// see below for algebraic expressions L9:=[ [ 0.408248290463863016366214012451, 0.000000000000000000000000000000, 0.000000000000000000000000000000, -0.577350269189625764509148780502, 0.000000000000000000000000000000, 0.000000000000000000000000000000, -0.577350269189625764509148780502, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.408248290463863016366214012451 ], [ 0.450693909432998661639902658434, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.684653196881457641821212228502, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.000000000000000000000000000000, -0.572821961869480000823505899216, 0.000000000000000000000000000000 ], [ 0.450693909432998661639902658434, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.684653196881457641821212228502, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.572821961869480000823505899216, 0.000000000000000000000000000000 ], [ 0.000000000000000000000000000000, 0.612372435695794524549321018677, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.707106781186547524400844362105, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.353553390593273762200422181052 ], [ 0.000000000000000000000000000000, 0.612372435695794524549321018677, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.707106781186547524400844362105, 0.000000000000000000000000000000, 0.000000000000000000000000000000, -0.353553390593273762200422181052 ], [ 0.408248290463863016366214012451, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.000000000000000000000000000000, -0.774596669241483377035853079957, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.483045891539647952457288805239 ], [ 0.408248290463863016366214012451, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.774596669241483377035853079957, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.000000000000000000000000000000, -0.483045891539647952457288805239 ], [ 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.745355992499929898803057889577, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.235702260395515841466948120702, 0.000000000000000000000000000000, 0.000000000000000000000000000000, -0.623609564462323564263958122052, 0.000000000000000000000000000000 ], [ 0.000000000000000000000000000000, 0.597614304667196819984408589847, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.577350269189625764509148780502, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.556348640264186790983913566314, 0.000000000000000000000000000000 ], [ 0.000000000000000000000000000000, 0.597614304667196819984408589847, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.000000000000000000000000000000, -0.577350269189625764509148780502, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.556348640264186790983913566314, 0.000000000000000000000000000000 ], [ 0.000000000000000000000000000000, 0.530994244053591519514741515969, 0.000000000000000000000000000000, 0.458831467741123531814419156196*I, 0.000000000000000000000000000000, 0.397359707119513141959825516155, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.591290165781890828708668604467, 0.000000000000000000000000000000 ], [ 0.000000000000000000000000000000, 0.530994244053591519514741515969, 0.000000000000000000000000000000, -0.458831467741123531814419156196, 0.000000000000000000000000000000, -0.397359707119513141959825516155, 0.000000000000000000000000000000, 0.000000000000000000000000000000, -0.591290165781890828708668604467, 0.000000000000000000000000000000 ], [ 0.000000000000000000000000000000, 0.530994244053591519514741515969, 0.000000000000000000000000000000, 0.458831467741123531814419156196, 0.000000000000000000000000000000, -0.397359707119513141959825516155, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.591290165781890828708668604467, 0.000000000000000000000000000000 ], [ 0.267261241912424384684553480879, 0.000000000000000000000000000000, -0.654653670707977143798292456247, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.000000000000000000000000000000, -0.654653670707977143798292456247, 0.000000000000000000000000000000, -0.267261241912424384684553480879 ], [ 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.739264247686155010763856007791, 0.000000000000000000000000000000, 0.000000000000000000000000000000, -0.215665546406876831453870141256, 0.000000000000000000000000000000, 0.494151845601762279785301969958, 0.000000000000000000000000000000, 0.403473292392964474705232425046 ], [ 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.739264247686155010763856007791, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.215665546406876831453870141256, 0.000000000000000000000000000000, 0.494151845601762279785301969958*I, 0.000000000000000000000000000000, 0.403473292392964474705232425046 ], [ 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.739264247686155010763856007791, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.215665546406876831453870141256, 0.000000000000000000000000000000, 0.494151845601762279785301969958, 0.000000000000000000000000000000, -0.403473292392964474705232425046 ], [ 0.408248290463863016366214012451, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.577350269189625764509148780502, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.577350269189625764509148780502, 0.000000000000000000000000000000, 0.000000000000000000000000000000, -0.408248290463863016366214012451 ], [ 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.731925054711399884515788477180, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.000000000000000000000000000000, -0.500000000000000000000000000000, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.462910049886275730783283388292 ], [ 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.731925054711399884515788477180, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.500000000000000000000000000000, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.462910049886275730783283388292 ], [ 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.745355992499929898803057889577, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.235702260395515841466948120702, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.623609564462323564263958122052, 0.000000000000000000000000000000 ], [ 0.500000000000000000000000000000, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.000000000000000000000000000000, -0.707106781186547524400844362105, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.500000000000000000000000000000, 0.000000000000000000000000000000 ], [ 0.500000000000000000000000000000, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.000000000000000000000000000000, -0.707106781186547524400844362105, 0.000000000000000000000000000000, 0.000000000000000000000000000000, -0.500000000000000000000000000000, 0.000000000000000000000000000000 ], [ 0.235702260395515841466948120702, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.816496580927726032732428024902, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.000000000000000000000000000000, -0.527046276694729888666482257405 ], [ 0.235702260395515841466948120702, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.816496580927726032732428024902, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.527046276694729888666482257405 ], [ 0.000000000000000000000000000000, 0.360843918243516102818217987814, 0.000000000000000000000000000000, -0.726483157256778925372830330644, 0.000000000000000000000000000000, -0.270030862433660842956915309837, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.000000000000000000000000000000, -0.518745816582863848951498777782 ], [ 0.000000000000000000000000000000, 0.360843918243516102818217987814, 0.000000000000000000000000000000, -0.726483157256778925372830330644, 0.000000000000000000000000000000, -0.270030862433660842956915309837, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.518745816582863848951498777782 ], [ 0.000000000000000000000000000000, 0.547722557505166113456969782801, 0.000000000000000000000000000000, -0.447213595499957939281834733746, 0.000000000000000000000000000000, 0.000000000000000000000000000000, -0.447213595499957939281834733746, 0.000000000000000000000000000000, -0.547722557505166113456969782801, 0.000000000000000000000000000000 ], [ 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.707106781186547524400844362105, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.500000000000000000000000000000, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.000000000000000000000000000000, -0.500000000000000000000000000000 ], [ 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.707106781186547524400844362105, 0.000000000000000000000000000000, 0.000000000000000000000000000000, -0.500000000000000000000000000000, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.000000000000000000000000000000, -0.500000000000000000000000000000 ], [ 0.408248290463863016366214012451, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.577350269189625764509148780502, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.577350269189625764509148780502, 0.000000000000000000000000000000, 0.000000000000000000000000000000, 0.408248290463863016366214012451 ] ] where I:=Sqrt(-1); // squared length of the multipoles [ 0.10000, 0.00000, 0.00000, 0.0064155, 0.022378, 0.32435, 0.10455, 0.0011396, 0.27308, 0.16810 ] [ 0.10000, 0.00000, 0.00000, 0.0065559, 0.023298, 0.32201, 0.10514, 0.0030004, 0.27156, 0.16843 ] [ 0.10000, 0.00000, 0.00000, 0.0065559, 0.12485, 0.098599, 0.17115, 0.18370, 0.10400, 0.21114 ] [ 0.10000, 0.00000, 0.00000, 0.0090326, 0.030682, 0.30032, 0.10994, 0.020073, 0.25938, 0.17057 ] [ 0.10000, 0.00000, 0.00000, 0.0090326, 0.030682, 0.30032, 0.10994, 0.020073, 0.25938, 0.17057 ] [ 0.10000, 0.00000, 0.00000, 0.011655, 0.12587, 0.085128, 0.17182, 0.19362, 0.10231, 0.20959 ] [ 0.10000, 0.00000, 0.00000, 0.011655, 0.12587, 0.085128, 0.17182, 0.19362, 0.10231, 0.20959 ] [ 0.10000, 0.00000, 0.00000, 0.022097, 0.13056, 0.051847, 0.17486, 0.21855, 0.094575, 0.20751 ] [ 0.10000, 0.00000, 0.00000, 0.028231, 0.11406, 0.074644, 0.16414, 0.19893, 0.12179, 0.19819 ] [ 0.10000, 0.00000, 0.00000, 0.028231, 0.11406, 0.074644, 0.16414, 0.19893, 0.12179, 0.19819 ] [ 0.10000, 0.00000, 0.00000, 0.031491, 0.11311, 0.069565, 0.16352, 0.20242, 0.12336, 0.19652 ] [ 0.10000, 0.00000, 0.00000, 0.031491, 0.11311, 0.069565, 0.16352, 0.20242, 0.12336, 0.19652 ] [ 0.10000, 0.00000, 0.00000, 0.031491, 0.11311, 0.069565, 0.16352, 0.20242, 0.12336, 0.19652 ] [ 0.10000, 0.00000, 0.00000, 0.041958, 0.050350, 0.18462, 0.12273, 0.10736, 0.22692, 0.16606 ] [ 0.10000, 0.00000, 0.00000, 0.047949, 0.065920, 0.13718, 0.13285, 0.14458, 0.20123, 0.17029 ] [ 0.10000, 0.00000, 0.00000, 0.047949, 0.065920, 0.13718, 0.13285, 0.14458, 0.20123, 0.17029 ] [ 0.10000, 0.00000, 0.00000, 0.047949, 0.065920, 0.13718, 0.13285, 0.14458, 0.20123, 0.17029 ] [ 0.10000, 0.00000, 0.00000, 0.051800, 0.10629, 0.039886, 0.15909, 0.22254, 0.13461, 0.18577 ] [ 0.10000, 0.00000, 0.00000, 0.051865, 0.097552, 0.058974, 0.15341, 0.20709, 0.14904, 0.18207 ] [ 0.10000, 0.00000, 0.00000, 0.051865, 0.097552, 0.058974, 0.15341, 0.20709, 0.14904, 0.18207 ] [ 0.10000, 0.00000, 0.00000, 0.055805, 0.10167, 0.041253, 0.15608, 0.22067, 0.14225, 0.18227 ] [ 0.10000, 0.00000, 0.00000, 0.079254, 0.078321, 0.041026, 0.14091, 0.21637, 0.18077, 0.16335 ] [ 0.10000, 0.00000, 0.00000, 0.079254, 0.078321, 0.041026, 0.14091, 0.21637, 0.18077, 0.16335 ] [ 0.10000, 0.00000, 0.00000, 0.10852, 0.036364, 0.068946, 0.11364, 0.18819, 0.25000, 0.13434 ] [ 0.10000, 0.00000, 0.00000, 0.10852, 0.036364, 0.068946, 0.11364, 0.18819, 0.25000, 0.13434 ] [ 0.10000, 0.00000, 0.00000, 0.11335, 0.048740, 0.031097, 0.12168, 0.21788, 0.22958, 0.13767 ] [ 0.10000, 0.00000, 0.00000, 0.11335, 0.060194, 0.0058993, 0.12913, 0.23826, 0.21068, 0.14249 ] [ 0.10000, 0.00000, 0.00000, 0.11655, 0.050350, 0.020513, 0.12273, 0.22583, 0.22692, 0.13710 ] [ 0.10000, 0.00000, 0.00000, 0.12383, 0.029808, 0.049679, 0.10938, 0.20085, 0.26082, 0.12564 ] [ 0.10000, 0.00000, 0.00000, 0.12383, 0.029808, 0.049679, 0.10938, 0.20085, 0.26082, 0.12564 ] [ 0.10000, 0.00000, 0.00000, 0.14070, 0.022378, 0.028929, 0.10455, 0.21441, 0.27308, 0.11596 ] // cumulative squared lengths A_M: [ 0.00000, 0.00000, 0.0064155, 0.028793, 0.35314, 0.45769, 0.45882, 0.73190, 0.90000 ] [ 0.00000, 0.00000, 0.0065559, 0.029854, 0.35187, 0.45701, 0.46001, 0.73157, 0.90000 ] [ 0.00000, 0.00000, 0.0065559, 0.13141, 0.23001, 0.40116, 0.58486, 0.68886, 0.90000 ] [ 0.00000, 0.00000, 0.0090326, 0.039714, 0.34003, 0.44998, 0.47005, 0.72942, 0.90000 ] [ 0.00000, 0.00000, 0.0090326, 0.039714, 0.34003, 0.44998, 0.47005, 0.72942, 0.90000 ] [ 0.00000, 0.00000, 0.011655, 0.13753, 0.22266, 0.39447, 0.58810, 0.69041, 0.90000 ] [ 0.00000, 0.00000, 0.011655, 0.13753, 0.22266, 0.39447, 0.58810, 0.69041, 0.90000 ] [ 0.00000, 0.00000, 0.022097, 0.15266, 0.20450, 0.37937, 0.59792, 0.69249, 0.90000 ] [ 0.00000, 0.00000, 0.028231, 0.14229, 0.21694, 0.38108, 0.58002, 0.70181, 0.90000 ] [ 0.00000, 0.00000, 0.028231, 0.14229, 0.21694, 0.38108, 0.58002, 0.70181, 0.90000 ] [ 0.00000, 0.00000, 0.031491, 0.14460, 0.21417, 0.37769, 0.58011, 0.70348, 0.90000 ] [ 0.00000, 0.00000, 0.031491, 0.14460, 0.21417, 0.37769, 0.58011, 0.70348, 0.90000 ] [ 0.00000, 0.00000, 0.031491, 0.14460, 0.21417, 0.37769, 0.58011, 0.70348, 0.90000 ] [ 0.00000, 0.00000, 0.041958, 0.092308, 0.27692, 0.39965, 0.50701, 0.73394, 0.90000 ] [ 0.00000, 0.00000, 0.047949, 0.11387, 0.25105, 0.38390, 0.52848, 0.72971, 0.90000 ] [ 0.00000, 0.00000, 0.047949, 0.11387, 0.25105, 0.38390, 0.52848, 0.72971, 0.90000 ] [ 0.00000, 0.00000, 0.047949, 0.11387, 0.25105, 0.38390, 0.52848, 0.72971, 0.90000 ] [ 0.00000, 0.00000, 0.051800, 0.15809, 0.19798, 0.35707, 0.57961, 0.71423, 0.90000 ] [ 0.00000, 0.00000, 0.051865, 0.14942, 0.20839, 0.36180, 0.56889, 0.71793, 0.90000 ] [ 0.00000, 0.00000, 0.051865, 0.14942, 0.20839, 0.36180, 0.56889, 0.71793, 0.90000 ] [ 0.00000, 0.00000, 0.055805, 0.15747, 0.19872, 0.35481, 0.57548, 0.71773, 0.90000 ] [ 0.00000, 0.00000, 0.079254, 0.15758, 0.19860, 0.33951, 0.55589, 0.73665, 0.90000 ] [ 0.00000, 0.00000, 0.079254, 0.15758, 0.19860, 0.33951, 0.55589, 0.73665, 0.90000 ] [ 0.00000, 0.00000, 0.10852, 0.14488, 0.21383, 0.32747, 0.51566, 0.76566, 0.90000 ] [ 0.00000, 0.00000, 0.10852, 0.14488, 0.21383, 0.32747, 0.51566, 0.76566, 0.90000 ] [ 0.00000, 0.00000, 0.11335, 0.16209, 0.19319, 0.31487, 0.53275, 0.76233, 0.90000 ] [ 0.00000, 0.00000, 0.11335, 0.17354, 0.17944, 0.30857, 0.54683, 0.75751, 0.90000 ] [ 0.00000, 0.00000, 0.11655, 0.16690, 0.18741, 0.31014, 0.53597, 0.76289, 0.90000 ] [ 0.00000, 0.00000, 0.12383, 0.15364, 0.20332, 0.31270, 0.51355, 0.77436, 0.90000 ] [ 0.00000, 0.00000, 0.12383, 0.15364, 0.20332, 0.31270, 0.51355, 0.77436, 0.90000 ] [ 0.00000, 0.00000, 0.14070, 0.16307, 0.19200, 0.29655, 0.51096, 0.78403, 0.90000 ] //====================================================================== // exact values //====================================================================== L9_exact:=[ [1/6*Sqrt(2)*Sqrt(3),0,0,-1/3*Sqrt(3),0,0,-1/3*Sqrt(3),0,0,1/6*Sqrt(2)*Sqrt(3)], [1/8*Sqrt(13),0,0,0,1/8*Sqrt(2)*Sqrt(3)*Sqrt(5),0,0,0,-1/8*Sqrt(3)*Sqrt(7),0], [1/8*Sqrt(13),0,0,0,1/8*Sqrt(2)*Sqrt(3)*Sqrt(5),0,0,0,1/8*Sqrt(3)*Sqrt(7),0], [0,1/4*Sqrt(2)*Sqrt(3),0,0,0,0,1/2*Sqrt(2),0,0,1/4*Sqrt(2)], [0,1/4*Sqrt(2)*Sqrt(3),0,0,0,0,1/2*Sqrt(2),0,0,-1/4*Sqrt(2)], [1/6*Sqrt(2)*Sqrt(3),0,0,0,-1/5*Sqrt(3)*Sqrt(5),0,0,0,0,1/30*Sqrt(2)*Sqrt(3)*Sqrt(5)*Sqrt(7)], [1/6*Sqrt(2)*Sqrt(3),0,0,0,1/5*Sqrt(3)*Sqrt(5),0,0,0,0,-1/30*Sqrt(2)*Sqrt(3)*Sqrt(5)*Sqrt(7)], [0,0,1/3*Sqrt(5),0,0,1/6*Sqrt(2),0,0,-1/6*Sqrt(2)*Sqrt(7),0], [0,1/14*Sqrt(2)*Sqrt(5)*Sqrt(7),0,0,0,1/3*Sqrt(3),0,0,1/42*Sqrt(2)*Sqrt(3)*Sqrt(7)*Sqrt(13),0], [0,1/14*Sqrt(2)*Sqrt(5)*Sqrt(7),0,0,0,-1/3*Sqrt(3),0,0,1/42*Sqrt(2)*Sqrt(3)*Sqrt(7)*Sqrt(13),0], [0,5/266*Sqrt(2)*Sqrt(3)*Sqrt(7)*Sqrt(19),0,2/19*Sqrt(-1)*Sqrt(19),0,1/19*Sqrt(3)*Sqrt(19),0,0,1/266*Sqrt(2)*Sqrt(3)*Sqrt(7)*Sqrt(19)*Sqrt(31),0], [0,5/266*Sqrt(2)*Sqrt(3)*Sqrt(7)*Sqrt(19),0,-2/19*Sqrt(19),0,-1/19*Sqrt(3)*Sqrt(19),0,0,-1/266*Sqrt(2)*Sqrt(3)*Sqrt(7)*Sqrt(19)*Sqrt(31),0], [0,5/266*Sqrt(2)*Sqrt(3)*Sqrt(7)*Sqrt(19),0,2/19*Sqrt(19),0,-1/19*Sqrt(3)*Sqrt(19),0,0,1/266*Sqrt(2)*Sqrt(3)*Sqrt(7)*Sqrt(19)*Sqrt(31),0], [1/14*Sqrt(2)*Sqrt(7),0,-1/7*Sqrt(3)*Sqrt(7),0,0,0,0,-1/7*Sqrt(3)*Sqrt(7),0,-1/14*Sqrt(2)*Sqrt(7)], [0,0,1/86*Sqrt(2)*Sqrt(43)*Sqrt(47),0,0,-1/43*Sqrt(2)*Sqrt(43),0,1/86*Sqrt(2)*Sqrt(3)*Sqrt(7)*Sqrt(43),0,1/43*Sqrt(7)*Sqrt(43)], [0,0,1/86*Sqrt(2)*Sqrt(43)*Sqrt(47),0,0,1/43*Sqrt(2)*Sqrt(43),0,1/86*Sqrt(-1)*Sqrt(2)*Sqrt(3)*Sqrt(7)*Sqrt(43),0,1/43*Sqrt(7)*Sqrt(43)], [0,0,1/86*Sqrt(2)*Sqrt(43)*Sqrt(47),0,0,1/43*Sqrt(2)*Sqrt(43),0,1/86*Sqrt(2)*Sqrt(3)*Sqrt(7)*Sqrt(43),0,-1/43*Sqrt(7)*Sqrt(43)], [1/6*Sqrt(2)*Sqrt(3),0,0, 1/3*Sqrt(3),0,0, 1/3*Sqrt(3),0,0,-1/6*Sqrt(2)*Sqrt(3)], [0,0,1/14*Sqrt(3)*Sqrt(5)*Sqrt(7),0,0,0,-1/2,0,0,1/14*Sqrt(2)*Sqrt(3)*Sqrt(7)], [0,0,1/14*Sqrt(3)*Sqrt(5)*Sqrt(7),0,0,0,1/2,0,0,1/14*Sqrt(2)*Sqrt(3)*Sqrt(7)], [0,0,1/3*Sqrt(5),0,0,1/6*Sqrt(2),0,0,1/6*Sqrt(2)*Sqrt(7),0], [1/2,0,0,0,0,-1/2*Sqrt(2),0,0, 1/2,0], [1/2,0,0,0,0,-1/2*Sqrt(2),0,0,-1/2,0], [1/6*Sqrt(2),0,0,1/3*Sqrt(2)*Sqrt(3),0,0,0,0,0,-1/6*Sqrt(2)*Sqrt(5)], [1/6*Sqrt(2),0,0,1/3*Sqrt(2)*Sqrt(3),0,0,0,0,0,1/6*Sqrt(2)*Sqrt(5)], [0,5/24*Sqrt(3),0,-1/6*Sqrt(19),0,-1/24*Sqrt(2)*Sqrt(3)*Sqrt(7),0,0,0,-1/24*Sqrt(5)*Sqrt(31)], [0,5/24*Sqrt(3),0,-1/6*Sqrt(19),0,-1/24*Sqrt(2)*Sqrt(3)*Sqrt(7),0,0,0,1/24*Sqrt(5)*Sqrt(31)], [0,1/10*Sqrt(2)*Sqrt(3)*Sqrt(5),0,-1/5*Sqrt(5),0,0,-1/5*Sqrt(5),0,-1/10*Sqrt(2)*Sqrt(3)*Sqrt(5),0], [0,0,1/2*Sqrt(2),0,0,1/2,0,0,0,-1/2], [0,0,1/2*Sqrt(2),0,0,-1/2,0,0,0,-1/2], [1/6*Sqrt(2)*Sqrt(3),0,0, 1/3*Sqrt(3),0,0, 1/3*Sqrt(3),0,0,1/6*Sqrt(2)*Sqrt(3)] ]; // squared length of the multipoles [ 1/10, 0, 0, -40/3861*Sqrt(2)*Sqrt(3)*Sqrt(7) + 284/3861, 16/715, 8/351*Sqrt(2)*Sqrt(3)*Sqrt(7) + 62/351, 23/220, -40/2431*Sqrt(2)*Sqrt(3)*Sqrt(7) + 262/2431, 71/260, 8/1989*Sqrt(2)*Sqrt(3)*Sqrt(7) + 565/3978 ] [ 1/10, 0, 0, 15/2288, -225/73216*Sqrt(3)*Sqrt(7)*Sqrt(13) + 27117/366080, 45/6656*Sqrt(3)*Sqrt(7)*Sqrt(13) + 6999/33280, -45/22528*Sqrt(3)*Sqrt(7)*Sqrt(13) + 15561/112640, -2475/452608*Sqrt(3)*Sqrt(7)*Sqrt(13) + 464769/4978688, 135/26624*Sqrt(3)*Sqrt(7)*Sqrt(13) + 24997/133120, -45/34816*Sqrt(3)*Sqrt(7)*Sqrt(13) + 429493/2263040 ] [ 1/10, 0, 0, 15/2288, 225/73216*Sqrt(3)*Sqrt(7)*Sqrt(13) + 27117/366080, -45/6656*Sqrt(3)*Sqrt(7)*Sqrt(13) + 6999/33280, 45/22528*Sqrt(3)*Sqrt(7)*Sqrt(13) + 15561/112640, 2475/452608*Sqrt(3)*Sqrt(7)*Sqrt(13) + 464769/4978688, -135/26624*Sqrt(3)*Sqrt(7)*Sqrt(13) + 24997/133120, 45/34816*Sqrt(3)*Sqrt(7)*Sqrt(13) + 429493/2263040 ] [ 1/10, 0, 0, 31/3432, 27/880, 937/3120, 387/3520, 3123/155584, 83/320, 12063/70720 ] [ 1/10, 0, 0, 31/3432, 27/880, 937/3120, 387/3520, 3123/155584, 83/320, 12063/70720 ] [ 1/10, 0, 0, 5/429, 18/143, 83/975, 189/1100, 4707/24310, 133/1300, 1158/5525 ] [ 1/10, 0, 0, 5/429, 18/143, 83/975, 189/1100, 4707/24310, 133/1300, 1158/5525 ] [ 1/10, 0, 0, -70/34749*Sqrt(2)*Sqrt(5)*Sqrt(7) + 2707/69498, 20/11583*Sqrt(2)*Sqrt(5)*Sqrt(7) + 26899/231660, 2/3159*Sqrt(2)*Sqrt(5)*Sqrt(7) + 2941/63180, 1/891*Sqrt(2)*Sqrt(5)*Sqrt(7) + 2359/14256, -25/196911*Sqrt(2)*Sqrt(5)*Sqrt(7) + 691895/3150576, -1/351*Sqrt(2)*Sqrt(5)*Sqrt(7) + 665/5616, 1/663*Sqrt(2)*Sqrt(5)*Sqrt(7) + 10337/53040 ] [ 1/10, 0, 0, 109/3861, 734/6435, 131/1755, 65/396, 8705/43758, 19/156, 219/1105 ] [ 1/10, 0, 0, 109/3861, 734/6435, 131/1755, 65/396, 8705/43758, 19/156, 219/1105 ] [ 1/10, 0, 0, 4877/154869, 29196/258115, 4897/70395, 12987/79420, 177642/877591, 11579/93860, 156789/797810 ] [ 1/10, 0, 0, 4877/154869, 29196/258115, 4897/70395, 12987/79420, 177642/877591, 11579/93860, 156789/797810 ] [ 1/10, 0, 0, 4877/154869, 29196/258115, 4897/70395, 12987/79420, 177642/877591, 11579/93860, 156789/797810 ] [ 1/10, 0, 0, 6/143, 36/715, 12/65, 27/220, 261/2431, 59/260, 367/2210 ] [ 1/10, 0, 0, 152137/3172884, 87149/1322035, 107/780, 2702/20339, 1299797/8989838, 4837/24037, 695839/4086290 ] [ 1/10, 0, 0, 152137/3172884, 87149/1322035, 107/780, 2702/20339, 1299797/8989838, 4837/24037, 695839/4086290 ] [ 1/10, 0, 0, 152137/3172884, 87149/1322035, 107/780, 2702/20339, 1299797/8989838, 4837/24037, 695839/4086290 ] [ 1/10, 0, 0, 200/3861, 76/715, 14/351, 7/44, 541/2431, 7/52, 739/3978 ] [ 1/10, 0, 0, 89/1716, 279/2860, 23/390, 27/176, 8055/38896, 31/208, 3219/17680 ] [ 1/10, 0, 0, 89/1716, 279/2860, 23/390, 27/176, 8055/38896, 31/208, 3219/17680 ] [ 1/10, 0, 0, 70/34749*Sqrt(2)*Sqrt(5)*Sqrt(7) + 2707/69498, -20/11583*Sqrt(2)*Sqrt(5)*Sqrt(7) + 26899/231660, -2/3159*Sqrt(2)*Sqrt(5)*Sqrt(7) + 2941/63180, -1/891*Sqrt(2)*Sqrt(5)*Sqrt(7) + 2359/14256, 25/196911*Sqrt(2)*Sqrt(5)*Sqrt(7) + 691895/3150576, 1/351*Sqrt(2)*Sqrt(5)*Sqrt(7) + 665/5616, -1/663*Sqrt(2)*Sqrt(5)*Sqrt(7) + 10337/53040 ] [ 1/10, 0, 0, 34/429, 56/715, 8/195, 31/220, 526/2431, 47/260, 361/2210 ] [ 1/10, 0, 0, 34/429, 56/715, 8/195, 31/220, 526/2431, 47/260, 361/2210 ] [ 1/10, 0, 0, 419/3861, 2/55, 121/1755, 5/44, 915/4862, 1/4, 1336/9945 ] [ 1/10, 0, 0, 419/3861, 2/55, 121/1755, 5/44, 915/4862, 1/4, 1336/9945 ] [ 1/10, 0, 0, 15755/138996, -175/658944*Sqrt(3)*Sqrt(5)*Sqrt(31) + 179453/3294720, 35/59904*Sqrt(3)*Sqrt(5)*Sqrt(31) + 149597/8087040, -35/202752*Sqrt(3)*Sqrt(5)*Sqrt(31) + 127129/1013760, -1925/4073472*Sqrt(3)*Sqrt(5)*Sqrt(31) + 601153/2635776, 35/79872*Sqrt(3)*Sqrt(5)*Sqrt(31) + 87911/399360, -35/313344*Sqrt(3)*Sqrt(5)*Sqrt(31) + 1510429/10782720 ] [ 1/10, 0, 0, 15755/138996, 175/658944*Sqrt(3)*Sqrt(5)*Sqrt(31) + 179453/3294720, -35/59904*Sqrt(3)*Sqrt(5)*Sqrt(31) + 149597/8087040, 35/202752*Sqrt(3)*Sqrt(5)*Sqrt(31) + 127129/1013760, 1925/4073472*Sqrt(3)*Sqrt(5)*Sqrt(31) + 601153/2635776, -35/79872*Sqrt(3)*Sqrt(5)*Sqrt(31) + 87911/399360, 35/313344*Sqrt(3)*Sqrt(5)*Sqrt(31) + 1510429/10782720 ] [ 1/10, 0, 0, 50/429, 36/715, 4/195, 27/220, 549/2431, 59/260, 303/2210 ] [ 1/10, 0, 0, 425/3432, 31/1040, 31/624, 7/64, 31249/155584, 217/832, 1777/14144 ] [ 1/10, 0, 0, 425/3432, 31/1040, 31/624, 7/64, 31249/155584, 217/832, 1777/14144 ] [ 1/10, 0, 0, 40/3861*Sqrt(2)*Sqrt(3)*Sqrt(7) + 284/3861, 16/715, -8/351*Sqrt(2)*Sqrt(3)*Sqrt(7) + 62/351, 23/220, 40/2431*Sqrt(2)*Sqrt(3)*Sqrt(7) + 262/2431, 71/260, -8/1989*Sqrt(2)*Sqrt(3)*Sqrt(7) + 565/3978 ] // cumulative squared lengths A_M: [ 0, 0, -40/3861*Sqrt(2)*Sqrt(3)*Sqrt(7) + 284/3861, -40/3861*Sqrt(2)*Sqrt(3)*Sqrt(7) + 1852/19305, 16/1287*Sqrt(2)*Sqrt(3)*Sqrt(7) + 1754/6435, 16/1287*Sqrt(2)*Sqrt(3)*Sqrt(7) + 9707/25740, -8/1989*Sqrt(2)*Sqrt(3)*Sqrt(7) + 19289/39780, -8/1989*Sqrt(2)*Sqrt(3)*Sqrt(7) + 7538/9945, 9/10 ] [ 0, 0, 15/2288, -225/73216*Sqrt(3)*Sqrt(7)*Sqrt(13) + 29517/366080, 135/36608*Sqrt(3)*Sqrt(7)*Sqrt(13) + 53253/183040, 45/26624*Sqrt(3)*Sqrt(7)*Sqrt(13) + 628317/1464320, -855/226304*Sqrt(3)*Sqrt(7)*Sqrt(13) + 591147/1131520, 45/34816*Sqrt(3)*Sqrt(7)*Sqrt(13) + 1607243/2263040, 9/10 ] [ 0, 0, 15/2288, 225/73216*Sqrt(3)*Sqrt(7)*Sqrt(13) + 29517/366080, -135/36608*Sqrt(3)*Sqrt(7)*Sqrt(13) + 53253/183040, -45/26624*Sqrt(3)*Sqrt(7)*Sqrt(13) + 628317/1464320, 855/226304*Sqrt(3)*Sqrt(7)*Sqrt(13) + 591147/1131520, -45/34816*Sqrt(3)*Sqrt(7)*Sqrt(13) + 1607243/2263040, 9/10 ] [ 0, 0, 31/3432, 1363/34320, 389/1144, 20591/45760, 16621/35360, 10317/14144, 9/10 ] [ 0, 0, 31/3432, 1363/34320, 389/1144, 20591/45760, 16621/35360, 10317/14144, 9/10 ] [ 0, 0, 5/429, 59/429, 796/3575, 5641/14300, 12997/22100, 7629/11050, 9/10 ] [ 0, 0, 5/429, 59/429, 796/3575, 5641/14300, 12997/22100, 7629/11050, 9/10 ] [ 0, 0, -70/34749*Sqrt(2)*Sqrt(5)*Sqrt(7) + 2707/69498, -10/34749*Sqrt(2)*Sqrt(5)*Sqrt(7) + 9797/63180, 4/11583*Sqrt(2)*Sqrt(5)*Sqrt(7) + 2123/10530, 17/11583*Sqrt(2)*Sqrt(5)*Sqrt(7) + 340159/926640, 8/5967*Sqrt(2)*Sqrt(5)*Sqrt(7) + 140033/238680, -1/663*Sqrt(2)*Sqrt(5)*Sqrt(7) + 37399/53040, 9/10 ] [ 0, 0, 109/3861, 2747/19305, 1396/6435, 9809/25740, 7691/13260, 1551/2210, 9/10 ] [ 0, 0, 109/3861, 2747/19305, 1396/6435, 9809/25740, 7691/13260, 1551/2210, 9/10 ] [ 0, 0, 4877/154869, 111973/774345, 11056/51623, 389951/1032460, 925637/1595620, 56124/79781, 9/10 ] [ 0, 0, 4877/154869, 111973/774345, 11056/51623, 389951/1032460, 925637/1595620, 56124/79781, 9/10 ] [ 0, 0, 4877/154869, 111973/774345, 11056/51623, 389951/1032460, 925637/1595620, 56124/79781, 9/10 ] [ 0, 0, 6/143, 6/65, 18/65, 1143/2860, 2241/4420, 811/1105, 9/10 ] [ 0, 0, 152137/3172884, 977/8580, 359/1430, 1015051/2644070, 1079766/2043145, 1490911/2043145, 9/10 ] [ 0, 0, 152137/3172884, 977/8580, 359/1430, 1015051/2644070, 1079766/2043145, 1490911/2043145, 9/10 ] [ 0, 0, 152137/3172884, 977/8580, 359/1430, 1015051/2644070, 1079766/2043145, 1490911/2043145, 9/10 ] [ 0, 0, 200/3861, 3052/19305, 98/495, 707/1980, 23057/39780, 7103/9945, 9/10 ] [ 0, 0, 89/1716, 641/4290, 149/715, 4139/11440, 5029/8840, 12693/17680, 9/10 ] [ 0, 0, 89/1716, 641/4290, 149/715, 4139/11440, 5029/8840, 12693/17680, 9/10 ] [ 0, 0, 70/34749*Sqrt(2)*Sqrt(5)*Sqrt(7) + 2707/69498, 10/34749*Sqrt(2)*Sqrt(5)*Sqrt(7) + 9797/63180, -4/11583*Sqrt(2)*Sqrt(5)*Sqrt(7) + 2123/10530, -17/11583*Sqrt(2)*Sqrt(5)*Sqrt(7) + 340159/926640, -8/5967*Sqrt(2)*Sqrt(5)*Sqrt(7) + 140033/238680, 1/663*Sqrt(2)*Sqrt(5)*Sqrt(7) + 37399/53040, 9/10 ] [ 0, 0, 34/429, 26/165, 142/715, 971/2860, 189/340, 814/1105, 9/10 ] [ 0, 0, 34/429, 26/165, 142/715, 971/2860, 189/340, 814/1105, 9/10 ] [ 0, 0, 419/3861, 2797/19305, 1376/6435, 8429/25740, 20513/39780, 15229/19890, 9/10 ] [ 0, 0, 419/3861, 2797/19305, 1376/6435, 8429/25740, 20513/39780, 15229/19890, 9/10 ] [ 0, 0, 15755/138996, -175/658944*Sqrt(3)*Sqrt(5)*Sqrt(31) + 14928431/88957440, 35/109824*Sqrt(3)*Sqrt(5)*Sqrt(31) + 2762333/14826240, 35/239616*Sqrt(3)*Sqrt(5)*Sqrt(31) + 36972757/118609920, -665/2036736*Sqrt(3)*Sqrt(5)*Sqrt(31) + 2910211/5391360, 35/313344*Sqrt(3)*Sqrt(5)*Sqrt(31) + 8194019/10782720, 9/10 ] [ 0, 0, 15755/138996, 175/658944*Sqrt(3)*Sqrt(5)*Sqrt(31) + 14928431/88957440, -35/109824*Sqrt(3)*Sqrt(5)*Sqrt(31) + 2762333/14826240, -35/239616*Sqrt(3)*Sqrt(5)*Sqrt(31) + 36972757/118609920, 665/2036736*Sqrt(3)*Sqrt(5)*Sqrt(31) + 2910211/5391360, -35/313344*Sqrt(3)*Sqrt(5)*Sqrt(31) + 8194019/10782720, 9/10 ] [ 0, 0, 50/429, 358/2145, 134/715, 887/2860, 2369/4420, 843/1105, 9/10 ] [ 0, 0, 425/3432, 5273/34320, 1163/5720, 14309/45760, 18159/35360, 54763/70720, 9/10 ] [ 0, 0, 425/3432, 5273/34320, 1163/5720, 14309/45760, 18159/35360, 54763/70720, 9/10 ] [ 0, 0, 40/3861*Sqrt(2)*Sqrt(3)*Sqrt(7) + 284/3861, 40/3861*Sqrt(2)*Sqrt(3)*Sqrt(7) + 1852/19305, -16/1287*Sqrt(2)*Sqrt(3)*Sqrt(7) + 1754/6435, -16/1287*Sqrt(2)*Sqrt(3)*Sqrt(7) + 9707/25740, 8/1989*Sqrt(2)*Sqrt(3)*Sqrt(7) + 19289/39780, 8/1989*Sqrt(2)*Sqrt(3)*Sqrt(7) + 7538/9945, 9/10 ]