// exact solution below

L8:=[
[ 0.456435464587638427880808152334, 0.0, 0.0, 0.0, 0.763762615825973334431341198955, 0.0, 0.0, 0.0, -0.456435464587638427880808152334 ],
[ 0.456435464587638427880808152334, 0.0, 0.0, 0.0, 0.763762615825973334431341198955, 0.0, 0.0, 0.0,  0.456435464587638427880808152334 ]
];

// squared length of the multipoles
[ 0.11111, 0.00000, 0.00000, 0.00000, 0.13326, 0.29914, 0.050505, 0.18697, 0.21902 ]
[ 0.11111, 0.00000, 0.00000, 0.00000, 0.22844, 0.00000, 0.40404,  0.00000, 0.25641 ]

// cumulative squared lengths A_M:
         [ 0.00000, 0.00000, 0.00000, 0.13326, 0.43240, 0.48291,  0.66987, 0.88889 ]
         [ 0.00000, 0.00000, 0.00000, 0.22844, 0.22844, 0.63248,  0.63248, 0.88889 ]


L8_algebraic:=[
[ 1/12*sqrt(5)*sqrt(3)*sqrt(2), 0, 0, 0, 1/6*sqrt(7)*sqrt(3), 0, 0, 0, -1/12*sqrt(5)*sqrt(3)*sqrt(2) ],
[ 1/12*sqrt(5)*sqrt(3)*sqrt(2), 0, 0, 0, 1/6*sqrt(7)*sqrt(3), 0, 0, 0,  1/12*sqrt(5)*sqrt(3)*sqrt(2) ]
];


// squared length of the multipoles
[ 1/9, 0, 0, 0, 343/2574,  35/117,   5/99,  175/936, 205/936 ]
[ 1/9, 0, 0, 0,  98/429,    0,      40/99,    0,      10/39 ]

// cumulative squared lengths A_M:
     [ 0, 0, 0, 343/2574, 371/858, 113/234, 209/312,   8/9 ]
     [ 0, 0, 0,  98/429,   98/429,  74/117,  74/117,   8/9 ]