.8935966646215763, 0.9930272225438518, 0.9930272225438518, 0.9930272225438518, 0.9930272225438518, 0.9930272225438518, 0.9930272225438518, 0.9930272225438518, 0.9930272225438518, 0.9930272225438518, 0.9930272225438518, 0.9930272225438518, 0.9930272225438518, 1.0249403615485184, 1.0249403615485184, 1.0249403615485184, 1.0249403615485184, 1.0249403615485184, 1.0249403615485184, 1.0249403615485184, 1.0249403615485184, 1.0249403615485184, 1.0249403615485184, 1.0249403615485184, 1.0249403615485184, 1.0346957277725266, 1.0346957277725266, 1.0346957277725266, 1.0346957277725266, 1.0346957277725266, 1.0346957277725266, 1.0346957277725266, 1.0346957277725266, 1.0346957277725266, 1.0346957277725266, 1.0346957277725266, 1.0346957277725266, 1.053740023513527, 1.053740023513527, 1.053740023513527, 1.053740023513527, 1.053740023513527, 1.053740023513527, 1.053740023513527, 1.053740023513527, 1.053740023513527, 1.053740023513527, 1.053740023513527, 1.053740023513527]

symmetries =  [1.         0.         0.         0.         0.         0.
 0.         0.         0.         0.         0.         0.
 0.         0.05187415 0.08413808 0.05052016]
symmetry error = 0.0
minimal relative intensity = 0.984542 rms = 0.00329519

N = 64, l_max = 13

best uniform solution: 96.13

64 points (theta,phi) 32 different weights:
theta: [3.0327054032853336, 0.10888725030341564, 0.7059886503029189, 2.4356040032873945, 1.9746110818212514, 1.1669815717688026, 1.4297266365946104, 1.711866016993206, 2.8004969662281707, 0.34109568736327284, 1.6666342904916334, 1.4749583630975382, 2.4047570661713853, 0.7368355874180538, 1.775615067662451, 1.3659775859250636, 1.8918908772663194, 1.2497017763228888, 2.4454893398466626, 0.6961033137443569, 1.3167195557234872, 1.8248730978652279, 2.6871106514331884, 0.45448200215805273, 2.010736011794007, 1.1308566417988202, 1.6610201547584007, 1.4805724988316986, 0.38035095134157926, 2.7612417022510996, 2.1274754607508997, 1.0141171928357489, 1.9013166056200554, 1.2402760479692863, 1.1407182454009417, 2.0008744081890364, 2.090066136382095, 1.051526517207934, 0.5500171220833715, 2.591575531503467, 1.317501867077761, 1.8240907865095635, 0.5043613755674253, 2.6372312780234974, 0.7437425658893998, 2.397850087700489, 0.928245932559469, 2.213346721028753, 0.8057840187248153, 2.335808634863557, 1.7344371602463107, 1.4071554933437893, 1.9248103018647362, 1.2167823517243634, 2.261237493700175, 0.8803551598886876, 2.2003081255048453, 0.9412845280829485, 1.4854953442029422, 1.656097309386756, 2.156094744973228, 0.9854979086154283, 1.5780114589659913, 1.5635811946244544]
phi: [0.6697581536949991, 3.8113508072895135, 0.21600333575512387, 3.3575959893449423, 4.048843826547785, 0.9072511729552043, 1.21311161765841, 4.354704271246731, 3.019599201552422, 6.161191855145521, 2.8330844485656104, 5.974677102154328, 3.9983184606828894, 0.8567258070912509, 4.781880544555741, 1.640287890965838, 0.30328332902336064, 3.444875982616206, 5.105723280742008, 1.9641306271530858, 4.204820343939534, 1.0632276903503244, 5.848195082150941, 2.706602428563319, 3.5612568023361817, 0.41966414874763125, 3.286876287846199, 0.1452836342567494, 1.3210410950393612, 4.462633748628474, 0.7070037968403527, 3.8485964504288335, 6.100471821473459, 2.9588791678818254, 2.0366365328735054, 5.178229186464689, 2.994733529356041, 6.1363261829437645, 3.887346348404249, 0.7457536948135495, 4.6971429471194535, 1.555550293529704, 4.932826819591848, 1.791234166003894, 5.674162895814854, 2.5325702422250207, 2.56827238218539, 5.709865035772619, 3.232929761225942, 0.09133710763514717, 5.590489988416283, 2.4488973348273584, 2.442124267240045, 5.583716920831607, 1.3137994070163082, 4.455392060607762, 4.571222682075825, 1.4296300284867773, 5.170912349953086, 2.0293196963645825, 1.9444923470696673, 5.086085000655982, 0.6621883132503205, 3.80378096684075]
weights: [0.8719563354221114, 0.8719563354221114, 0.8848591329105633, 0.8848591329105633, 0.9043062352982658, 0.9043062352982658, 0.9055317281690395, 0.9055317281690395, 0.9202594393721593, 0.9202594393721593, 0.9213708535886851, 0.9213708535886851, 0.9237012540701632, 0.9237012540701632, 0.9375790446960531, 0.9375790446960531, 0.937800285494615, 0.937800285494615, 0.9569854844110837, 0.9569854844110837, 0.961360025525758, 0.961360025525758, 0.9837541728181975, 0.9837541728181975, 0.9923722300893235, 0.9923722300893235, 0.994925812885059, 0.994925812885059, 1.0063048720199328, 1.0063048720199328, 1.007252862189682, 1.007252862189682, 1.0155543990636628, 1.0155543990636628, 1.0195763018504427, 1.0195763018504427, 1.0254208976051387, 1.0254208976051387, 1.0322280451309636, 1.0322280451309636, 1.0437656010206509, 1.0437656010206509, 1.0454002760393402, 1.0454002760393402, 1.048303236944561, 1.048303236944561, 1.0578375724636782, 1.0578375724636782, 1.0599048003069254, 1.0599048003069254, 1.0618939243523398, 1.0618939243523398, 1.0628187150811512, 1.0628187150811512, 1.0717345463787362, 1.0717345463787362, 1.0753046860654494, 1.0753046860654494, 1.0786334854657063, 1.0786334854657063, 1.0953533975147731, 1.0953533975147731, 1.095950345755789, 1.095950345755789]

symmetries =  [1.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
 0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
 0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
 0.00000000e+00 0.00000000e+00 1.38057504e-01 7.24437063e-24
 4.42682722e-02]
symmetry error = 0.0
minimal relative intensity = 0.985146 rms = 0.00255375

N = 72, l_max = 14

best uniform solution: 108.14

72 points (theta,phi) 2 different weights:
theta: [1.570796326794894, 1.5707963267948994, 1.5707963267948994, 1.570796326794894, 2.1243706856920457, 1.0172219678977474, 1.0172219678977474, 2.1243706856920457, 0.553574358897149, 2.5880182946926444, 2.5880182946926444, 0.553574358897149, 1.1491880742954592, 1.992404579294334, 1.992404579294334, 1.1491880742954592, 1.2171645701120242, 1.924428083477769, 1.924428083477769, 1.2171645701120242, 1.045409483694764, 2.096183169895029, 2.096183169895029, 1.045409483694764, 0.7040578522196116, 2.4375348013701816, 2.4375348013701816, 0.7040578522196116, 0.8494277341123436, 2.2921649194774494, 2.2921649194774494, 0.8494277341123436, 2.7221843035747035, 0.41940835001508975, 0.41940835001508975, 2.7221843035747035, 1.310828575047656, 1.830764078542137, 1.830764078542137, 1.310828575047656, 2.3000023719275777, 0.8415902816622157, 0.8415902816622157, 2.3000023719275777, 1.2494562217538965, 1.8921364318358966, 1.8921364318358966, 1.2494562217538965, 1.7266673281712817, 1.4149253254185115, 1.4149253254185115, 1.7266673281712817, 1.4191070012698126, 1.7224856523199807, 1.7224856523199807, 1.4191070012698126, 2.9232201418418287, 0.21837251174796468, 0.21837251174796468, 2.9232201418418287, 0.6139373949535742, 2.5276552586362193, 2.5276552586362193, 0.6139373949535742, 2.1736833987774284, 0.9679092548123646, 0.9679092548123646, 2.1736833987774284, 1.6727818147410136, 1.4688108388487795, 1.4688108388487795, 1.6727818147410136]
phi: [2.1243706856920457, 4.15881462148754, 1.0172219678977474, 5.265963339281839, 3.3134750731082e-15, 6.283185307179583, 3.14159265358979, 3.1415926535897962, 4.712388980384695, 1.5707963267948912, 4.712388980384684, 1.5707963267949019, 0.9888124407285062, 5.29437286645108, 2.152780212861287, 4.130405094318299, 2.3606150997793494, 3.922570207400237, 0.7809775538104439, 5.5022077533691425, 0.49272141503149497, 5.790463892148091, 2.648871238558298, 3.634314068621288, 0.8864164746259138, 5.396768832553672, 2.2551761789638793, 4.028009128215707, 5.191722370928203, 1.0914629362513832, 4.233055589841176, 2.05012971733841, 3.824730121200286, 2.4584551859793002, 5.600047839569093, 0.6831374676104929, 5.950248391530775, 0.3329369156488108, 3.474529569238604, 2.8086557379409824, 3.624553208138237, 2.6586320990413492, 5.800224752631142, 0.48296055454844417, 4.9867343042770065, 1.2964510029025795, 4.438043656492372, 1.8451416506872138, 6.129619756331894, 0.15356555084769227, 3.2951582044374854, 2.988027102742101, 1.413099631286041, 4.870085675893545, 1.728493022303752, 4.554692284875834, 2.3427063559058823, 3.940478951273704, 0.7988862976839112, 5.484299009495675, 3.3192493760002035, 2.9639359311793827, 6.105528584769176, 0.17765672241041075, 1.6947122593271187, 4.588473047852467, 1.4468803942626745, 4.836304912916912, 5.676698233760886, 0.6064870734187001, 3.748079727008493, 2.535105580171093]
weights: [0.8928571428599884, 0.8928571428599884, 0.8928571428599884, 0.8928571428599884, 0.8928571428599884, 0.8928571428599884, 0.8928571428599884, 0.8928571428599884, 0.8928571428599884, 0.8928571428599884, 0.8928571428599884, 0.8928571428599884, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025, 1.0214285714280025]

symmetries =  [1.         0.         0.         0.         0.         0.
 0.         0.         0.         0.         0.         0.
 0.         0.         0.         0.10261877 0.06483876 0.        ]
symmetry error = 0.0
minimal relative intensity = 0.995359 rms = 0.00105520

N = 82, l_max = 15

best uniform solution: 120.15

82 points (theta,phi) 9 different weights:
theta: [1.1780665441084595, 1.8727667522291274, 1.2688259013603038, 1.9635261094810712, 2.3240135762943477, 0.8175790772935139, 1.796529901701773, 1.34506275188719, 1.9410537833068344, 1.2005388702819786, 2.1927542803853335, 1.7422902881185955, 0.948838373202883, 1.0549956980525999, 2.0865969555371926, 1.3993023654722425, 1.5447847521561515, 2.084679555297012, 1.05691309829292, 1.5968079014336218, 2.5268471963162926, 0.6147454572727806, 1.5690079084675428, 1.5962104077559596, 1.545382245834525, 2.13109833910967, 0.45281453586308407, 2.6887781177247367, 2.176894223350758, 1.0104943144799807, 1.5725847451218538, 0.9646984302398104, 0.7197280960586272, 1.7339812669945274, 2.421864557528717, 2.54659019400401, 1.281065201298467, 1.8605274522919668, 1.4076113865961255, 1.6968515734678835, 0.5950024595860973, 1.4447410801204281, 2.1369154545421574, 1.0046771990481318, 0.47089529776844524, 0.2717887104469008, 2.2488831549108936, 0.7317403462756299, 2.869803943142536, 0.8927094986791368, 2.6706973558216336, 2.4098523073153673, 2.5303342335368915, 1.2447626655197515, 0.31847689303000426, 1.427405735271761, 1.0099466081094988, 1.8968299880702055, 2.8231157605586494, 2.1316460454804127, 0.6112584200523596, 1.714186918317773, 1.8656477842093309, 1.1529090868517184, 2.963122433628948, 2.275749455258106, 1.9886835667386205, 0.17847021995983262, 0.865843198330646, 2.557927660501902, 1.2759448693785493, 0.5836649930873238, 2.1702114494787503, 1.883381816339183, 1.6582871066197689, 1.361302736083453, 1.258210837250771, 1.483305546969469, 0.7277211538362046, 0.9713812041093857, 1.780289917506091, 2.413871499754068]
phi: [1.7358955249131438, 6.208373671326277, 3.066781017736231, 4.877488178502092, 5.590990310410589, 2.4493976568220597, 0.6366929957241607, 3.778285649314073, 4.132045248444283, 0.9904525948544873, 6.039272967060883, 0.28026417248137603, 2.8976803134723395, 1.3663140985446849, 4.507906752133057, 3.421856826069971, 0.9080932391863042, 5.265652143967327, 2.124059490379633, 4.049685892776887, 2.276049765687967, 5.417642419277431, 2.8493843137280925, 4.909886174437318, 1.7682935208479793, 0.8055495882509658, 2.355772265246279, 5.497364918834902, 3.809509920204968, 3.947142241840919, 5.990976967317326, 0.6679172666143366, 1.106600804819238, 3.221629918989531, 4.248193458408639, 6.27836280309479, 4.280812392454213, 1.1392197388628078, 0.08003726540060967, 5.304168936045312, 3.1367701495044487, 2.162576282454944, 2.402435482749884, 5.5440281363391355, 4.676589545095048, 5.9408343948993405, 1.8477899429770932, 6.039367377517849, 2.799241741307621, 4.989382596566041, 1.5349968915046168, 2.897774723928845, 0.7282919561333857, 4.709408888234912, 1.286900001952457, 5.613796099577057, 0.18661096908249605, 1.567816234644565, 4.428492655545502, 3.328203622672468, 3.869884609724023, 2.4722034459870015, 2.0432513440373, 5.99095451821658, 0.4122230096643072, 1.2969254185664196, 2.8493618646283165, 3.5538156632523323, 4.438518072156438, 3.550914757575052, 5.18484399762681, 0.4093221039856536, 0.3046341487870286, 5.704487711056117, 4.494301193794052, 0.5175523903032693, 2.562895057464858, 1.3527085402035592, 1.7661096654555961, 3.4462268023749814, 3.659145043892744, 4.90770231904655]
weights: [0.902416441564556, 0.902416441564556, 0.902416441564556, 0.902416441564556, 0.902416441564556, 0.902416441564556, 0.902416441564556, 0.902416441564556, 0.902416441564556, 0.902416441564556, 0.9061287607824264, 0.9061287607824264, 0.9061287607824264, 0.9061287607824264, 0.9061287607824264, 0.9061287607824264, 0.9061287607824264, 0.9061287607824264, 0.9061287607824264, 0.9061287607824264, 0.9139518852823021, 0.9139518852823021, 0.947228248658056, 0.947228248658056, 0.947228248658056, 0.947228248658056, 0.947228248658056, 0.947228248658056, 0.947228248658056, 0.947228248658056, 0.947228248658056, 0.947228248658056, 1.0066500392065465, 1.0066500392065465, 1.0066500392065465, 1.0066500392065465, 1.0066500392065465, 1.0066500392065465, 1.0066500392065465, 1.0066500392065465, 1.0066500392065465, 1.0066500392065465, 1.0433696152455139, 1.0433696152455139, 1.0433696152455139, 1.0433696152455139, 1.0433696152455139, 1.0433696152455139, 1.0433696152455139, 1.0433696152455139, 1.0433696152455139, 1.0433696152455139, 1.0531280280695285, 1.0531280280695285, 1.0531280280695285, 1.0531280280695285, 1.0531280280695285, 1.0531280280695285, 1.0531280280695285, 1.0531280280695285, 1.0531280280695285, 1.0531280280695285, 1.056353666233679, 1.056353666233679, 1.056353666233679, 1.056353666233679, 1.056353666233679, 1.056353666233679, 1.056353666233679, 1.056353666233679, 1.056353666233679, 1.056353666233679, 1.1019348231832335, 1.1019348231832335, 1.1019348231832335, 1.1019348231832335, 1.1019348231832335, 1.1019348231832335, 1.1019348231832335, 1.1019348231832335, 1.1019348231832335, 1.1019348231832335]

symmetries =  [1.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
 0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
 0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
 0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
 1.31284829e-01 1.51199470e-24 2.23242329e-02]
symmetry error = 0.0
minimal relative intensity = 0.998525 rms = 0.00024952

N = 98/100, l_max = 16

best uniform solution: 144.16

98 points (theta,phi) 98 different weights:
theta: [0.9323910358739712, 1.224957016424637, 0.5840790026733846, 2.3440306427709188, 2.0488573407926376, 0.6497927436524878, 1.348176237567165, 1.5361670023945369, 2.158090129764281, 1.379175427333143, 0.9190207009642966, 0.4704626448103172, 0.8927158116849769, 0.7672405826201764, 0.8123747842049011, 2.7162333931835194, 2.03631779049205, 2.052065919958169, 1.2342167845308996, 1.4796772241330436, 2.306192058661394, 1.8108323619846431, 0.712495247055939, 1.6088350147299513, 1.6900324131189413, 1.1678551618478301, 1.8298626567983836, 1.3880062019162758, 0.7842982261058593, 0.868116161631316, 0.5935736439801248, 1.0410649856123855, 2.4827962155627654, 0.3053956829890751, 0.06365318387468201, 2.6001348884621733, 0.6874386803818395, 0.3765017572160208, 1.746108663300137, 0.3363808562625645, 1.0540295366587067, 1.1305668854063369, 1.4304415064171534, 2.3652820110141173, 1.4297086633848475, 2.1326137416215607, 1.24286650868835, 1.1198164731778253, 1.509517866245914, 2.707709328537596, 0.6798470715199116, 1.4904908428421415, 1.7922636077336767, 1.7043751292483955, 2.155775175444147, 0.42348069098328295, 2.483331934446563, 2.2267919461165837, 0.8186475162867349, 0.42891824257405115, 0.7544234378707442, 1.435560363540185, 3.0479499092044824, 1.1048325705350543, 1.6250756070842265, 1.8497199959822739, 2.01468785742746, 1.7798174806423914, 1.0909001627821486, 1.7482973498893746, 1.7598497883687643, 2.397786235840658, 1.962979770049404, 1.6690273601671222, 1.270145568352709, 1.5356717399199575, 1.4389814782958605, 2.1384039901705116, 2.153292725911169, 1.8120670102616618, 2.792690358360503, 2.772247901416808, 1.787862305183858, 1.413193353232087, 1.9229223829901405, 2.141589394972872, 1.0336968961369333, 1.4324598514840314, 2.4308164656419127, 1.0649783498684886, 2.1207541051079932, 1.0494941281675716, 2.372671819464391, 1.216544665865211, 1.9737738500990234, 2.827662627516585, 2.5173181569406387, 2.501129917369996]
phi: [0.4382007592183331, 0.4624728865010383, 4.511650366199369, 4.294773149174043, 3.616051773069461, 0.5481152335107643, 3.6871424611160664, 5.356904216750224, 3.252105575140095, 6.098164909978604, 5.369369543820563, 1.8814491416963157, 3.0451718887632095, 1.5721207527111667, 2.090938724676596, 5.417186072089224, 4.399321865493645, 1.5226873153014746, 5.498940279643875, 2.3232892956876974, 3.8663507819797713, 5.537449529045664, 1.0872841808897968, 5.041496913015921, 6.239054779041313, 0.06788881929656645, 5.918302552208254, 2.6597950625454723, 4.970733346208466, 4.293828916815192, 3.2538249473892384, 2.6347600158124043, 3.3729594902383324, 2.7128888317059894, 5.385778398563208, 4.689583482579179, 5.707510483349736, 3.9692680377526033, 0.529913583895833, 1.0666042355084586, 4.681058738943975, 2.2414337212300395, 3.338409111743036, 5.274340764569209, 0.7267851971257249, 6.120718701103252, 3.0344651929450217, 3.9480980447046283, 0.26065994193899644, 3.9297304307075342, 2.5681491307330835, 5.771313120087283, 3.273997918961421, 3.660796122424421, 5.646207845998554, 5.214106509599846, 5.938389214527281, 4.809022442848946, 6.247461951696783, 6.263647560401598, 3.7932265252070545, 1.9537216108360946, 4.762081909539062, 1.7814281330341895, 2.9429648779284685, 4.758476048678712, 2.8825585147141712, 2.5645031357060657, 1.329891572784862, 1.3122842379393274, 1.7524958132475383, 2.7886860397346926, 0.21078322181819903, 4.390720695945503, 4.324563212527045, 4.026150354269433, 4.703089934217138, 1.1083028804876665, 1.9377672384178788, 2.147880542614598, 2.817960204854373, 0.23930271384451368, 0.9198584847042418, 1.1041638582206064, 4.02322298946828, 2.4144291069173094, 3.459251074152418, 1.5327235986592613, 1.5141004769211799, 5.866468790331191, 0.6416344097116372, 0.8792801361678493, 0.257100057412616, 5.070159931407837, 5.181717343824007, 1.4989486053906773, 2.159813049768234, 0.8748865435884975]
weights: [0.8020752818291782, 0.804489631117914, 0.8301395033447078, 0.8546198119750402, 0.8749221299785479, 0.8758402762357662, 0.8763147242760301, 0.877272851188724, 0.8796338664837198, 0.8842526986577645, 0.8871533348847781, 0.8903850875992206, 0.8925472108824948, 0.8933096779631647, 0.9012044049667471, 0.9079450673193807, 0.9110217196507332, 0.9212704427745182, 0.9213934234929934, 0.9272775660761104, 0.9311914415178062, 0.9318594562587437, 0.9420939209625777, 0.9476582966508537, 0.9485269084407524, 0.9494389368748102, 0.9538067760664879, 0.9540920242173758, 0.957355592680913, 0.9581957966230673, 0.9637551165308438, 0.9667669435486683, 0.9670171848157699, 0.9682160006333772, 0.9751756589284004, 0.9772730396201385, 0.9809255513683522, 0.9845553284966649, 0.9864667657411219, 0.9871343630673728, 0.9967074252830955, 0.997813561443109, 0.9979630813174424, 1.000121174142392, 1.0010428368543818, 1.0015976869664194, 1.0025118038478469, 1.0043020785590306, 1.0051289743508942, 1.0098123373941845, 1.0098735240089607, 1.012514158267619, 1.0141053798799566, 1.0173284773545659, 1.0191254314573488, 1.0256049734796227, 1.0272711559383536, 1.03060683112727, 1.0312606032523748, 1.0348494820736105, 1.0352400679302571, 1.0379002299446343, 1.0398820831710172, 1.0416311780845346, 1.0418688313477598, 1.046719431207292, 1.0470486997546185, 1.0506474906098657, 1.0534082408255112, 1.0540478685827153, 1.0563217089293953, 1.0578850525259895, 1.0599277084812913, 1.0613995514109924, 1.0667998850897285, 1.0672885212604653, 1.0737914009231908, 1.074600652847195, 1.0752479052923778, 1.0783948458773946, 1.0841935077046796, 1.0849437156928108, 1.0854413992874095, 1.0866470381736058, 1.0888948853749285, 1.0901792070719503, 1.091125947542335, 1.0936959560689175, 1.0945610666888541, 1.0969972504230079, 1.0970521736993981, 1.1026300482715563, 1.1030026455920323, 1.1108987768944347, 1.1157372517521265, 1.1187572590983146, 1.1231375196718774, 1.1279382061554244]

symmetries =  [1.         0.         0.         0.         0.         0.
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symmetry error = 0.0
minimal relative intensity = 0.999155 rms = 0.00015824

100 points (theta,phi) 9 different weights:
theta: [0.9553166181241007, 2.1862760354656925, 2.1862760354656925, 0.9553166181241007, 1.4775483944502068, 1.6640442591395865, 1.6640442591395865, 1.4775483944502068, 2.0851502325954288, 1.0564424209943644, 1.0564424209943644, 2.0851502325954288, 2.61717800329836, 0.5244146502914335, 0.5244146502914335, 2.61717800329836, 0.9023062180220782, 2.239286435567715, 2.239286435567715, 0.9023062180220782, 2.367660426390921, 0.773932227198872, 0.773932227198872, 2.367660426390921, 1.8997787425370496, 1.2418139110527437, 1.2418139110527437, 1.8997787425370496, 1.3273264220659922, 1.814266231523801, 1.814266231523801, 1.3273264220659922, 1.0855535363690787, 2.0560391172207146, 2.0560391172207146, 1.0855535363690787, 2.5888315763184426, 0.5527610772713505, 0.5527610772713505, 2.5888315763184426, 1.6119571498718535, 1.5296355037179397, 1.5296355037179397, 1.6119571498718535, 2.4352447165101703, 0.706347937079623, 0.706347937079623, 2.4352447165101703, 2.2754291499616337, 0.8661635036281596, 0.8661635036281596, 2.2754291499616337, 1.636636249465063, 1.5049564041247303, 1.5049564041247303, 1.636636249465063, 0.19295046054223883, 2.9486421930475544, 2.9486421930475544, 0.19295046054223883, 1.7518999647863942, 1.389692688803399, 1.389692688803399, 1.7518999647863942, 0.7750058121646571, 2.366586841425136, 2.366586841425136, 0.7750058121646571, 0.954825963427193, 2.1867666901626004, 2.1867666901626004, 0.954825963427193, 1.976571147332747, 1.1650215062570464, 1.1650215062570464, 1.976571147332747, 1.2450673959280687, 1.8965252576617246, 1.8965252576617246, 1.2450673959280687, 1.6855068322233064, 1.4560858213664867, 1.4560858213664867, 1.6855068322233064, 0.34676135130367514, 2.794831302286118, 2.794831302286118, 0.34676135130367514, 1.2965829078240025, 1.8450097457657908, 1.8450097457657908, 1.2965829078240025, 0.5357471239306999, 2.6058455296590934, 2.6058455296590934, 0.5357471239306999, 1.1232632491258399, 2.0183294044639535, 2.0183294044639535, 1.1232632491258399]
phi: [3.9269908169870718, 2.3561944901925145, 5.497787143782308, 0.7853981633972785, 3.65841398838589, 2.6247713187936963, 5.766363972383489, 0.5168213347960972, 4.819545120778974, 1.4636401864006123, 4.605232839990405, 1.6779524671891808, 2.9545408022750337, 3.3286445049045525, 0.1870518513147594, 6.096133455864827, 4.288077053553995, 1.995108253625591, 5.136700907215384, 1.146484399964202, 5.802671795465733, 0.4805135117138532, 3.6221061653036464, 2.66107914187594, 2.4275101543678863, 3.8556751528117, 0.7140824992219067, 5.569102807957679, 2.6402570248673154, 3.642928282312271, 0.5013356287224778, 5.7818496784571085, 4.98841295664408, 1.294772350535506, 4.436365004125299, 1.8468203030542873, 3.618630638329542, 2.664554668850044, 5.806147322439837, 0.4770379847397492, 0.865442530632284, 5.4177427765473025, 2.2761501229575094, 4.007035184222077, 4.648948271223926, 1.6342370359556606, 4.7758296895454535, 1.5073556176341325, 0.05403854498753944, 6.229146762192046, 3.0875541086022538, 3.1956311985773325, 2.960091401775485, 3.323093905404101, 0.1815012518143082, 6.101684055365278, 0.35022122200175704, 5.932964085177829, 2.7913714315880362, 3.49181387559155, 1.6377325551555824, 4.645452752024004, 1.503860098434211, 4.779325208745375, 2.1701713935410902, 4.113013913638496, 0.9714212600487028, 5.311764047130883, 2.63681556107237, 3.6463697461072164, 0.5047770925174235, 5.7784082146621625, 4.032364090476916, 2.25082121670267, 5.392413870292463, 0.8907714368871232, 6.162077823213478, 0.12110748396610861, 3.2627001375559015, 3.0204851696236847, 1.2428320958460977, 5.040353211333489, 1.8987605577436955, 4.38442474943589, 1.9143012477705688, 4.368884059409018, 1.2272914058192244, 5.055893901360362, 1.1045479160449607, 5.178637391134625, 2.0370447375448326, 4.246140569634754, 1.011656255585284, 5.271529051594302, 2.129936398004509, 4.153248909175077, 3.4466744176782167, 2.8365108895013695, 5.978103543091162, 0.3050817640884237]
weights: [0.7450786787981862, 0.7450786787981862, 0.7450786787981862, 0.7450786787981862, 0.8108368527829571, 0.8108368527829571, 0.8108368527829571, 0.8108368527829571, 0.8108368527829571, 0.8108368527829571, 0.8108368527829571, 0.8108368527829571, 0.8108368527829571, 0.8108368527829571, 0.8108368527829571, 0.8108368527829571, 0.8812729220578084, 0.8812729220578084, 0.8812729220578084, 0.8812729220578084, 0.8812729220578084, 0.8812729220578084, 0.8812729220578084, 0.8812729220578084, 0.8812729220578084, 0.8812729220578084, 0.8812729220578084, 0.8812729220578084, 1.028657092938444, 1.028657092938444, 1.028657092938444, 1.028657092938444, 1.028657092938444, 1.028657092938444, 1.028657092938444, 1.028657092938444, 1.028657092938444, 1.028657092938444, 1.028657092938444, 1.028657092938444, 1.0376439538109412, 1.0376439538109412, 1.0376439538109412, 1.0376439538109412, 1.0376439538109412, 1.0376439538109412, 1.0376439538109412, 1.0376439538109412, 1.0376439538109412, 1.0376439538109412, 1.0376439538109412, 1.0376439538109412, 1.0464371368442293, 1.0464371368442293, 1.0464371368442293, 1.0464371368442293, 1.0464371368442293, 1.0464371368442293, 1.0464371368442293, 1.0464371368442293, 1.0464371368442293, 1.0464371368442293, 1.0464371368442293, 1.0464371368442293, 1.062580203388776, 1.062580203388776, 1.062580203388776, 1.062580203388776, 1.062580203388776, 1.062580203388776, 1.062580203388776, 1.062580203388776, 1.062580203388776, 1.062580203388776, 1.062580203388776, 1.062580203388776, 1.103254052761993, 1.103254052761993, 1.103254052761993, 1.103254052761993, 1.103254052761993, 1.103254052761993, 1.103254052761993, 1.103254052761993, 1.103254052761993, 1.103254052761993, 1.103254052761993, 1.103254052761993, 1.1142915591487894, 1.1142915591487894, 1.1142915591487894, 1.1142915591487894, 1.1142915591487894, 1.1142915591487894, 1.1142915591487894, 1.1142915591487894, 1.1142915591487894, 1.1142915591487894, 1.1142915591487894, 1.1142915591487894]

symmetries =  [1.         0.         0.         0.         0.         0.
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 0.         0.         0.         0.         0.         0.06130647
 0.04950395 0.01449863]
symmetry error = 0.0
minimal relative intensity = 0.999289 rms = 0.00015548

N = 108, l_max = 17

best uniform solution: 156.17

108 points (theta,phi) 9 different weights:
theta: [2.0102521355828493, 1.131340518006944, 1.131340518006944, 2.0102521355828493, 0.4937637807416594, 2.647828872848134, 2.647828872848134, 0.4937637807416594, 1.781192484388904, 1.3604001692008891, 1.3604001692008891, 1.781192484388904, 1.4186385117018732, 1.7229541418879202, 1.7229541418879202, 1.4186385117018732, 1.6161501329874435, 1.5254425206023496, 1.5254425206023496, 1.6161501329874435, 2.9827181909369025, 0.15887446265289062, 0.15887446265289062, 2.9827181909369025, 1.0596603894966892, 2.081932264093104, 2.081932264093104, 1.0596603894966892, 2.0608040896274704, 1.0807885639623227, 1.0807885639623227, 2.0608040896274704, 2.3954540046513575, 0.7461386489384357, 0.7461386489384357, 2.3954540046513575, 0.5399718355709607, 2.6016208180188327, 2.6016208180188327, 0.5399718355709607, 1.0435519027153588, 2.0980407508744348, 2.0980407508744348, 1.0435519027153588, 1.6765734720377325, 1.4650191815520606, 1.4650191815520606, 1.6765734720377325, 2.776311039921086, 0.36528161366870693, 0.36528161366870693, 2.776311039921086, 1.8656408140408611, 1.2759518395489322, 1.2759518395489322, 1.8656408140408611, 1.7800642275223313, 1.361528426067462, 1.361528426067462, 1.7800642275223313, 1.691336986034372, 1.4502556675554212, 1.4502556675554212, 1.691336986034372, 0.4231633022426989, 2.7184293513470945, 2.7184293513470945, 0.4231633022426989, 1.9743033373736858, 1.1672893162161073, 1.1672893162161073, 1.9743033373736858, 2.3629958066026195, 0.7785968469871737, 0.7785968469871737, 2.3629958066026195, 1.7469448014594668, 1.3946478521303265, 1.3946478521303265, 1.7469448014594668, 0.8229432349878096, 2.318649418601984, 2.318649418601984, 0.8229432349878096, 2.4427817605409192, 0.6988108930488742, 0.6988108930488742, 2.4427817605409192, 0.9050807369318496, 2.2365119166579435, 2.2365119166579435, 0.9050807369318496, 1.3898442575183483, 1.751748396071445, 1.751748396071445, 1.3898442575183483, 0.7412399496652162, 2.400352703924577, 2.400352703924577, 0.7412399496652162, 2.1011466175072675, 1.0404460360825258, 1.0404460360825258, 2.1011466175072675, 1.107111755102752, 2.0344808984870415, 2.0344808984870415, 1.107111755102752]
phi: [6.050311413404659, 0.23287389377492776, 3.3744665473647206, 2.9087187598148656, 4.25605658048924, 2.027128726690347, 5.1687213802801395, 1.1144639268994465, 5.162472175129273, 1.1207131320503132, 4.262305785640106, 2.02087952153948, 3.1874769584424514, 3.095708348737135, 6.2373010023269275, 0.045884304852658576, 1.7231119899294693, 4.560073317250117, 1.4184806636603238, 4.8647046435192625, 1.8614469890523693, 4.4217383181272165, 1.280145664537424, 5.003039642642162, 4.142427953844489, 2.140757353335097, 5.28235000692489, 1.000835300254696, 2.553943983970475, 3.729241323209111, 0.5876486696193178, 5.695536637560268, 5.5170966385801385, 0.7660886685994476, 3.907681322189241, 2.3755039849903454, 1.777631826637483, 4.505553480542103, 1.3639608269523102, 4.919224480227276, 3.0191154316752273, 3.264069875504359, 0.12247722191456575, 6.1607080852650205, 4.181868991895537, 2.1013163152840493, 5.242908968873842, 1.0402763383057438, 0.6206567747192571, 5.662528532460329, 2.520935878870536, 3.76224942830905, 4.49353321667376, 1.7896520905058262, 4.931244744095619, 1.3519405630839672, 3.443217610848156, 2.8399676963314304, 5.9815603499212235, 0.3016249572583628, 5.87655563919467, 0.4066296679849162, 3.548222321574709, 2.734962985604877, 3.438776020268243, 2.8444092869113433, 5.986001940501136, 0.2971833666784501, 4.843513912919328, 1.4396713942602577, 4.58126404785005, 1.7019212593295356, 2.8893996545958522, 3.393785652583734, 0.25219299899394126, 6.030992308185645, 5.520652485171387, 0.7625328220081992, 3.904125475597992, 2.379059831581594, 1.329438028478017, 4.953747278701569, 1.8121546251117764, 4.47103068206781, 3.425127709015811, 2.858057598163775, 5.999650251753568, 0.2835350554260184, 1.8016681613646852, 4.481517145814901, 1.339924492225108, 4.9432608149544786, 5.604367297411717, 0.678818009767869, 3.820410663357662, 2.4627746438219242, 5.43639133392275, 0.8467939732568356, 3.9883866268466286, 2.2947986803329576, 0.54505923235629, 5.738126074823296, 2.596533421233503, 3.686651885946083, 2.1718984876673995, 4.111286819512187, 0.9696941659223935, 5.313491141257193]
weights: [0.8603612086887933, 0.8603612086887933, 0.8603612086887933, 0.8603612086887933, 0.8603612086887933, 0.8603612086887933, 0.8603612086887933, 0.8603612086887933, 0.8603612086887933, 0.8603612086887933, 0.8603612086887933, 0.8603612086887933, 0.9523862543533034, 0.9523862543533034, 0.9523862543533034, 0.9523862543533034, 0.9523862543533034, 0.9523862543533034, 0.9523862543533034, 0.9523862543533034, 0.9523862543533034, 0.9523862543533034, 0.9523862543533034, 0.9523862543533034, 0.9908231523204949, 0.9908231523204949, 0.9908231523204949, 0.9908231523204949, 0.9908231523204949, 0.9908231523204949, 0.9908231523204949, 0.9908231523204949, 0.9908231523204949, 0.9908231523204949, 0.9908231523204949, 0.9908231523204949, 0.9929083628277344, 0.9929083628277344, 0.9929083628277344, 0.9929083628277344, 0.9929083628277344, 0.9929083628277344, 0.9929083628277344, 0.9929083628277344, 0.9929083628277344, 0.9929083628277344, 0.9929083628277344, 0.9929083628277344, 1.0204034622716214, 1.0204034622716214, 1.0204034622716214, 1.0204034622716214, 1.0204034622716214, 1.0204034622716214, 1.0204034622716214, 1.0204034622716214, 1.0204034622716214, 1.0204034622716214, 1.0204034622716214, 1.0204034622716214, 1.0225808280406439, 1.0225808280406439, 1.0225808280406439, 1.0225808280406439, 1.0225808280406439, 1.0225808280406439, 1.0225808280406439, 1.0225808280406439, 1.0225808280406439, 1.0225808280406439, 1.0225808280406439, 1.0225808280406439, 1.0233971669576758, 1.0233971669576758, 1.0233971669576758, 1.0233971669576758, 1.0233971669576758, 1.0233971669576758, 1.0233971669576758, 1.0233971669576758, 1.0233971669576758, 1.0233971669576758, 1.0233971669576758, 1.0233971669576758, 1.0673925569743303, 1.0673925569743303, 1.0673925569743303, 1.0673925569743303, 1.0673925569743303, 1.0673925569743303, 1.0673925569743303, 1.0673925569743303, 1.0673925569743303, 1.0673925569743303, 1.0673925569743303, 1.0673925569743303, 1.0697470075654032, 1.0697470075654032, 1.0697470075654032, 1.0697470075654032, 1.0697470075654032, 1.0697470075654032, 1.0697470075654032, 1.0697470075654032, 1.0697470075654032, 1.0697470075654032, 1.0697470075654032, 1.0697470075654032]

symmetries =  [1.         0.         0.         0.         0.         0.
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symmetry error = 0.0
minimal relative intensity = 0.999234 rms = 0.00014760

N = 126, l_max = 18

best uniform solution: 180.18

126 points (theta,phi) 15 different weights:
theta: [0.9553166181245383, 2.186276035465255, 2.1862760354653266, 2.067642194962834, 1.164382953680593, 1.0739504586269593, 1.0739504586269593, 2.067642194962834, 2.4737999208636423, 0.667792732726151, 0.667792732726151, 2.4737999208636423, 1.9772096999092004, 1.9772096999092004, 1.164382953680593, 0.9553166181244667, 2.206044448230611, 0.935548205359182, 0.935548205359182, 2.206044448230611, 1.7626340325694927, 1.3789586210203006, 1.3789586210203006, 1.7626340325694927, 0.6729004238149737, 2.4686922297748195, 2.4686922297748195, 0.6729004238149737, 1.8473202445380423, 1.2942724090517508, 1.2942724090517508, 1.8473202445380423, 0.9806492456650454, 2.160943407924748, 2.160943407924748, 0.9806492456650454, 2.473045417083598, 0.6685472365061953, 0.6685472365061953, 2.473045417083598, 1.989156742596952, 1.1524359109928413, 1.1524359109928413, 1.989156742596952, 0.4194282276663252, 2.722164425923468, 2.722164425923468, 0.4194282276663252, 1.598970261161374, 1.5426223924284193, 1.5426223924284193, 1.598970261161374, 0.9553166181244667, 2.1862760354653266, 1.489040593901871, 1.6525520596879224, 0.8409846886410011, 2.3006079649487923, 2.3006079649487923, 0.8409846886410011, 2.4050752863218525, 0.7365173672679405, 0.7365173672679405, 2.4050752863218525, 1.489040593901871, 1.6525520596879224, 1.6937114529716093, 1.447881200618184, 1.447881200618184, 1.6937114529716093, 1.4524370635747612, 1.689155590015032, 1.4524370635747612, 1.689155590015032, 2.9705368437337865, 0.1710558098560066, 2.9705368437337865, 0.1710558098560066, 2.8069159580672656, 0.33467669552252755, 1.759251029034774, 1.382341624555019, 1.382341624555019, 1.759251029034774, 1.8439811173279146, 1.2976115362618788, 1.2976115362618788, 1.8439811173279146, 0.33467669552252755, 2.8069159580672656, 2.5963558051970557, 0.5452368483927377, 0.5452368483927377, 2.5963558051970557, 1.1431559789658272, 1.998436674623966, 1.2541291840853799, 1.8874634695044132, 1.8874634695044132, 1.2541291840853799, 1.998436674623966, 1.1431559789658272, 1.6789742596303487, 1.4626183939594446, 2.645486016870482, 0.4961066367193113, 0.4961066367193113, 2.645486016870482, 1.0887435567805557, 2.0528490968092377, 2.0528490968092377, 1.0887435567805557, 1.4626183939594446, 1.6789742596303487, 0.8231750417202409, 2.3184176118695525, 2.3184176118695525, 0.8231750417202409, 1.9804278460820561, 1.161164807507737, 0.9074868882386998, 2.234105765351093, 2.234105765351093, 0.9074868882386998, 1.161164807507737, 1.9804278460820561]
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weights: [0.25477892850787254, 0.5116349893410405, 0.5158612850049387, 0.6629006512533326, 0.6629006512533326, 0.6629006512533326, 0.6629006512533326, 0.6629006512533326, 0.6629006512533326, 0.6629006512533326, 0.6629006512533326, 0.6629006512533326, 0.6629006512533326, 0.6629006512533326, 0.6629006512533326, 0.7727173458418637, 0.8962561345160687, 0.8962561345160687, 0.8962561345160687, 0.8962561345160687, 0.8962561345160687, 0.8962561345160687, 0.8962561345160687, 0.8962561345160687, 0.8962561345160687, 0.8962561345160687, 0.8962561345160687, 0.8962561345160687, 0.9624832222639739, 0.9624832222639739, 0.9624832222639739, 0.9624832222639739, 0.9624832222639739, 0.9624832222639739, 0.9624832222639739, 0.9624832222639739, 0.9624832222639739, 0.9624832222639739, 0.9624832222639739, 0.9624832222639739, 0.9738210618460261, 0.9738210618460261, 0.9738210618460261, 0.9738210618460261, 0.9738210618460261, 0.9738210618460261, 0.9738210618460261, 0.9738210618460261, 0.9738210618460261, 0.9738210618460261, 0.9738210618460261, 0.9738210618460261, 1.0274962743418, 1.0274962743418, 1.0569284474336578, 1.0569284474336578, 1.0569284474336578, 1.0569284474336578, 1.0569284474336578, 1.0569284474336578, 1.0569284474336578, 1.0569284474336578, 1.0569284474336578, 1.0569284474336578, 1.0569284474336578, 1.0569284474336578, 1.1107059050212142, 1.1107059050212142, 1.1107059050212142, 1.1107059050212142, 1.1107059050212142, 1.1107059050212142, 1.1107059050212142, 1.1107059050212142, 1.1107059050212142, 1.1107059050212142, 1.1107059050212142, 1.1107059050212142, 1.1145772549029134, 1.1145772549029134, 1.1145772549029134, 1.1145772549029134, 1.1145772549029134, 1.1145772549029134, 1.1145772549029134, 1.1145772549029134, 1.1145772549029134, 1.1145772549029134, 1.1145772549029134, 1.1145772549029134, 1.1150779084007316, 1.1150779084007316, 1.1150779084007316, 1.1150779084007316, 1.1150779084007316, 1.1150779084007316, 1.1150779084007316, 1.1150779084007316, 1.1150779084007316, 1.1150779084007316, 1.1150779084007316, 1.1150779084007316, 1.1228105276309237, 1.1228105276309237, 1.1228105276309237, 1.1228105276309237, 1.1228105276309237, 1.1228105276309237, 1.1228105276309237, 1.1228105276309237, 1.1228105276309237, 1.1228105276309237, 1.1228105276309237, 1.1228105276309237, 1.1419401286162183, 1.1419401286162183, 1.1419401286162183, 1.1419401286162183, 1.1419401286162183, 1.1419401286162183, 1.1419401286162183, 1.1419401286162183, 1.1419401286162183, 1.1419401286162183, 1.1419401286162183, 1.1419401286162183]

symmetries =  [1.         0.         0.         0.         0.         0.
 0.         0.         0.         0.         0.         0.
 0.         0.         0.         0.         0.         0.
 0.         0.06723385 0.02555896 0.01673601]
symmetry error = 0.0
minimal relative intensity = 0.999538 rms = 0.00008887

N = 132, l_max = 19

best uniform solution: 204.19

132 points (theta,phi) 3 different weights:
theta: [1.5707963267950238, 1.5707963267947695, 1.5707963267947695, 1.5707963267950238, 0.553574358897899, 2.5880182946918944, 2.5880182946918944, 0.553574358897899, 1.0172219678969978, 2.1243706856927957, 2.1243706856927957, 1.0172219678969978, 1.4976050337520537, 1.6439876198377397, 1.6439876198377397, 1.4976050337520537, 1.3134187045583376, 1.8281739490314557, 1.8281739490314557, 1.3134187045583376, 0.26803908522608905, 2.8735535683637043, 2.8735535683637043, 0.26803908522608905, 0.6054877901357145, 2.536104863454079, 2.536104863454079, 0.6054877901357145, 1.8742434700066268, 1.2673491835831665, 1.2673491835831665, 1.8742434700066268, 1.8027405044420433, 1.3388521491477499, 1.3388521491477499, 1.8027405044420433, 2.2981654067764015, 0.8434272468133919, 0.8434272468133919, 2.2981654067764015, 2.0764912507690347, 1.0651014028207584, 1.0651014028207584, 2.0764912507690347, 2.361253227213127, 0.7803394263766662, 0.7803394263766662, 2.361253227213127, 1.9456057712320352, 1.1959868823577582, 1.1959868823577582, 1.9456057712320352, 0.7406460241206059, 2.4009466294691872, 2.4009466294691872, 0.7406460241206059, 0.8426590105438804, 2.298933643045913, 2.298933643045913, 0.8426590105438804, 1.4590139017260522, 1.682578751863741, 1.682578751863741, 1.4590139017260522, 1.8272143206921447, 1.3143783328976484, 1.3143783328976484, 1.8272143206921447, 2.6800139676761177, 0.4615786859136756, 0.4615786859136756, 2.6800139676761177, 0.9418439606502862, 2.199748692939507, 2.199748692939507, 0.9418439606502862, 2.0021748791777574, 1.1394177744120357, 1.1394177744120357, 2.0021748791777574, 1.3800390499810915, 1.7615536036087018, 1.7615536036087018, 1.3800390499810915, 0.7598973341099453, 2.381695319479848, 2.381695319479848, 0.7598973341099453, 1.0802394852462442, 2.061353168343549, 2.061353168343549, 1.0802394852462442, 2.335266709934967, 0.8063259436548262, 0.8063259436548262, 2.335266709934967, 2.097344824550846, 1.0442478290389476, 1.0442478290389476, 2.097344824550846, 0.45356129582117977, 2.6880313577686135, 2.6880313577686135, 0.45356129582117977, 1.7078336512804537, 1.4337590023093396, 1.4337590023093396, 1.7078336512804537, 2.072327710740324, 1.0692649428494694, 1.0692649428494694, 2.072327710740324, 2.9586974725889315, 0.1828951810008618, 0.1828951810008618, 2.9586974725889315, 2.618224044560076, 0.5233686090297168, 0.5233686090297168, 2.618224044560076, 1.395920744396228, 1.745671909193565, 1.745671909193565, 1.395920744396228, 1.1647030812045267, 1.9768895723852666, 1.9768895723852666, 1.1647030812045267, 1.5177778194937113, 1.6238148340960818, 1.6238148340960818, 1.5177778194937113]
phi: [0.5535743588978989, 5.729610948281687, 2.5880182946918944, 3.695167012487692, 3.1415926535895515, 3.1415926535900347, 2.418923349795825e-13, 6.283185307179345, 4.712388980384839, 1.5707963267947471, 4.7123889803845405, 1.5707963267950462, 0.2580842833579041, 6.025101023821682, 2.883508370231889, 3.399676936947697, 1.4951073632213168, 4.788077943958269, 1.6464852903684766, 4.63670001681111, 0.27974715780034876, 6.003438149379237, 2.8618454957894444, 3.421339811390142, 3.6943097012082466, 2.5888756059713396, 5.730468259561133, 0.5527170476184536, 5.750780231626697, 0.5324050755528893, 3.673997729142682, 2.609187578036904, 3.8937313778775176, 2.3894539293020687, 5.531046582891862, 0.7521387242877248, 4.399556695570758, 1.8836286116088283, 5.0252212651986214, 1.257964041980965, 1.9193749064347827, 4.3638104007448035, 1.2222177471550106, 5.060967560024576, 0.33285478436922783, 5.950330522810359, 2.8087378692205656, 3.4744474379590207, 4.4363595306593, 1.8468257765202865, 4.988418430110079, 1.2947668770695067, 4.5463100377260695, 1.7368752694535163, 4.878467923043309, 1.404717384136277, 0.1500093124051802, 6.1331759947744064, 2.991583341184613, 3.2916019659949733, 3.9786382254207737, 2.3045470817588125, 5.446139735348606, 0.8370455718309808, 0.3881421332626837, 5.8950431739169025, 2.7534505203271094, 3.529734786852477, 0.6058525335553483, 5.677332773624238, 2.535740120034445, 3.747445187145141, 3.685013702523128, 2.598171604656458, 5.739764258246251, 0.5434210489333353, 5.416854777248093, 0.8663305299314932, 4.007923183521286, 2.2752621236583, 4.298338249585249, 1.9848470575943375, 5.126439711184131, 1.1567455959954558, 2.3239203913376283, 3.959264915841958, 0.817672262252165, 5.465513044927421, 5.318580389105739, 0.9646049180738474, 4.106197571663641, 2.176987735515946, 2.188690196328243, 4.094495110851343, 0.9529024572615502, 5.330282849918036, 3.717897812019414, 2.565287495160172, 5.706880148749965, 0.5763051584296215, 1.1232942687449117, 5.159891038434674, 2.018298384844882, 4.264886922334704, 4.2056613215936265, 2.0775239855859597, 5.219116639175753, 1.0640686680038332, 3.2980253280517142, 2.985159979127872, 6.126752632717665, 0.15643267446192116, 4.416729710958001, 1.8664555962215852, 5.008048249811378, 1.2751370573682081, 1.2939466911973065, 4.98923861598228, 1.8476459623924866, 4.4355393447871, 6.229344859512386, 0.05384044766720016, 3.195433101256993, 3.0877522059225933, 0.20788181519614665, 6.07530349198344, 2.9337108383936465, 3.3494744687859397, 4.887513180254119, 1.3956721269254664, 4.53726478051526, 1.7459205266643267]
weights: [0.8394383394376632, 0.8394383394376632, 0.8394383394376632, 0.8394383394376632, 0.8394383394376632, 0.8394383394376632, 0.8394383394376632, 0.8394383394376632, 0.8394383394376632, 0.8394383394376632, 0.8394383394376632, 0.8394383394376632, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 0.9953502694079881, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791, 1.0367620627044791]

symmetries =  [1.         0.         0.         0.         0.         0.
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 0.         0.         0.         0.         0.         0.
 0.         0.         0.02482586 0.06994034 0.02790389]
symmetry error = 0.0
minimal relative intensity = 0.999907 rms = 0.00002414

N = 150, l_max = 20

best uniform solution: 216.20

150 points (theta,phi) 17 different weights:
theta: [0.9553166181238711, 2.1862760354650614, 2.186276035465922, 0.955316618124732, 1.4488565525750694, 1.692736101014724, 1.692736101014724, 1.4488565525750694, 0.3748329417704343, 2.766759711819359, 2.766759711819359, 0.3748329417704343, 1.218216503904536, 1.923376149685257, 1.923376149685257, 1.218216503904536, 0.38981483080935847, 2.7517778227804346, 2.7517778227804346, 0.38981483080935847, 1.3481048677927214, 1.793487785797072, 1.793487785797072, 1.3481048677927214, 1.8852011965822728, 1.2563914570075205, 1.2563914570075205, 1.8852011965822728, 1.1024079009695882, 2.039184752620205, 2.039184752620205, 1.1024079009695882, 1.2295823980635832, 1.91201025552621, 1.91201025552621, 1.2295823980635832, 0.5967374501640564, 2.544855203425737, 2.544855203425737, 0.5967374501640564, 1.440812520953901, 1.700780132635892, 1.700780132635892, 1.440812520953901, 3.001659455380831, 0.1399331982089623, 0.1399331982089623, 3.001659455380831, 1.5192664760528887, 1.6223261775369044, 1.6223261775369044, 1.5192664760528887, 2.1862760354650614, 0.955316618124732, 1.8067446002993213, 1.334848053290472, 2.534412676432118, 0.6071799771576751, 0.6071799771576751, 2.534412676432118, 1.0233990294693855, 2.1181936241204076, 1.334848053290472, 1.8067446002993213, 2.1181936241204076, 1.0233990294693855, 2.2646421384278734, 0.8769505151619198, 0.8726738386729693, 2.268918814916824, 0.8769505151619198, 2.2646421384278734, 2.0061730860178333, 1.13541956757196, 1.13541956757196, 2.0061730860178333, 2.268918814916824, 0.8726738386729693, 1.4373653819591312, 1.704227271630662, 2.0476240863593427, 1.0939685672304504, 1.0939685672304504, 2.0476240863593427, 0.4982114099021298, 2.6433812436876636, 2.6433812436876636, 0.4982114099021298, 1.704227271630662, 1.4373653819591312, 2.0469990862675775, 1.0945935673222156, 2.003573108935463, 1.1380195446543302, 1.1380195446543302, 2.003573108935463, 2.4711769789386326, 0.6704156746511606, 1.0945935673222156, 2.0469990862675775, 0.6704156746511606, 2.4711769789386326, 0.5859987268297736, 2.55559392676002, 2.150392814897282, 0.9911998386925109, 2.55559392676002, 0.5859987268297736, 1.4940152012748016, 1.6475774523149918, 1.6475774523149918, 1.4940152012748016, 0.9911998386925109, 2.150392814897282, 2.3496980396588736, 0.7918946139309196, 2.2917520056779064, 0.8498406479118867, 0.8498406479118867, 2.2917520056779064, 1.8400462250921643, 1.301546428497629, 0.7918946139309196, 2.3496980396588736, 1.301546428497629, 1.8400462250921643, 2.836271341802047, 0.305321311787746, 1.475929161076473, 1.6656634925133202, 1.6656634925133202, 1.475929161076473, 2.836271341802047, 0.305321311787746, 1.281492918869404, 1.8600997347203894, 1.8600997347203894, 1.281492918869404, 1.659968665473981, 1.4816239881158122, 1.4816239881158122, 1.659968665473981, 0.8640738608205584, 2.277518792769235, 2.426850194481685, 0.7147424591081082, 0.7147424591081082, 2.426850194481685, 2.277518792769235, 0.8640738608205584]
phi: [2.356194490192209, 3.926990816986496, 3.9269908169873773, 2.35619449019309, 1.2154624362781967, 5.06772287090139, 1.9261302173115966, 4.357055089867989, 1.2321215327281185, 5.0510637744514675, 1.9094711208616748, 4.373714186317912, 3.271569131080278, 3.011616176099308, 6.153208829689101, 0.12997647749048508, 2.521424207363172, 3.761761099816414, 0.6201684462266212, 5.663016860952965, 5.960535890100844, 0.3226494170787424, 3.4642420706685355, 2.8189432365110507, 1.3364166488995115, 4.9467686582800745, 1.8051760046902816, 4.478009302489305, 0.38442084658515413, 5.898764460594432, 2.757171807004639, 3.526013500174947, 4.212796890979881, 2.070388416199705, 5.2119810697894975, 1.0712042373900885, 4.074518853115688, 2.2086664540638985, 5.350259107653692, 0.9329261995258948, 3.089624004849475, 3.193561302330111, 0.05196864874031805, 6.231216658439268, 1.1925545109360942, 5.090630796243492, 1.9490381426536991, 4.334147164525887, 4.842546533624475, 1.4406387735551116, 4.5822314271449045, 1.7009538800346815, 0.7853981633967033, 5.497787143782883, 5.718324749212426, 0.5648605579671595, 2.7194497781554983, 3.563735529024088, 0.4221428754342951, 5.861042431745291, 4.435081040946935, 1.8481042662326512, 3.7064532115569526, 2.5767320956226336, 4.989696919822444, 1.293488387357142, 5.702508007442313, 0.5806772997372731, 4.129361332645942, 2.153823974533644, 3.722269953327066, 2.5609153538525202, 2.353634976688913, 3.9295503304906734, 0.7879576769008803, 5.4952276302787055, 5.295416628123437, 0.9877686790561494, 2.052262619162259, 4.230922688017327, 3.2918957543464495, 2.9912895528331367, 6.13288220642293, 0.15030310075665662, 4.994517909812761, 1.2886673973668252, 4.430260050956618, 1.8529252562229679, 1.089330034427534, 5.193855272752052, 1.0793570561250498, 5.203828251054537, 3.6709316788964705, 2.6122536282831157, 5.753846281872908, 0.5293390253066776, 0.7409816343407658, 5.54220367283882, 2.0622355974647433, 4.2209497097148425, 2.4006110192490273, 3.882574287930559, 6.144036334071883, 0.13914897310770352, 1.478989155721619, 4.804196151457967, 3.2807416266974965, 3.0024436804820898, 5.701653319453496, 0.5815319877260906, 3.7231246413158834, 2.560060665863703, 1.6626034978681743, 4.620581809311412, 4.329316037093319, 1.9538692700862677, 3.5035688013582, 2.7796165058213864, 5.9212091594111795, 0.361976147768407, 3.9580943300817455, 2.3250909770978407, 5.09546192367606, 1.1877233835035255, 5.466683630687633, 0.8165016764919526, 2.8210096537896843, 3.462175653389902, 0.2906481301420128, 5.9925371770375735, 2.8509445234477804, 3.432240783731806, 5.962602307379477, 0.32058299980010896, 4.613395258871833, 1.6697900483077528, 4.811382701897546, 1.4718026052820403, 2.2809298157528275, 4.002255491426759, 0.8606628378369656, 5.422522469342621, 0.11736972959670589, 6.1658155775828805, 1.7070907962658572, 4.5760945109137285, 1.434501857323936, 4.84868344985565, 3.0242229239930873, 3.258962383186499]
weights: [0.46282374388171366, 0.48400722862296824, 0.4913450890171869, 0.5125285737499722, 0.9053311778137941, 0.9053311778137941, 0.9053311778137941, 0.9053311778137941, 0.9053311778137941, 0.9053311778137941, 0.9053311778137941, 0.9053311778137941, 0.9053311778137941, 0.9053311778137941, 0.9053311778137941, 0.9053311778137941, 0.9366711338673447, 0.9366711338673447, 0.9366711338673447, 0.9366711338673447, 0.9366711338673447, 0.9366711338673447, 0.9366711338673447, 0.9366711338673447, 0.9366711338673447, 0.9366711338673447, 0.9366711338673447, 0.9366711338673447, 0.9381528731873874, 0.9381528731873874, 0.9381528731873874, 0.9381528731873874, 0.9381528731873874, 0.9381528731873874, 0.9381528731873874, 0.9381528731873874, 0.9381528731873874, 0.9381528731873874, 0.9381528731873874, 0.9381528731873874, 0.9678340997470184, 0.9678340997470184, 0.9678340997470184, 0.9678340997470184, 0.9678340997470184, 0.9678340997470184, 0.9678340997470184, 0.9678340997470184, 0.9678340997470184, 0.9678340997470184, 0.9678340997470184, 0.9678340997470184, 0.9753523176274022, 0.9753523176274022, 1.0020023033591663, 1.0020023033591663, 1.0020023033591663, 1.0020023033591663, 1.0020023033591663, 1.0020023033591663, 1.0020023033591663, 1.0020023033591663, 1.0020023033591663, 1.0020023033591663, 1.0020023033591663, 1.0020023033591663, 1.0127190569547055, 1.0127190569547055, 1.0127190569547055, 1.0127190569547055, 1.0127190569547055, 1.0127190569547055, 1.0127190569547055, 1.0127190569547055, 1.0127190569547055, 1.0127190569547055, 1.0127190569547055, 1.0127190569547055, 1.021841503636969, 1.021841503636969, 1.021841503636969, 1.021841503636969, 1.021841503636969, 1.021841503636969, 1.021841503636969, 1.021841503636969, 1.021841503636969, 1.021841503636969, 1.021841503636969, 1.021841503636969, 1.0241223405719746, 1.0241223405719746, 1.0241223405719746, 1.0241223405719746, 1.0241223405719746, 1.0241223405719746, 1.0241223405719746, 1.0241223405719746, 1.0241223405719746, 1.0241223405719746, 1.0241223405719746, 1.0241223405719746, 1.0519778574450065, 1.0519778574450065, 1.0519778574450065, 1.0519778574450065, 1.0519778574450065, 1.0519778574450065, 1.0519778574450065, 1.0519778574450065, 1.0519778574450065, 1.0519778574450065, 1.0519778574450065, 1.0519778574450065, 1.0751986557797482, 1.0751986557797482, 1.0751986557797482, 1.0751986557797482, 1.0751986557797482, 1.0751986557797482, 1.0751986557797482, 1.0751986557797482, 1.0751986557797482, 1.0751986557797482, 1.0751986557797482, 1.0751986557797482, 1.08489108807267, 1.08489108807267, 1.08489108807267, 1.08489108807267, 1.08489108807267, 1.08489108807267, 1.08489108807267, 1.08489108807267, 1.08489108807267, 1.08489108807267, 1.08489108807267, 1.08489108807267, 1.15414047035366, 1.15414047035366, 1.15414047035366, 1.15414047035366, 1.15414047035366, 1.15414047035366, 1.15414047035366, 1.15414047035366, 1.15414047035366, 1.15414047035366, 1.15414047035366, 1.15414047035366]

symmetries =  [1.         0.         0.         0.         0.         0.
 0.         0.         0.         0.         0.         0.
 0.         0.         0.         0.         0.         0.
 0.         0.         0.         0.055029   0.02344959 0.02075174]
symmetry error = 0.0
minimal relative intensity = 0.999937 rms = 0.00001155

N = 168, l_max = 21

best uniform solution: 240.21

168 points (theta,phi) 14 different weights:
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weights: [0.8267809740274884, 0.8267809740274884, 0.8267809740274884, 0.8267809740274884, 0.8267809740274884, 0.8267809740274884, 0.8267809740274884, 0.8267809740274884, 0.8267809740274884, 0.8267809740274884, 0.8267809740274884, 0.8267809740274884, 0.829599142203449, 0.829599142203449, 0.829599142203449, 0.829599142203449, 0.829599142203449, 0.829599142203449, 0.829599142203449, 0.829599142203449, 0.829599142203449, 0.829599142203449, 0.829599142203449, 0.829599142203449, 0.9189145330365971, 0.9189145330365971, 0.9189145330365971, 0.9189145330365971, 0.9189145330365971, 0.9189145330365971, 0.9189145330365971, 0.9189145330365971, 0.9189145330365971, 0.9189145330365971, 0.9189145330365971, 0.9189145330365971, 0.921650275068613, 0.921650275068613, 0.921650275068613, 0.921650275068613, 0.921650275068613, 0.921650275068613, 0.921650275068613, 0.921650275068613, 0.921650275068613, 0.921650275068613, 0.921650275068613, 0.921650275068613, 0.945166318103689, 0.945166318103689, 0.945166318103689, 0.945166318103689, 0.945166318103689, 0.945166318103689, 0.945166318103689, 0.945166318103689, 0.945166318103689, 0.945166318103689, 0.945166318103689, 0.945166318103689, 1.015449080924561, 1.015449080924561, 1.015449080924561, 1.015449080924561, 1.015449080924561, 1.015449080924561, 1.015449080924561, 1.015449080924561, 1.015449080924561, 1.015449080924561, 1.015449080924561, 1.015449080924561, 1.0339409342298937, 1.0339409342298937, 1.0339409342298937, 1.0339409342298937, 1.0339409342298937, 1.0339409342298937, 1.0339409342298937, 1.0339409342298937, 1.0339409342298937, 1.0339409342298937, 1.0339409342298937, 1.0339409342298937, 1.0354290409249134, 1.0354290409249134, 1.0354290409249134, 1.0354290409249134, 1.0354290409249134, 1.0354290409249134, 1.0354290409249134, 1.0354290409249134, 1.0354290409249134, 1.0354290409249134, 1.0354290409249134, 1.0354290409249134, 1.051139547234463, 1.051139547234463, 1.051139547234463, 1.051139547234463, 1.051139547234463, 1.051139547234463, 1.051139547234463, 1.051139547234463, 1.051139547234463, 1.051139547234463, 1.051139547234463, 1.051139547234463, 1.0689995476113516, 1.0689995476113516, 1.0689995476113516, 1.0689995476113516, 1.0689995476113516, 1.0689995476113516, 1.0689995476113516, 1.0689995476113516, 1.0689995476113516, 1.0689995476113516, 1.0689995476113516, 1.0689995476113516, 1.0819425182081583, 1.0819425182081583, 1.0819425182081583, 1.0819425182081583, 1.0819425182081583, 1.0819425182081583, 1.0819425182081583, 1.0819425182081583, 1.0819425182081583, 1.0819425182081583, 1.0819425182081583, 1.0819425182081583, 1.0857741187618235, 1.0857741187618235, 1.0857741187618235, 1.0857741187618235, 1.0857741187618235, 1.0857741187618235, 1.0857741187618235, 1.0857741187618235, 1.0857741187618235, 1.0857741187618235, 1.0857741187618235, 1.0857741187618235, 1.0859038293575851, 1.0859038293575851, 1.0859038293575851, 1.0859038293575851, 1.0859038293575851, 1.0859038293575851, 1.0859038293575851, 1.0859038293575851, 1.0859038293575851, 1.0859038293575851, 1.0859038293575851, 1.0859038293575851, 1.0993101403074141, 1.0993101403074141, 1.0993101403074141, 1.0993101403074141, 1.0993101403074141, 1.0993101403074141, 1.0993101403074141, 1.0993101403074141, 1.0993101403074141, 1.0993101403074141, 1.0993101403074141, 1.0993101403074141]

symmetries =  [1.         0.         0.         0.         0.         0.
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 0.         0.         0.         0.         0.05442633 0.00292924
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symmetry error = 0.0
minimal relative intensity = 0.999915 rms = 0.00001471

N = 180, l_max = 22:

best uniform solution: 266.22

180 points (theta,phi) 15 different weights:
theta: [1.0694616637229344, 2.0721309898668587, 2.0721309898668587, 1.0694616637229344, 2.4745548730316203, 0.667037780558173, 0.667037780558173, 2.4745548730316203, 1.9709669412663824, 1.1706257123234107, 1.1706257123234107, 1.9709669412663824, 0.9236471327820441, 2.217945520807749, 2.217945520807749, 0.9236471327820441, 1.1119586712234535, 2.02963398236634, 2.02963398236634, 1.1119586712234535, 0.845212564160403, 2.29638008942939, 2.29638008942939, 0.845212564160403, 1.3271743092137043, 1.8144183443760888, 1.8144183443760888, 1.3271743092137043, 0.7050742351730399, 2.436518418416753, 2.436518418416753, 0.7050742351730399, 2.2162071993122163, 0.9253854542775771, 0.9253854542775771, 2.2162071993122163, 0.9683874256074632, 2.17320522798233, 2.17320522798233, 0.9683874256074632, 2.5390941631010038, 0.6024984904887895, 0.6024984904887895, 2.5390941631010038, 1.5616496679564986, 1.5799429856332947, 1.5799429856332947, 1.5616496679564986, 2.0596210344122783, 1.0819716191775148, 1.0819716191775148, 2.0596210344122783, 2.3334057497197764, 0.8081869038700168, 0.8081869038700168, 2.3334057497197764, 2.1529073825698495, 0.988685271019944, 0.988685271019944, 2.1529073825698495, 1.9152540133073996, 1.2263386402823935, 1.2263386402823935, 1.9152540133073996, 1.5963568311864536, 1.5452358224033398, 1.5452358224033398, 1.5963568311864536, 2.7961087215487614, 0.345483932041032, 0.345483932041032, 2.7961087215487614, 2.622808088925906, 0.5187845646638876, 0.5187845646638876, 2.622808088925906, 1.0859844135005372, 2.055608240089256, 2.055608240089256, 1.0859844135005372, 1.7408702334143682, 1.4007224201754251, 1.4007224201754251, 1.7408702334143682, 2.3434670347217863, 0.7981256188680069, 0.7981256188680069, 2.3434670347217863, 2.3276346324337296, 0.8139580211560636, 0.8139580211560636, 2.3276346324337296, 1.3662211612802924, 1.7753714923095008, 1.7753714923095008, 1.3662211612802924, 2.465741566541754, 0.6758510870480394, 0.6758510870480394, 2.465741566541754, 2.2441045638737958, 0.8974880897159974, 0.8974880897159974, 2.2441045638737958, 1.6206230295475899, 1.5209696240422035, 1.5209696240422035, 1.6206230295475899, 2.1481289975338225, 0.9934636560559709, 0.9934636560559709, 2.1481289975338225, 0.6789260233921652, 2.462666630197628, 2.462666630197628, 0.6789260233921652, 1.2550218094623429, 1.8865708441274505, 1.8865708441274505, 1.2550218094623429, 0.13798011659946785, 3.0036125369903255, 3.0036125369903255, 0.13798011659946785, 1.4792319584622189, 1.6623606951275745, 1.6623606951275745, 1.4792319584622189, 1.6737270386963397, 1.4678656148934535, 1.4678656148934535, 1.6737270386963397, 2.6100689780598465, 0.5315236755299467, 0.5315236755299467, 2.6100689780598465, 1.3244895585309868, 1.8171030950588065, 1.8171030950588065, 1.3244895585309868, 2.0312414758566204, 1.1103511777331727, 1.1103511777331727, 2.0312414758566204, 1.9387082169738812, 1.202884436615912, 1.202884436615912, 1.9387082169738812, 0.4786621045728022, 2.662930549016991, 2.662930549016991, 0.4786621045728022, 1.2789486765659934, 1.8626439770238, 1.8626439770238, 1.2789486765659934, 1.1893877046181582, 1.9522049489716349, 1.9522049489716349, 1.1893877046181582, 1.6642546845036188, 1.4773379690861743, 1.4773379690861743, 1.6642546845036188, 0.39385219838236946, 2.7477404552074236, 2.7477404552074236, 0.39385219838236946, 0.27512831498362017, 2.866464338606173, 2.866464338606173, 0.27512831498362017, 1.362401532593612, 1.7791911209961813, 1.7791911209961813, 1.362401532593612, 1.3938012814032787, 1.7477913721865146, 1.7477913721865146, 1.3938012814032787]
phi: [0.4603293565035941, 5.822855950675992, 2.681263297086199, 3.601922010093387, 2.251972432629995, 4.031212874549592, 0.8896202209597984, 5.3935650862197875, 2.1197832360669713, 4.163402071112615, 1.0218094175228218, 5.261375889656764, 4.12383908342675, 2.1593462237528365, 5.300938877342629, 0.9822464298369568, 0.7375401895607402, 5.545645117618846, 2.404052464029053, 3.879132843150533, 3.7751704180243015, 2.5080148891552847, 5.649607542745078, 0.6335777644345088, 5.614659390090769, 0.6685259170888175, 3.8101185706786103, 2.473066736500976, 2.7602144087658127, 3.5229708984137735, 0.38137824482398064, 5.901807062355606, 4.405644034091154, 1.8775412730884327, 5.019133926678226, 1.2640513805013607, 4.723489730722699, 1.5596955764568867, 4.70128823004668, 1.5818970771329064, 3.157733262780489, 3.1254520443990974, 6.26704469798889, 0.01614060919069603, 2.539154985031115, 3.7440303221484714, 0.6024376685586785, 5.6807476386209075, 3.813804180732337, 2.4693811264472494, 5.6109737800370425, 0.6722115271425436, 0.7068844252822215, 5.576300881897365, 2.4347082283075716, 3.8484770788720146, 0.9737806147387331, 5.309404692440853, 2.16781203885106, 4.115373268328526, 1.543640283390513, 4.7395450237890735, 1.5979523701992802, 4.685232936980306, 0.34457491700971526, 5.938610390169871, 2.797017736580078, 3.486167570599508, 3.2171334460407053, 3.066051861138881, 6.207644514728674, 0.0755407924509122, 1.91916055149843, 4.364024755681156, 1.2224321020913633, 5.060753205088223, 3.334079205908941, 2.949106101270645, 6.090698754860438, 0.19248655231914819, 5.204927647703959, 1.0782576594756272, 4.21985031306542, 2.063334994114166, 5.995523028429524, 0.2876622787500619, 3.429254932339855, 2.8539303748397313, 1.8540005287268906, 4.429184778452695, 1.2875921248629028, 4.995593182316684, 3.918740854223359, 2.3644444529562274, 5.506037106546021, 0.7771482006335657, 4.632686415627793, 1.6504988915517933, 4.792091545141586, 1.4910937620379998, 3.2053457888517825, 3.077839518327804, 6.219432171917597, 0.06375313526198968, 4.038089158941206, 2.2450961482383804, 5.386688801828173, 0.8964965053514129, 3.5212721377524674, 2.761913169427119, 5.9035058230169115, 0.3796794841626744, 5.229699340607554, 1.0534859665720326, 4.1950786201618255, 2.0881066870177607, 2.182399272919814, 4.100786034259772, 0.9591933806699791, 5.323991926509607, 5.439596365343218, 0.8435889418363687, 3.9851815954261616, 2.2980037117534247, 1.6741615839056612, 4.609023723273925, 1.467431069684132, 4.8157542374954545, 0.09205294591243589, 6.19113236126715, 3.0495397076773574, 3.233645599502229, 5.7813222236648585, 0.5018630835147282, 3.6434557371045213, 2.639729570075065, 0.4759407783817402, 5.807244528797846, 2.665651875208053, 3.617533431971533, 4.98803534344231, 1.2951499637372763, 4.436742617327069, 1.846442689852517, 4.398923189299026, 1.8842621178805596, 5.025854771470352, 1.2573305357092337, 5.608464775360454, 0.6747205318191327, 3.8163131854089256, 2.4668721217706606, 2.7566043282950536, 3.5265809788845326, 0.3849883252947394, 5.898196981884847, 6.182467519898546, 0.10071778728104003, 3.242310440870833, 3.0408748663087533, 1.187629221364918, 5.0955560858146685, 1.9539634322248753, 4.329221874954711, 1.8164457017177253, 4.466739605461861, 1.325146951872068, 4.958038355307519, 0.8656961041224297, 5.417489203057157, 2.2758965494673635, 4.007288757712223, 0.18095194781573742, 6.102233359363849, 2.960640705774056, 3.3225446014055304, 1.3590442347570904, 4.924141072422496, 1.782548418832703, 4.500636888346883]
weights: [0.8356503130052707, 0.8356503130052707, 0.8356503130052707, 0.8356503130052707, 0.8356503130052707, 0.8356503130052707, 0.8356503130052707, 0.8356503130052707, 0.8356503130052707, 0.8356503130052707, 0.8356503130052707, 0.8356503130052707, 0.9224436883584962, 0.9224436883584962, 0.9224436883584962, 0.9224436883584962, 0.9224436883584962, 0.9224436883584962, 0.9224436883584962, 0.9224436883584962, 0.9224436883584962, 0.9224436883584962, 0.9224436883584962, 0.9224436883584962, 0.9272606671503179, 0.9272606671503179, 0.9272606671503179, 0.9272606671503179, 0.9272606671503179, 0.9272606671503179, 0.9272606671503179, 0.9272606671503179, 0.9272606671503179, 0.9272606671503179, 0.9272606671503179, 0.9272606671503179, 0.9375108753867002, 0.9375108753867002, 0.9375108753867002, 0.9375108753867002, 0.9375108753867002, 0.9375108753867002, 0.9375108753867002, 0.9375108753867002, 0.9375108753867002, 0.9375108753867002, 0.9375108753867002, 0.9375108753867002, 0.9564541780270265, 0.9564541780270265, 0.9564541780270265, 0.9564541780270265, 0.9564541780270265, 0.9564541780270265, 0.9564541780270265, 0.9564541780270265, 0.9564541780270265, 0.9564541780270265, 0.9564541780270265, 0.9564541780270265, 0.9700919256720218, 0.9700919256720218, 0.9700919256720218, 0.9700919256720218, 0.9700919256720218, 0.9700919256720218, 0.9700919256720218, 0.9700919256720218, 0.9700919256720218, 0.9700919256720218, 0.9700919256720218, 0.9700919256720218, 0.9872790407535827, 0.9872790407535827, 0.9872790407535827, 0.9872790407535827, 0.9872790407535827, 0.9872790407535827, 0.9872790407535827, 0.9872790407535827, 0.9872790407535827, 0.9872790407535827, 0.9872790407535827, 0.9872790407535827, 1.0170286018029293, 1.0170286018029293, 1.0170286018029293, 1.0170286018029293, 1.0170286018029293, 1.0170286018029293, 1.0170286018029293, 1.0170286018029293, 1.0170286018029293, 1.0170286018029293, 1.0170286018029293, 1.0170286018029293, 1.0285631801654453, 1.0285631801654453, 1.0285631801654453, 1.0285631801654453, 1.0285631801654453, 1.0285631801654453, 1.0285631801654453, 1.0285631801654453, 1.0285631801654453, 1.0285631801654453, 1.0285631801654453, 1.0285631801654453, 1.04540144692826, 1.04540144692826, 1.04540144692826, 1.04540144692826, 1.04540144692826, 1.04540144692826, 1.04540144692826, 1.04540144692826, 1.04540144692826, 1.04540144692826, 1.04540144692826, 1.04540144692826, 1.0455065532071963, 1.0455065532071963, 1.0455065532071963, 1.0455065532071963, 1.0455065532071963, 1.0455065532071963, 1.0455065532071963, 1.0455065532071963, 1.0455065532071963, 1.0455065532071963, 1.0455065532071963, 1.0455065532071963, 1.0635717263859783, 1.0635717263859783, 1.0635717263859783, 1.0635717263859783, 1.0635717263859783, 1.0635717263859783, 1.0635717263859783, 1.0635717263859783, 1.0635717263859783, 1.0635717263859783, 1.0635717263859783, 1.0635717263859783, 1.0757332273159554, 1.0757332273159554, 1.0757332273159554, 1.0757332273159554, 1.0757332273159554, 1.0757332273159554, 1.0757332273159554, 1.0757332273159554, 1.0757332273159554, 1.0757332273159554, 1.0757332273159554, 1.0757332273159554, 1.083848454431528, 1.083848454431528, 1.083848454431528, 1.083848454431528, 1.083848454431528, 1.083848454431528, 1.083848454431528, 1.083848454431528, 1.083848454431528, 1.083848454431528, 1.083848454431528, 1.083848454431528, 1.1036561214092937, 1.1036561214092937, 1.1036561214092937, 1.1036561214092937, 1.1036561214092937, 1.1036561214092937, 1.1036561214092937, 1.1036561214092937, 1.1036561214092937, 1.1036561214092937, 1.1036561214092937, 1.1036561214092937]

symmetries =  [1.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
 0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
 0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
 0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
 0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
 0.00000000e+00 0.00000000e+00 0.00000000e+00 3.82841069e-02
 3.50973954e-02 7.33076061e-04]
symmetry error = 0.0
minimal relative intensity = 0.999948 rms = 0.00000885

N = 192, l_max = 23:

best uniform solution: 278.23

192 points (theta,phi) 4 different weights:
theta: [1.5707963267935952, 1.5707963267961982, 1.5707963267961982, 1.5707963267935952, 1.0172219678993604, 2.124370685690433, 2.124370685690433, 1.0172219678993604, 0.5535743588955363, 2.588018294694257, 2.588018294694257, 0.5535743588955363, 2.1464652349718594, 0.9951274186179337, 0.9951274186179337, 2.1464652349718594, 2.503549185843261, 0.6380434677465324, 0.6380434677465324, 2.503549185843261, 1.3267147292639176, 1.8148779243258755, 1.8148779243258755, 1.3267147292639176, 0.7992449725143074, 2.342347681075486, 2.342347681075486, 0.7992449725143074, 2.7306273587309704, 0.41096529485882277, 0.41096529485882277, 2.7306273587309704, 1.7431839770582105, 1.3984086765315829, 1.3984086765315829, 1.7431839770582105, 1.93991423789508, 1.2016784156947133, 1.2016784156947133, 1.93991423789508, 1.6066884025715498, 1.5349042510182436, 1.5349042510182436, 1.6066884025715498, 2.3687531618924687, 0.7728394916973245, 0.7728394916973245, 2.3687531618924687, 1.7920854421396617, 1.3495072114501316, 1.3495072114501316, 1.7920854421396617, 0.35133711226740955, 2.7902555413223835, 2.7902555413223835, 0.35133711226740955, 1.23988132212481, 1.9017113314649832, 1.9017113314649832, 1.23988132212481, 1.4570784237609438, 1.6845142298288494, 1.6845142298288494, 1.4570784237609438, 0.9242189870895275, 2.217373666500266, 2.217373666500266, 0.9242189870895275, 0.6960540389542376, 2.4455386146355558, 2.4455386146355558, 0.6960540389542376, 1.0658765344148418, 2.0757161191749516, 2.0757161191749516, 1.0658765344148418, 0.7984031295940905, 2.3431895239957026, 2.3431895239957026, 0.7984031295940905, 1.6593112355362603, 1.482281418053533, 1.482281418053533, 1.6593112355362603, 2.026179888972515, 1.1154127646172782, 1.1154127646172782, 2.026179888972515, 0.8933947658043098, 2.2481978877854836, 2.2481978877854836, 0.8933947658043098, 1.2207977393465268, 1.9207949142432663, 1.9207949142432663, 1.2207977393465268, 0.24693194716432712, 2.894660706425466, 2.894660706425466, 0.24693194716432712, 1.7309605063133866, 1.4106321472764067, 1.4106321472764067, 1.7309605063133866, 1.800701446577565, 1.3408912070122283, 1.3408912070122283, 1.800701446577565, 1.0143097139670758, 2.1272829396227175, 2.1272829396227175, 1.0143097139670758, 2.0782228735666113, 1.063369780023182, 1.063369780023182, 2.0782228735666113, 0.8721032725558162, 2.269489381033977, 2.269489381033977, 0.8721032725558162, 0.5133531538291254, 2.6282394997606677, 2.6282394997606677, 0.5133531538291254, 1.641937236346669, 1.4996554172431242, 1.4996554172431242, 1.641937236346669, 0.3878178477303071, 2.753774805859486, 2.753774805859486, 0.3878178477303071, 2.513339623039212, 0.6282530305505813, 0.6282530305505813, 2.513339623039212, 1.1442981742373886, 1.9972944793524046, 1.9972944793524046, 1.1442981742373886, 1.1853770147850173, 1.956215638804776, 1.956215638804776, 1.1853770147850173, 0.5153586774838607, 2.6262339761059326, 2.6262339761059326, 0.5153586774838607, 2.3925957691743966, 0.7489968844153966, 0.7489968844153966, 2.3925957691743966, 2.232478002231715, 0.9091146513580781, 0.9091146513580781, 2.232478002231715, 1.2730055011948624, 1.868587152394931, 1.868587152394931, 1.2730055011948624, 1.2464523149230375, 1.8951403386667556, 1.8951403386667556, 1.2464523149230375, 1.1401204520635007, 2.0014722015262927, 2.0014722015262927, 1.1401204520635007, 2.1854884420180656, 0.9561042115717276, 0.9561042115717276, 2.1854884420180656, 2.487224681924875, 0.6543679716649181, 0.6543679716649181, 2.487224681924875, 1.766657176594097, 1.3749354769956963, 1.3749354769956963, 1.766657176594097, 3.002861200035678, 0.1387314535541154, 0.1387314535541154, 3.002861200035678, 1.6318948459458, 1.509697807643993, 1.509697807643993, 1.6318948459458, 1.6951929803788715, 1.4463996732109219, 1.4463996732109219, 1.6951929803788715]
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weights: [0.7996569902556139, 0.7996569902556139, 0.7996569902556139, 0.7996569902556139, 0.7996569902556139, 0.7996569902556139, 0.7996569902556139, 0.7996569902556139, 0.7996569902556139, 0.7996569902556139, 0.7996569902556139, 0.7996569902556139, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 0.9749144267519279, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0271129860257913, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585, 1.0380411891711585]

symmetries =  [1.         0.         0.         0.         0.         0.
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symmetry error = 0.0
minimal relative intensity = 0.999974 rms = 0.00000467

N = 216, l_max = 24:

best uniform solution: 314.24

216 points (theta,phi) 18 different weights:
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symmetries =  [1.         0.         0.         0.         0.         0.
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 0.         0.         0.         0.         0.         0.
 0.         0.         0.         0.         0.         0.
 0.         0.03516697 0.01730018 0.00939737]
symmetry error = 0.0
minimal relative intensity = 0.999976 rms = 0.00000395

N = 228, l_max = 25:

best uniform solution: 328.25

228 points (theta,phi) 19 different weights:
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symmetries =  [1.         0.         0.         0.         0.         0.
 0.         0.         0.         0.         0.         0.
 0.         0.         0.         0.         0.         0.
 0.         0.         0.         0.         0.         0.
 0.         0.         0.03028827 0.02204193 0.01180584]
symmetry error = 0.0
minimal relative intensity = 0.999978 rms = 0.00000355

N = 252, l_max = 26:

best uniform solution: 366.26

252 points (theta,phi) 21 different weights:
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symmetries =  [1.         0.         0.         0.         0.         0.
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symmetry error = 0.0
minimal relative intensity = 0.999989 rms = 0.00000166

N = 276, l_max = 27:

best uniform solution: 380.27

276 points (theta,phi) 23 different weights:
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symmetries =  [1.         0.         0.         0.         0.         0.
 0.         0.         0.         0.         0.         0.
 0.         0.         0.         0.         0.         0.
 0.         0.         0.         0.         0.         0.
 0.         0.         0.         0.         0.02035446 0.02764865
 0.00806988]
symmetry error = 0.0
minimal relative intensity = 0.999988 rms = 0.00000161