// see below for algebraic expressions L15:=[ [ 0.319801074533415651447656598651, 0.0, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.630656223886891270306351315733, 0.000000000000000000000000000000, 0.0, 0.0, 0.0, 0.0, 0.000000000000000000000000000000, 0.630656223886891270306351315733, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.0, 0.319801074533415651447656598651 ], [ 0.319801074533415651447656598651, 0.0, 0.000000000000000000000000000000, 0.000000000000000000000000000000, -0.630656223886891270306351315733, 0.000000000000000000000000000000, 0.0, 0.0, 0.0, 0.0, 0.000000000000000000000000000000, -0.630656223886891270306351315733, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.0, 0.319801074533415651447656598651 ], [ 0.387298334620741688517926539978, 0.0, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.000000000000000000000000000000, -0.591607978309961604256732829156, 0.0, 0.0, 0.0, 0.0, -0.591607978309961604256732829156, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.0, 0.387298334620741688517926539978 ], [ 0.387298334620741688517926539978, 0.0, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.591607978309961604256732829156, 0.0, 0.0, 0.0, 0.0, 0.591607978309961604256732829156, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.0, 0.387298334620741688517926539978 ], [ 0.117851130197757920733474060351, 0.0, 0.000000000000000000000000000000, 0.697216688778396289981810021487*I, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.0, 0.0, 0.0, 0.0, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.697216688778396289981810021487*I, 0.000000000000000000000000000000, 0.0, 0.117851130197757920733474060351 ], [ 0.000000000000000000000000000000, 0.0, 0.559016994374947424102293417183*I, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.433012701892219323381861585377, 0.0, 0.0, 0.0, 0.0, 0.433012701892219323381861585377, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.559016994374947424102293417183*I, 0.0, 0.000000000000000000000000000000 ], [ 0.000000000000000000000000000000, 0.0, 0.559016994374947424102293417183*I, 0.000000000000000000000000000000, 0.000000000000000000000000000000, -0.433012701892219323381861585377, 0.0, 0.0, 0.0, 0.0, -0.433012701892219323381861585377, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.559016994374947424102293417183*I, 0.0, 0.000000000000000000000000000000 ] ] where I:=Sqrt(-1); // squared length of the multipoles [ 0.062500, 0.0, 0.0, 0.0, 0.0039210, 0.00000, 0.12873, 0.070268, 0.095619, 0.045774, 0.054520, 0.070664, 0.10654, 0.17545, 0.048176, 0.13784 ], [ 0.062500, 0.0, 0.0, 0.0, 0.0039210, 0.00000, 0.12873, 0.070268, 0.095619, 0.045774, 0.054520, 0.070664, 0.10654, 0.17545, 0.048176, 0.13784 ], [ 0.062500, 0.0, 0.0, 0.0, 0.020422, 0.022568, 0.033553, 0.16845, 0.033832, 0.023095, 0.065875, 0.14603, 0.14077, 0.017762, 0.14305, 0.12209 ], [ 0.062500, 0.0, 0.0, 0.0, 0.020422, 0.072804, 0.033553, 0.0046347, 0.033832, 0.23890, 0.065875, 0.00090344, 0.14077, 0.067698, 0.14305, 0.11506 ], [ 0.062500, 0.0, 0.0, 0.0, 0.077655, 0.00000, 0.056028, 0.00000, 0.057528, 0.12916, 0.096473, 0.085819, 0.13664, 0.11380, 0.013173, 0.17122 ], [ 0.062500, 0.0, 0.0, 0.0, 0.099514, 0.012659, 0.00072562, 0.0084089, 0.064569, 0.084784, 0.14075, 0.14636, 0.097241, 0.075511, 0.034705, 0.17228 ], [ 0.062500, 0.0, 0.0, 0.0, 0.099514, 0.012659, 0.00072562, 0.0084089, 0.064569, 0.084784, 0.14075, 0.14636, 0.097241, 0.075511, 0.034705, 0.17228 ] // cumulative squared lengths A_M: [ 0.0, 0.0, 0.0, 0.0039210, 0.0039210, 0.13265, 0.20292, 0.29853, 0.34431, 0.39883, 0.46949, 0.57603, 0.75148, 0.79966, 0.93750 ], [ 0.0, 0.0, 0.0, 0.0039210, 0.0039210, 0.13265, 0.20292, 0.29853, 0.34431, 0.39883, 0.46949, 0.57603, 0.75148, 0.79966, 0.93750 ], [ 0.0, 0.0, 0.0, 0.020422, 0.042990, 0.076542, 0.24499, 0.27882, 0.30192, 0.36779, 0.51382, 0.65460, 0.67236, 0.81541, 0.93750 ], [ 0.0, 0.0, 0.0, 0.020422, 0.093226, 0.12678, 0.13141, 0.16525, 0.40414, 0.47002, 0.47092, 0.61169, 0.67939, 0.82243, 0.93750 ], [ 0.0, 0.0, 0.0, 0.077655, 0.077655, 0.13368, 0.13368, 0.19121, 0.32037, 0.41684, 0.50266, 0.63931, 0.75311, 0.76628, 0.93750 ], [ 0.0, 0.0, 0.0, 0.099514, 0.11217, 0.11290, 0.12131, 0.18588, 0.27066, 0.41140, 0.55776, 0.65501, 0.73051, 0.76522, 0.93750 ], [ 0.0, 0.0, 0.0, 0.099514, 0.11217, 0.11290, 0.12131, 0.18588, 0.27066, 0.41140, 0.55776, 0.65501, 0.73051, 0.76522, 0.93750 ] //====================================================================== // exact values //====================================================================== L15_algebraic:=[ [ 3/44*sqrt(11)*sqrt(2), 0, 0, 0, 1/44*sqrt(11)*sqrt(7)*sqrt(5)*sqrt(2), 0, 0, 0, 0, 0, 0, 1/44*sqrt(11)*sqrt(7)*sqrt(5)*sqrt(2), 0, 0, 0, 3/44*sqrt(11)*sqrt(2) ], [ 3/44*sqrt(11)*sqrt(2), 0, 0, 0, -1/44*sqrt(11)*sqrt(7)*sqrt(5)*sqrt(2), 0, 0, 0, 0, 0, 0, -1/44*sqrt(11)*sqrt(7)*sqrt(5)*sqrt(2), 0, 0, 0, 3/44*sqrt(11)*sqrt(2) ], [ 1/10*sqrt(5)*sqrt(3), 0, 0, 0, 0, -1/10*sqrt(7)*sqrt(5), 0, 0, 0, 0, -1/10*sqrt(7)*sqrt(5), 0, 0, 0, 0, 1/10*sqrt(5)*sqrt(3) ], [ 1/10*sqrt(5)*sqrt(3), 0, 0, 0, 0, 1/10*sqrt(7)*sqrt(5), 0, 0, 0, 0, 1/10*sqrt(7)*sqrt(5), 0, 0, 0, 0, 1/10*sqrt(5)*sqrt(3) ], [ 1/12*sqrt(2), 0, 0, 1/12*sqrt(-1)*sqrt(7)*sqrt(5)*sqrt(2), 0, 0, 0, 0, 0, 0, 0, 0, 1/12*sqrt(-1)*sqrt(7)*sqrt(5)*sqrt(2), 0, 0, 1/12*sqrt(2) ], [ 0, 0, 1/4*sqrt(-1)*sqrt(5), 0, 0, 1/4*sqrt(3), 0, 0, 0, 0, 1/4*sqrt(3), 0, 0, 1/4*sqrt(-1)*sqrt(5), 0, 0 ], [ 0, 0, 1/4*sqrt(-1)*sqrt(5), 0, 0, -1/4*sqrt(3), 0, 0, 0, 0, -1/4*sqrt(3), 0, 0, 1/4*sqrt(-1)*sqrt(5), 0, 0 ] ]; // squared length of the multipoles [ 1/16, 0, 0, 0, 63749/16258528, 0, 160995/1250656, 91875/1307504, 95605/999856, 1575/34408, 178213/3268760, 3899/55176, 27580625/258885792, 113281/645656, 3668251/76142880, 533993/3873936 ], [ 1/16, 0, 0, 0, 63749/16258528, 0, 160995/1250656, 91875/1307504, 95605/999856, 1575/34408, 178213/3268760, 3899/55176, 27580625/258885792, 113281/645656, 3668251/76142880, 533993/3873936 ], [ 1/16, 0, 0, 0, 343/16796, -441/209950*sqrt(2)0*sqrt(11) + 40047/839800, 51/1520, 1323/193154*sqrt(2)0*sqrt(11) + 8358/96577, 961/28405, -441/48875*sqrt(2)0*sqrt(11) + 51219/391000, 2303/34960, 49/8075*sqrt(2)0*sqrt(11) + 2373/32300, 150595/1069776, -1323/633650*sqrt(2)0*sqrt(11) + 13538/316825, 2813/19665, 49/166750*sqrt(2)0*sqrt(11) + 19773/166750 ], [ 1/16, 0, 0, 0, 343/16796, 441/209950*sqrt(2)0*sqrt(11) + 40047/839800, 51/1520, -1323/193154*sqrt(2)0*sqrt(11) + 8358/96577, 961/28405, 441/48875*sqrt(2)0*sqrt(11) + 51219/391000, 2303/34960, -49/8075*sqrt(2)0*sqrt(11) + 2373/32300, 150595/1069776, 1323/633650*sqrt(2)0*sqrt(11) + 13538/316825, 2813/19665, -49/166750*sqrt(2)0*sqrt(11) + 19773/166750 ], [ 1/16, 0, 0, 0, 187817/2418624, 0, 93815/1674432, 0, 141185/2454192, 1925/14904, 516019/5348880, 1001/11664, 47361875/346607424, 2429/21344, 1342897/101943360, 115831/676512 ], [ 1/16, 0, 0, 0, 26743/268736, 1701/134368, 15/20672, 51975/6180928, 93905/1454336, 297027/3503360, 1338361/9509120, 254141/1736448, 1664425/17116416, 244979/3244288, 174713/5034240, 441259/2561280 ], [ 1/16, 0, 0, 0, 26743/268736, 1701/134368, 15/20672, 51975/6180928, 93905/1454336, 297027/3503360, 1338361/9509120, 254141/1736448, 1664425/17116416, 244979/3244288, 174713/5034240, 441259/2561280 ] // cumulative squared lengths A_M: [ 0, 0, 0, 63749/16258528, 63749/16258528, 539171/4064632, 37939991/186973072, 2146851/7191272, 1238013/3595636, 14340473/35956360, 12660991/26967270, 43860629/76142880, 1659379261/2208143520, 1548911/1936968, 15/16 ], [ 0, 0, 0, 63749/16258528, 63749/16258528, 539171/4064632, 37939991/186973072, 2146851/7191272, 1238013/3595636, 14340473/35956360, 12660991/26967270, 43860629/76142880, 1659379261/2208143520, 1548911/1936968, 15/16 ], [ 0, 0, 0, 343/16796, -441/209950*sqrt(2)0*sqrt(11) + 57197/839800, -441/209950*sqrt(2)0*sqrt(11) + 170749/1679600, 882/185725*sqrt(2)0*sqrt(11) + 7270427/38630800, 882/185725*sqrt(2)0*sqrt(11) + 659799/2971600, -3969/928625*sqrt(2)0*sqrt(11) + 5245317/14858000, -3969/928625*sqrt(2)0*sqrt(11) + 1556023/3714500, 98/54625*sqrt(2)0*sqrt(11) + 914459/1857250, 98/54625*sqrt(2)0*sqrt(11) + 4980319/7866000, -49/166750*sqrt(2)0*sqrt(11) + 154176611/228114000, -49/166750*sqrt(2)0*sqrt(11) + 1092441/1334000, 15/16 ], [ 0, 0, 0, 343/16796, 441/209950*sqrt(2)0*sqrt(11) + 57197/839800, 441/209950*sqrt(2)0*sqrt(11) + 170749/1679600, -882/185725*sqrt(2)0*sqrt(11) + 7270427/38630800, -882/185725*sqrt(2)0*sqrt(11) + 659799/2971600, 3969/928625*sqrt(2)0*sqrt(11) + 5245317/14858000, 3969/928625*sqrt(2)0*sqrt(11) + 1556023/3714500, -98/54625*sqrt(2)0*sqrt(11) + 914459/1857250, -98/54625*sqrt(2)0*sqrt(11) + 4980319/7866000, 49/166750*sqrt(2)0*sqrt(11) + 154176611/228114000, 49/166750*sqrt(2)0*sqrt(11) + 1092441/1334000, 15/16 ], [ 0, 0, 0, 187817/2418624, 187817/2418624, 727487/5441904, 727487/5441904, 460243/2406996, 514087/1604664, 6688927/16046640, 108891587/216629640, 65173063/101943360, 2226459617/2956357440, 518399/676512, 15/16 ], [ 0, 0, 0, 26743/268736, 30145/268736, 7585/67184, 749795/6180928, 353505/1901824, 4504047/16640960, 5476943/13312768, 397787/713184, 659489/1006848, 2666249/3649824, 1959941/2561280, 15/16 ], [ 0, 0, 0, 26743/268736, 30145/268736, 7585/67184, 749795/6180928, 353505/1901824, 4504047/16640960, 5476943/13312768, 397787/713184, 659489/1006848, 2666249/3649824, 1959941/2561280, 15/16 ]