// exact solution below

L5:=[
[ 0.000000000000000000000000000000, 0.707106781186547524400844362105, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.707106781186547524400844362105, 0.000000000000000000000000000000 ],
[ 0.612372435695794524549321018677, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.790569415042094832999723386108, 0.000000000000000000000000000000 ],
[ 0.408248290463863016366214012451, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.912870929175276855761616304668, 0.000000000000000000000000000000, 0.000000000000000000000000000000 ],
[ 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.866025403784438646763723170753, 0.000000000000000000000000000000, 0.500000000000000000000000000000, 0.000000000000000000000000000000 ],
[ 0.707106781186547524400844362105, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.707106781186547524400844362105 ]
];

// squared length of the multipoles
[ 0.16667, 0.00000, 0.011905, 0.22222,  0.32143,  0.27778 ]
[ 0.16667, 0.00000, 0.018601, 0.21701,  0.31473,  0.28299 ]
[ 0.16667, 0.00000, 0.074405, 0.17361,  0.25893,  0.32639 ]
[ 0.16667, 0.00000, 0.24628,  0.039930, 0.087053, 0.46007 ]
[ 0.16667, 0.00000, 0.29762,  0.00000,  0.035714, 0.50000 ]

// cumulative squared lengths A_M:
         [ 0.00000, 0.011905, 0.23413,  0.55556,  0.83334 ]
         [ 0.00000, 0.018601, 0.23561,  0.55035,  0.83334 ]
         [ 0.00000, 0.074405, 0.24802,  0.50694,  0.83334 ]
         [ 0.00000, 0.24628,  0.28621,  0.37326,  0.83334 ]
         [ 0.00000, 0.29762,  0.29762,  0.33333,  0.83334 ]


L5_algebraic:=[
[                   0, 1/2*sqrt(2),           0,                           0,         1/2*sqrt(2), 0 ],
[ 1/4*sqrt(3)*sqrt(2), 0,                     0,                           0, 1/4*sqrt(5)*sqrt(2), 0 ],
[ 1/6*sqrt(3)*sqrt(2), 0,                     0, 1/6*sqrt(5)*sqrt(3)*sqrt(2),                   0, 0 ],
[                   0, 0,           1/2*sqrt(3),                           0,                 1/2, 0 ],
[         1/2*sqrt(2), 0,                     0,                           0,                   0, 1/2*sqrt(2) ]
];


// squared length of the multipoles
[ 1/6, 0,   1/84,      2/9,     9/28,    5/18  ]
[ 1/6, 0,  25/1344, 125/576,  141/448, 163/576 ]
[ 1/6, 0,  25/336,   25/144,   29/112,  47/144 ]
[ 1/6, 0, 331/1344,  23/576,   39/448, 265/576 ]
[ 1/6, 0,  25/84,     0,       1/28,     1/2   ]


// cumulative squared lengths A_M:
     [ 0,   1/84,    59/252,    5/9,     5/6 ]
     [ 0,  25/1344, 475/2016, 317/576,   5/6 ]
     [ 0,  25/336,  125/504,   73/144,   5/6 ]
     [ 0, 331/1344, 577/2016, 215/576,   5/6 ]
     [ 0,  25/84,    25/84,     1/3,     5/6 ]