/* This code can be loaded, or copied and paste using cpaste, into Sage. It will load the data associated to the HMF, including the field, level, and Hecke and Atkin-Lehner eigenvalue data. */ P. = PolynomialRing(QQ) g = P([41, 9, -14, -1, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([1, 1, 1]) primes_array = [ [4, 2, 1/6*w^3 - 7/3*w - 17/6],\ [4, 2, w - 3],\ [5, 5, -1/6*w^3 + w^2 + 1/3*w - 25/6],\ [9, 3, 1/6*w^3 - 7/3*w + 13/6],\ [9, 3, -w - 2],\ [11, 11, -1/6*w^3 + 7/3*w + 11/6],\ [11, 11, -w + 2],\ [41, 41, -w],\ [41, 41, 1/6*w^3 - 7/3*w + 1/6],\ [59, 59, -1/6*w^3 + 1/3*w + 23/6],\ [59, 59, -1/3*w^3 + 11/3*w + 11/3],\ [61, 61, -5/3*w^3 - 3*w^2 + 46/3*w + 79/3],\ [61, 61, 5/6*w^3 + w^2 - 23/3*w - 55/6],\ [61, 61, w^3 + 2*w^2 - 9*w - 17],\ [61, 61, -1/6*w^3 + 10/3*w + 35/6],\ [71, 71, -1/2*w^3 + 2*w^2 + 4*w - 33/2],\ [71, 71, 1/6*w^3 - w^2 - 4/3*w + 67/6],\ [79, 79, 1/2*w^3 - 3*w + 1/2],\ [79, 79, -2/3*w^3 + 19/3*w - 2/3],\ [89, 89, 1/2*w^3 + w^2 - 5*w - 15/2],\ [89, 89, 1/3*w^3 - 11/3*w + 7/3],\ [101, 101, -5/6*w^3 - w^2 + 23/3*w + 61/6],\ [101, 101, 1/2*w^3 - w^2 - 3*w + 9/2],\ [109, 109, 1/2*w^3 - w^2 - 4*w + 11/2],\ [109, 109, -2/3*w^3 - w^2 + 16/3*w + 28/3],\ [121, 11, -1/2*w^3 + 4*w + 1/2],\ [131, 131, 2*w - 7],\ [131, 131, -7/6*w^3 - 2*w^2 + 28/3*w + 89/6],\ [139, 139, -1/6*w^3 + w^2 + 4/3*w - 31/6],\ [139, 139, 1/3*w^3 + w^2 - 8/3*w - 29/3],\ [149, 149, 1/6*w^3 - 2*w^2 + 5/3*w + 37/6],\ [151, 151, -7/6*w^3 + 3*w^2 + 28/3*w - 133/6],\ [151, 151, 5/3*w^3 + 3*w^2 - 40/3*w - 67/3],\ [169, 13, -1/6*w^3 - 2/3*w + 29/6],\ [169, 13, 7/6*w^3 - 4*w^2 - 19/3*w + 139/6],\ [181, 181, -2/3*w^3 + 13/3*w + 10/3],\ [181, 181, 5/6*w^3 - 23/3*w - 19/6],\ [191, 191, 5/6*w^3 + w^2 - 32/3*w - 97/6],\ [191, 191, w^3 - 9*w - 5],\ [191, 191, -w^2 + 4*w - 2],\ [191, 191, 5/6*w^3 + w^2 - 17/3*w - 43/6],\ [199, 199, 1/2*w^3 - 2*w - 7/2],\ [199, 199, 7/6*w^3 - 5*w^2 - 7/3*w + 109/6],\ [211, 211, -1/6*w^3 + w^2 + 7/3*w - 79/6],\ [211, 211, 1/2*w^3 + 2*w^2 - 6*w - 37/2],\ [229, 229, -1/2*w^3 + 5*w - 5/2],\ [229, 229, 1/2*w^3 - 2*w^2 - 2*w + 23/2],\ [229, 229, 1/6*w^3 + w^2 - 10/3*w - 53/6],\ [229, 229, -7/6*w^3 - 2*w^2 + 34/3*w + 107/6],\ [239, 239, 2*w^3 + 3*w^2 - 22*w - 36],\ [239, 239, -7/3*w^3 - 3*w^2 + 74/3*w + 113/3],\ [241, 241, -1/3*w^3 - w^2 + 2/3*w + 11/3],\ [241, 241, -1/2*w^3 + w^2 + 6*w - 23/2],\ [271, 271, -5/6*w^3 - w^2 + 20/3*w + 49/6],\ [271, 271, 1/6*w^3 - 1/3*w - 35/6],\ [281, 281, -1/6*w^3 - w^2 + 7/3*w + 11/6],\ [281, 281, -w^3 + 9*w - 1],\ [281, 281, -1/2*w^3 + 8*w + 19/2],\ [281, 281, w^3 - 4*w^2 - 4*w + 20],\ [311, 311, 2*w^2 - 13],\ [311, 311, -1/3*w^3 - 2*w^2 + 8/3*w + 50/3],\ [331, 331, 1/2*w^3 + w^2 - 4*w - 5/2],\ [331, 331, -1/2*w^3 - w^2 + 5*w + 27/2],\ [331, 331, 1/6*w^3 - 2*w^2 + 2/3*w + 73/6],\ [331, 331, 5/6*w^3 + 2*w^2 - 26/3*w - 103/6],\ [349, 349, -1/2*w^3 - w^2 + 4*w + 21/2],\ [349, 349, 1/6*w^3 + w^2 - 1/3*w - 59/6],\ [361, 19, 2/3*w^3 - 16/3*w - 7/3],\ [361, 19, 1/6*w^3 - 4/3*w - 29/6],\ [379, 379, 1/6*w^3 + w^2 - 10/3*w - 17/6],\ [379, 379, -5/6*w^3 - 2*w^2 + 23/3*w + 115/6],\ [379, 379, -1/3*w^3 + 2*w^2 + 5/3*w - 31/3],\ [379, 379, 1/3*w^3 + w^2 - 14/3*w - 35/3],\ [389, 389, -1/3*w^3 + 2/3*w + 14/3],\ [389, 389, 1/2*w^3 + 2*w^2 - 4*w - 35/2],\ [401, 401, 1/3*w^3 + 2*w^2 - 14/3*w - 38/3],\ [401, 401, 5/6*w^3 - 4*w^2 - 5/3*w + 101/6],\ [401, 401, -1/3*w^3 - 2*w^2 + 14/3*w + 50/3],\ [401, 401, -5/6*w^3 + 2*w^2 + 5/3*w - 29/6],\ [409, 409, -1/2*w^3 + 6*w - 7/2],\ [409, 409, 5/6*w^3 + 2*w^2 - 26/3*w - 91/6],\ [439, 439, 1/3*w^3 + 2*w^2 - 14/3*w - 32/3],\ [439, 439, 1/3*w^3 + 2*w^2 - 14/3*w - 56/3],\ [449, 449, -11/6*w^3 - 3*w^2 + 47/3*w + 157/6],\ [449, 449, -7/6*w^3 + 3*w^2 + 25/3*w - 109/6],\ [461, 461, 1/6*w^3 + 2*w^2 - 4/3*w - 77/6],\ [461, 461, 2/3*w^3 + 2*w^2 - 22/3*w - 55/3],\ [479, 479, 2/3*w^3 - 2*w^2 - 13/3*w + 32/3],\ [479, 479, -7/6*w^3 - 2*w^2 + 31/3*w + 113/6],\ [491, 491, 1/3*w^3 + 2*w^2 - 11/3*w - 35/3],\ [491, 491, 1/6*w^3 + 2*w^2 - 7/3*w - 107/6],\ [499, 499, 4/3*w^3 + 2*w^2 - 38/3*w - 53/3],\ [499, 499, 5/6*w^3 - 4*w^2 - 2/3*w + 101/6],\ [509, 509, -2/3*w^3 + 31/3*w + 40/3],\ [509, 509, 1/6*w^3 - 19/3*w + 85/6],\ [521, 521, -2/3*w^3 - 2*w^2 + 19/3*w + 52/3],\ [521, 521, -w^2 - 2*w + 10],\ [529, 23, -1/3*w^3 + 2*w^2 + 5/3*w - 49/3],\ [529, 23, 5/6*w^3 + 2*w^2 - 23/3*w - 79/6],\ [541, 541, -1/6*w^3 + 10/3*w - 13/6],\ [541, 541, 1/6*w^3 - 10/3*w - 11/6],\ [569, 569, 1/2*w^3 + w^2 - 5*w - 9/2],\ [569, 569, -1/6*w^3 + w^2 + 1/3*w - 61/6],\ [599, 599, 1/6*w^3 + w^2 - 1/3*w - 65/6],\ [599, 599, -1/6*w^3 + w^2 + 7/3*w - 25/6],\ [619, 619, 7/2*w^3 + 6*w^2 - 33*w - 113/2],\ [619, 619, -7/6*w^3 - 2*w^2 + 31/3*w + 89/6],\ [631, 631, -1/2*w^3 + 3*w^2 + w - 29/2],\ [631, 631, -1/6*w^3 + w^2 - 8/3*w + 29/6],\ [631, 631, -2/3*w^3 + 25/3*w + 16/3],\ [631, 631, -11/6*w^3 - 3*w^2 + 47/3*w + 145/6],\ [659, 659, 5/6*w^3 - 3*w^2 - 20/3*w + 155/6],\ [659, 659, 7/6*w^3 - 2*w^2 - 16/3*w + 49/6],\ [659, 659, -7/6*w^3 + 34/3*w + 29/6],\ [659, 659, -5/6*w^3 + w^2 + 14/3*w - 35/6],\ [661, 661, 1/3*w^3 - 20/3*w + 28/3],\ [661, 661, 1/3*w^3 - 20/3*w - 26/3],\ [691, 691, 1/2*w^3 + 2*w^2 - 5*w - 35/2],\ [691, 691, 2*w^2 - 15],\ [691, 691, 1/2*w^3 + 2*w^2 - 6*w - 31/2],\ [691, 691, 2*w^2 - w - 12],\ [701, 701, 1/3*w^3 + 2*w^2 - 11/3*w - 53/3],\ [701, 701, 1/6*w^3 + 2*w^2 - 7/3*w - 71/6],\ [709, 709, -11/6*w^3 - 3*w^2 + 50/3*w + 151/6],\ [709, 709, 7/6*w^3 + w^2 - 25/3*w - 71/6],\ [709, 709, 7/6*w^3 - w^2 - 31/3*w + 19/6],\ [709, 709, 13/6*w^3 + 4*w^2 - 58/3*w - 209/6],\ [739, 739, 7/6*w^3 - 4*w^2 - 28/3*w + 199/6],\ [739, 739, 5/3*w^3 - 5*w^2 - 40/3*w + 119/3],\ [739, 739, 5/6*w^3 - 4*w^2 - 8/3*w + 107/6],\ [739, 739, -5/6*w^3 + w^2 + 26/3*w - 35/6],\ [809, 809, 11/6*w^3 + 3*w^2 - 56/3*w - 175/6],\ [809, 809, -5/6*w^3 + 2*w^2 + 17/3*w - 89/6],\ [809, 809, -4/3*w^3 + 4*w^2 + 14/3*w - 49/3],\ [809, 809, -5/6*w^3 + w^2 + 11/3*w - 23/6],\ [811, 811, -1/3*w^3 + 17/3*w + 17/3],\ [811, 811, -1/6*w^3 + 13/3*w - 37/6],\ [821, 821, w^3 - w^2 - 8*w + 3],\ [821, 821, -2*w^3 - 3*w^2 + 18*w + 28],\ [821, 821, 7/6*w^3 + w^2 - 28/3*w - 71/6],\ [821, 821, 1/2*w^3 - w^2 - 7*w + 35/2],\ [829, 829, 5/6*w^3 + w^2 - 20/3*w - 73/6],\ [829, 829, 1/6*w^3 + w^2 - 1/3*w - 71/6],\ [839, 839, 1/2*w^3 - 10*w + 33/2],\ [839, 839, 1/2*w^3 - 10*w - 31/2],\ [841, 29, 5/6*w^3 - 20/3*w - 19/6],\ [841, 29, 1/6*w^3 - 4/3*w - 35/6],\ [859, 859, -1/6*w^3 + 2*w^2 + 7/3*w - 121/6],\ [859, 859, -2*w^3 - 4*w^2 + 16*w + 27],\ [859, 859, 1/2*w^3 - w^2 - 4*w - 1/2],\ [859, 859, 4/3*w^3 - 4*w^2 - 32/3*w + 97/3],\ [881, 881, -5/6*w^3 + w^2 + 14/3*w - 11/6],\ [881, 881, 4/3*w^3 + w^2 - 38/3*w - 38/3],\ [911, 911, 2*w^2 - 3*w - 4],\ [911, 911, -2/3*w^3 + 19/3*w - 20/3],\ [911, 911, 2/3*w^3 + w^2 - 22/3*w - 46/3],\ [911, 911, -2/3*w^3 + 4*w^2 - 2/3*w - 35/3],\ [919, 919, -1/6*w^3 + 3*w^2 - 11/3*w - 43/6],\ [919, 919, -13/6*w^3 - 4*w^2 + 55/3*w + 179/6],\ [919, 919, 5/6*w^3 - 2*w^2 - 8/3*w + 47/6],\ [919, 919, 4/3*w^3 - 4*w^2 - 29/3*w + 88/3],\ [929, 929, -7/6*w^3 + 3*w^2 + 10/3*w - 61/6],\ [929, 929, -1/2*w^3 + 2*w^2 + 4*w - 25/2],\ [929, 929, -2/3*w^3 + 22/3*w + 19/3],\ [929, 929, 2*w^2 - 4*w - 3],\ [961, 31, 5/6*w^3 - 20/3*w - 13/6],\ [961, 31, -5/6*w^3 + 20/3*w + 7/6],\ [971, 971, -7/6*w^3 + 6*w^2 + 4/3*w - 139/6],\ [971, 971, -2/3*w^3 + 2*w^2 + 1/3*w - 14/3],\ [991, 991, 7/3*w^3 + 4*w^2 - 59/3*w - 95/3],\ [991, 991, -1/6*w^3 + w^2 - 11/3*w + 47/6],\ [1009, 1009, -7/6*w^3 - w^2 + 31/3*w + 35/6],\ [1009, 1009, -5/6*w^3 + w^2 + 17/3*w - 53/6],\ [1021, 1021, -11/6*w^3 - 3*w^2 + 44/3*w + 139/6],\ [1021, 1021, 4/3*w^3 - 3*w^2 - 32/3*w + 64/3],\ [1021, 1021, -w^3 + 11*w + 3],\ [1021, 1021, 1/2*w^3 - w - 7/2],\ [1031, 1031, -w^3 + 6*w + 2],\ [1031, 1031, 4/3*w^3 - 38/3*w - 5/3],\ [1049, 1049, -w^3 + 4*w^2 + 3*w - 19],\ [1049, 1049, -5/2*w^3 - 4*w^2 + 25*w + 79/2],\ [1049, 1049, -2*w^3 - 4*w^2 + 18*w + 31],\ [1049, 1049, -w^3 + 4*w^2 + 6*w - 28],\ [1051, 1051, -3/2*w^3 - 2*w^2 + 13*w + 33/2],\ [1051, 1051, 1/6*w^3 - w^2 + 5/3*w - 17/6],\ [1061, 1061, 7/6*w^3 - 5*w^2 - 13/3*w + 157/6],\ [1061, 1061, -1/2*w^3 + 9*w + 23/2],\ [1091, 1091, -4/3*w^3 + 2*w^2 + 17/3*w - 16/3],\ [1091, 1091, -1/6*w^3 + 2*w^2 - 8/3*w - 13/6],\ [1109, 1109, -1/6*w^3 + 2*w^2 - 2/3*w - 91/6],\ [1109, 1109, -5/6*w^3 - 2*w^2 + 26/3*w + 85/6],\ [1151, 1151, -1/2*w^3 + 5*w - 13/2],\ [1151, 1151, -7/6*w^3 - w^2 + 31/3*w + 59/6],\ [1171, 1171, 5/6*w^3 - w^2 - 17/3*w + 23/6],\ [1171, 1171, 7/6*w^3 + w^2 - 31/3*w - 65/6],\ [1181, 1181, w^2 - 6*w + 10],\ [1181, 1181, -7/6*w^3 - w^2 + 46/3*w + 143/6],\ [1229, 1229, -5/3*w^3 - 3*w^2 + 46/3*w + 73/3],\ [1229, 1229, 1/6*w^3 - w^2 - 13/3*w + 109/6],\ [1229, 1229, 5/6*w^3 - 3*w^2 - 14/3*w + 119/6],\ [1229, 1229, -1/6*w^3 + w^2 + 13/3*w + 17/6],\ [1231, 1231, -2/3*w^3 + 25/3*w - 8/3],\ [1231, 1231, -1/6*w^3 - 5/3*w - 13/6],\ [1231, 1231, 1/2*w^3 - w - 3/2],\ [1231, 1231, w^3 - 11*w - 1],\ [1249, 1249, 1/3*w^3 + 3*w^2 - 14/3*w - 89/3],\ [1249, 1249, 7/6*w^3 - 2*w^2 - 25/3*w + 43/6],\ [1279, 1279, -17/6*w^3 - 5*w^2 + 80/3*w + 265/6],\ [1279, 1279, 4/3*w^3 - 5*w^2 - 20/3*w + 88/3],\ [1289, 1289, 5/6*w^3 + 2*w^2 - 23/3*w - 127/6],\ [1289, 1289, 1/2*w^3 + w^2 - 2*w - 21/2],\ [1291, 1291, -1/3*w^3 + 26/3*w - 46/3],\ [1291, 1291, -2/3*w^3 + 34/3*w + 43/3],\ [1301, 1301, 19/6*w^3 + 5*w^2 - 88/3*w - 275/6],\ [1301, 1301, -5/3*w^3 + 5*w^2 + 28/3*w - 83/3],\ [1321, 1321, -2*w^3 - 4*w^2 + 19*w + 34],\ [1321, 1321, -7/6*w^3 + 2*w^2 + 28/3*w - 97/6],\ [1321, 1321, 5/3*w^3 + 2*w^2 - 46/3*w - 64/3],\ [1321, 1321, 5/6*w^3 - 4*w^2 - 11/3*w + 149/6],\ [1381, 1381, -7/6*w^3 - 3*w^2 + 34/3*w + 167/6],\ [1381, 1381, 1/3*w^3 - 3*w^2 - 2/3*w + 49/3],\ [1399, 1399, 7/6*w^3 - 4*w^2 - 22/3*w + 163/6],\ [1399, 1399, 13/6*w^3 + 4*w^2 - 58/3*w - 191/6],\ [1409, 1409, -3/2*w^3 + 4*w^2 + 10*w - 45/2],\ [1409, 1409, 2*w^2 - 9],\ [1409, 1409, -5/2*w^3 - 4*w^2 + 22*w + 73/2],\ [1409, 1409, 1/3*w^3 + 2*w^2 - 8/3*w - 62/3],\ [1439, 1439, 7/6*w^3 - 37/3*w + 7/6],\ [1439, 1439, -1/2*w^3 - 3*w^2 + 7*w + 33/2],\ [1459, 1459, 2*w^2 - 2*w - 17],\ [1459, 1459, -5/6*w^3 + 26/3*w + 55/6],\ [1459, 1459, 2/3*w^3 + 2*w^2 - 22/3*w - 37/3],\ [1459, 1459, -1/2*w^3 + 2*w + 19/2],\ [1511, 1511, 3/2*w^3 - 6*w^2 - 8*w + 75/2],\ [1511, 1511, -19/6*w^3 - 6*w^2 + 88/3*w + 305/6],\ [1531, 1531, 5/2*w^3 + 3*w^2 - 29*w - 87/2],\ [1531, 1531, -13/6*w^3 - 3*w^2 + 61/3*w + 179/6],\ [1549, 1549, 1/2*w^3 + 2*w^2 - 6*w - 21/2],\ [1549, 1549, w^3 - 10*w + 2],\ [1579, 1579, -1/2*w^3 + w^2 + 3*w - 21/2],\ [1579, 1579, 5/6*w^3 + w^2 - 23/3*w - 25/6],\ [1601, 1601, 1/3*w^3 + w^2 - 20/3*w - 5/3],\ [1601, 1601, -7/6*w^3 - 3*w^2 + 31/3*w + 131/6],\ [1609, 1609, 5/6*w^3 - w^2 - 23/3*w + 23/6],\ [1609, 1609, -11/6*w^3 - 3*w^2 + 50/3*w + 175/6],\ [1609, 1609, -5/6*w^3 - w^2 + 17/3*w + 67/6],\ [1609, 1609, -w^3 + 3*w^2 + 6*w - 15],\ [1619, 1619, 1/3*w^3 + 2*w^2 - 5/3*w - 53/3],\ [1619, 1619, 2/3*w^3 - w^2 - 16/3*w + 2/3],\ [1619, 1619, 3/2*w^3 + 2*w^2 - 12*w - 39/2],\ [1619, 1619, 1/3*w^3 - 2*w^2 - 14/3*w + 70/3],\ [1621, 1621, 1/6*w^3 + 3*w^2 - 4/3*w - 113/6],\ [1621, 1621, -1/3*w^3 - 3*w^2 + 8/3*w + 77/3],\ [1681, 41, w^3 - 8*w - 4],\ [1699, 1699, 5/6*w^3 + w^2 - 29/3*w - 37/6],\ [1699, 1699, -1/6*w^3 + w^2 - 5/3*w - 49/6],\ [1709, 1709, -1/6*w^3 - 2*w^2 + 13/3*w + 35/6],\ [1709, 1709, -3/2*w^3 - w^2 + 14*w + 15/2],\ [1721, 1721, -5/6*w^3 - w^2 + 35/3*w + 103/6],\ [1721, 1721, 5/3*w^3 + 2*w^2 - 43/3*w - 61/3],\ [1721, 1721, -7/6*w^3 + 2*w^2 + 25/3*w - 55/6],\ [1721, 1721, -1/6*w^3 - w^2 + 19/3*w - 19/6],\ [1741, 1741, 1/6*w^3 - 3*w^2 + 5/3*w + 115/6],\ [1741, 1741, -1/2*w^3 + 4*w^2 - w - 25/2],\ [1741, 1741, 7/6*w^3 + 3*w^2 - 37/3*w - 149/6],\ [1741, 1741, -11/6*w^3 + 6*w^2 + 17/3*w - 131/6],\ [1789, 1789, 1/6*w^3 - w^2 - 13/3*w + 103/6],\ [1789, 1789, w^3 + 2*w^2 - 8*w - 18],\ [1789, 1789, -2*w^3 - 3*w^2 + 18*w + 30],\ [1789, 1789, -2/3*w^3 + 2*w^2 + 16/3*w - 35/3],\ [1801, 1801, -7/6*w^3 - w^2 + 28/3*w + 47/6],\ [1801, 1801, -w^3 + w^2 + 8*w - 7],\ [1811, 1811, 4/3*w^3 - 29/3*w - 26/3],\ [1811, 1811, -7/6*w^3 + w^2 + 31/3*w - 43/6],\ [1831, 1831, -1/3*w^3 + 2/3*w + 32/3],\ [1831, 1831, -3/2*w^3 - 2*w^2 + 12*w + 33/2],\ [1849, 43, -7/6*w^3 - 2*w^2 + 31/3*w + 125/6],\ [1849, 43, -1/2*w^3 + 3*w^2 + 5*w - 43/2],\ [1871, 1871, -3/2*w^3 - 2*w^2 + 12*w + 29/2],\ [1871, 1871, 7/6*w^3 - 2*w^2 - 28/3*w + 91/6],\ [1879, 1879, 3/2*w^3 + 3*w^2 - 13*w - 53/2],\ [1879, 1879, -5/6*w^3 + 3*w^2 + 17/3*w - 107/6],\ [1889, 1889, -1/2*w^3 - w^2 + 6*w + 5/2],\ [1889, 1889, w^2 - 2*w - 12],\ [1901, 1901, w^2 + 2*w + 2],\ [1901, 1901, 1/2*w^3 - 3*w^2 + w + 13/2],\ [1901, 1901, -1/6*w^3 + 4*w^2 - 11/3*w - 73/6],\ [1901, 1901, 1/6*w^3 - w^2 - 10/3*w + 103/6],\ [1931, 1931, 7/6*w^3 - w^2 - 28/3*w + 13/6],\ [1931, 1931, 4/3*w^3 + w^2 - 32/3*w - 38/3],\ [1931, 1931, -7/6*w^3 - 3*w^2 + 31/3*w + 137/6],\ [1931, 1931, 1/2*w^3 - 3*w^2 - 3*w + 43/2],\ [1949, 1949, 4/3*w^3 - 3*w^2 - 32/3*w + 58/3],\ [1949, 1949, 11/6*w^3 + 3*w^2 - 44/3*w - 151/6],\ [1951, 1951, 11/6*w^3 - 5*w^2 - 20/3*w + 113/6],\ [1951, 1951, -4/3*w^3 + 26/3*w - 1/3],\ [1999, 1999, -1/6*w^3 + 3*w^2 - 2/3*w - 151/6],\ [1999, 1999, 1/2*w^3 + 2*w^2 - 4*w - 49/2]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^6 - 17*x^4 + 56*x^2 - 4 K. = NumberField(heckePol) hecke_eigenvalues_array = [e, e, 1/8*e^5 - 19/8*e^3 + 39/4*e, 3/8*e^5 - 49/8*e^3 + 73/4*e, 3/8*e^5 - 49/8*e^3 + 73/4*e, -1/8*e^4 + 11/8*e^2 - 11/4, -1/8*e^4 + 11/8*e^2 - 11/4, 1/8*e^4 - 19/8*e^2 + 15/4, 1/8*e^4 - 19/8*e^2 + 15/4, -3/4*e^5 + 53/4*e^3 - 91/2*e, -3/4*e^5 + 53/4*e^3 - 91/2*e, -3/8*e^4 + 41/8*e^2 - 29/4, -3/8*e^4 + 41/8*e^2 - 29/4, 1/8*e^4 - 11/8*e^2 - 5/4, 1/8*e^4 - 11/8*e^2 - 5/4, 3/8*e^4 - 49/8*e^2 + 41/4, 3/8*e^4 - 49/8*e^2 + 41/4, 5/8*e^5 - 87/8*e^3 + 135/4*e, 5/8*e^5 - 87/8*e^3 + 135/4*e, 3/8*e^5 - 41/8*e^3 + 37/4*e, 3/8*e^5 - 41/8*e^3 + 37/4*e, -3/8*e^4 + 25/8*e^2 + 19/4, -3/8*e^4 + 25/8*e^2 + 19/4, e^5 - 17*e^3 + 56*e, e^5 - 17*e^3 + 56*e, -1/4*e^4 + 11/4*e^2 + 29/2, -5/8*e^4 + 79/8*e^2 - 67/4, -5/8*e^4 + 79/8*e^2 - 67/4, -7/8*e^5 + 117/8*e^3 - 177/4*e, -7/8*e^5 + 117/8*e^3 - 177/4*e, -1/4*e^5 + 11/4*e^3 + 1/2*e, -1/8*e^4 + 3/8*e^2 + 25/4, -1/8*e^4 + 3/8*e^2 + 25/4, 1/4*e^5 - 15/4*e^3 + 21/2*e, 1/4*e^5 - 15/4*e^3 + 21/2*e, 1/8*e^4 + 5/8*e^2 - 37/4, 1/8*e^4 + 5/8*e^2 - 37/4, -1/8*e^4 + 19/8*e^2 - 15/4, 7/8*e^4 - 85/8*e^2 + 33/4, -1/8*e^4 + 19/8*e^2 - 15/4, 7/8*e^4 - 85/8*e^2 + 33/4, -e^5 + 16*e^3 - 47*e, -e^5 + 16*e^3 - 47*e, -1/4*e^4 + 19/4*e^2 - 47/2, -1/4*e^4 + 19/4*e^2 - 47/2, -3/8*e^5 + 33/8*e^3 + 7/4*e, -7/8*e^5 + 125/8*e^3 - 205/4*e, -3/8*e^5 + 33/8*e^3 + 7/4*e, -7/8*e^5 + 125/8*e^3 - 205/4*e, -7/4*e^5 + 117/4*e^3 - 193/2*e, -7/4*e^5 + 117/4*e^3 - 193/2*e, -7/8*e^4 + 101/8*e^2 - 57/4, -7/8*e^4 + 101/8*e^2 - 57/4, -3/4*e^4 + 41/4*e^2 - 45/2, -3/4*e^4 + 41/4*e^2 - 45/2, -1/2*e^4 + 11/2*e^2 - 11, -1/2*e^4 + 11/2*e^2 - 11, -3/8*e^4 + 65/8*e^2 - 73/4, -3/8*e^4 + 65/8*e^2 - 73/4, -9/8*e^4 + 107/8*e^2 - 7/4, -9/8*e^4 + 107/8*e^2 - 7/4, -1/8*e^4 + 11/8*e^2 + 21/4, -1/8*e^4 + 11/8*e^2 + 21/4, 7/8*e^4 - 93/8*e^2 + 69/4, 7/8*e^4 - 93/8*e^2 + 69/4, 13/8*e^5 - 207/8*e^3 + 287/4*e, 13/8*e^5 - 207/8*e^3 + 287/4*e, 1/8*e^4 - 27/8*e^2 + 147/4, 1/8*e^4 - 27/8*e^2 + 147/4, 3/8*e^5 - 49/8*e^3 + 57/4*e, 9/8*e^5 - 147/8*e^3 + 187/4*e, 9/8*e^5 - 147/8*e^3 + 187/4*e, 3/8*e^5 - 49/8*e^3 + 57/4*e, -1/4*e^5 + 23/4*e^3 - 53/2*e, -1/4*e^5 + 23/4*e^3 - 53/2*e, 9/8*e^4 - 139/8*e^2 + 143/4, 9/8*e^4 - 107/8*e^2 + 55/4, 9/8*e^4 - 139/8*e^2 + 143/4, 9/8*e^4 - 107/8*e^2 + 55/4, 7/4*e^5 - 125/4*e^3 + 241/2*e, 7/4*e^5 - 125/4*e^3 + 241/2*e, -3/8*e^5 + 57/8*e^3 - 125/4*e, -3/8*e^5 + 57/8*e^3 - 125/4*e, 21/8*e^5 - 343/8*e^3 + 503/4*e, 21/8*e^5 - 343/8*e^3 + 503/4*e, 1/8*e^4 - 43/8*e^2 + 123/4, 1/8*e^4 - 43/8*e^2 + 123/4, -9/8*e^5 + 163/8*e^3 - 323/4*e, -9/8*e^5 + 163/8*e^3 - 323/4*e, -1/4*e^4 + 11/4*e^2 - 11/2, -1/4*e^4 + 11/4*e^2 - 11/2, 7/8*e^5 - 133/8*e^3 + 265/4*e, 7/8*e^5 - 133/8*e^3 + 265/4*e, 2*e^3 - 26*e, 2*e^3 - 26*e, -5/8*e^4 + 55/8*e^2 - 79/4, -5/8*e^4 + 55/8*e^2 - 79/4, -11/4*e^5 + 181/4*e^3 - 251/2*e, -11/4*e^5 + 181/4*e^3 - 251/2*e, -13/8*e^4 + 191/8*e^2 - 135/4, -13/8*e^4 + 191/8*e^2 - 135/4, 11/8*e^5 - 185/8*e^3 + 345/4*e, 11/8*e^5 - 185/8*e^3 + 345/4*e, 25/8*e^5 - 427/8*e^3 + 735/4*e, 25/8*e^5 - 427/8*e^3 + 735/4*e, 19/8*e^5 - 321/8*e^3 + 521/4*e, 19/8*e^5 - 321/8*e^3 + 521/4*e, -5/8*e^4 + 87/8*e^2 - 39/4, 15/8*e^4 - 205/8*e^2 + 169/4, -5/8*e^4 + 87/8*e^2 - 39/4, 15/8*e^4 - 205/8*e^2 + 169/4, e^5 - 15*e^3 + 38*e, -11/4*e^5 + 193/4*e^3 - 349/2*e, e^5 - 15*e^3 + 38*e, -11/4*e^5 + 193/4*e^3 - 349/2*e, 1/8*e^4 - 27/8*e^2 + 99/4, 1/8*e^4 - 27/8*e^2 + 99/4, -11/8*e^4 + 145/8*e^2 - 53/4, -1/4*e^4 + 19/4*e^2 - 23/2, -1/4*e^4 + 19/4*e^2 - 23/2, -11/8*e^4 + 145/8*e^2 - 53/4, 1/8*e^4 - 11/8*e^2 + 59/4, 1/8*e^4 - 11/8*e^2 + 59/4, 7/4*e^5 - 129/4*e^3 + 255/2*e, -5/8*e^5 + 87/8*e^3 - 127/4*e, -5/8*e^5 + 87/8*e^3 - 127/4*e, 7/4*e^5 - 129/4*e^3 + 255/2*e, -9/4*e^5 + 151/4*e^3 - 261/2*e, 7/8*e^5 - 109/8*e^3 + 157/4*e, 7/8*e^5 - 109/8*e^3 + 157/4*e, -9/4*e^5 + 151/4*e^3 - 261/2*e, -23/8*e^5 + 397/8*e^3 - 637/4*e, -23/8*e^5 + 397/8*e^3 - 637/4*e, -9/4*e^5 + 151/4*e^3 - 233/2*e, -9/4*e^5 + 151/4*e^3 - 233/2*e, -7/8*e^4 + 45/8*e^2 + 115/4, -7/8*e^4 + 45/8*e^2 + 115/4, -7/8*e^4 + 109/8*e^2 - 21/4, -5/4*e^4 + 55/4*e^2 - 55/2, -7/8*e^4 + 109/8*e^2 - 21/4, -5/4*e^4 + 55/4*e^2 - 55/2, 17/8*e^5 - 291/8*e^3 + 471/4*e, 17/8*e^5 - 291/8*e^3 + 471/4*e, 5/8*e^5 - 87/8*e^3 + 119/4*e, 5/8*e^5 - 87/8*e^3 + 119/4*e, 9/8*e^4 - 155/8*e^2 + 231/4, -7/8*e^4 + 109/8*e^2 + 59/4, 9/4*e^5 - 143/4*e^3 + 177/2*e, -9/4*e^5 + 151/4*e^3 - 213/2*e, 9/4*e^5 - 143/4*e^3 + 177/2*e, -9/4*e^5 + 151/4*e^3 - 213/2*e, 3/8*e^4 - 1/8*e^2 - 127/4, 3/8*e^4 - 1/8*e^2 - 127/4, 9/8*e^4 - 115/8*e^2 + 11/4, 1/8*e^4 - 11/8*e^2 - 37/4, 1/8*e^4 - 11/8*e^2 - 37/4, 9/8*e^4 - 115/8*e^2 + 11/4, -27/8*e^5 + 425/8*e^3 - 585/4*e, 2*e^5 - 35*e^3 + 121*e, -27/8*e^5 + 425/8*e^3 - 585/4*e, 2*e^5 - 35*e^3 + 121*e, 4*e^5 - 66*e^3 + 194*e, -7/4*e^5 + 109/4*e^3 - 129/2*e, -7/4*e^5 + 109/4*e^3 - 129/2*e, 4*e^5 - 66*e^3 + 194*e, -9/4*e^4 + 99/4*e^2 - 11/2, e^4 - 9*e^2 + 16, 1/2*e^4 + 1/2*e^2 - 55, 1/2*e^4 + 1/2*e^2 - 55, -7/8*e^4 + 53/8*e^2 + 111/4, -7/8*e^4 + 53/8*e^2 + 111/4, 1/8*e^5 + 21/8*e^3 - 157/4*e, 1/8*e^5 + 21/8*e^3 - 157/4*e, -5/8*e^4 + 31/8*e^2 + 37/4, -5/8*e^4 + 31/8*e^2 + 37/4, 15/8*e^4 - 173/8*e^2 + 81/4, 15/8*e^4 - 173/8*e^2 + 81/4, -3/8*e^4 - 7/8*e^2 + 83/4, -3/8*e^4 - 7/8*e^2 + 83/4, -13/8*e^5 + 231/8*e^3 - 403/4*e, -13/8*e^5 + 231/8*e^3 - 403/4*e, -9/2*e^5 + 147/2*e^3 - 219*e, -9/2*e^5 + 147/2*e^3 - 219*e, 5/8*e^4 - 39/8*e^2 - 65/4, 5/8*e^4 - 39/8*e^2 - 65/4, -3/8*e^4 - 7/8*e^2 + 203/4, -3/8*e^4 - 7/8*e^2 + 203/4, 17/8*e^4 - 275/8*e^2 + 231/4, 17/8*e^4 - 275/8*e^2 + 231/4, -29/8*e^5 + 495/8*e^3 - 839/4*e, -29/8*e^5 + 495/8*e^3 - 839/4*e, 13/8*e^4 - 167/8*e^2 + 59/4, 13/8*e^4 - 167/8*e^2 + 59/4, 17/8*e^4 - 179/8*e^2 - 25/4, 17/8*e^4 - 179/8*e^2 - 25/4, 1/8*e^4 - 43/8*e^2 + 195/4, 1/8*e^4 - 43/8*e^2 + 195/4, -5*e^5 + 83*e^3 - 264*e, -2*e^5 + 33*e^3 - 109*e, -5*e^5 + 83*e^3 - 264*e, -2*e^5 + 33*e^3 - 109*e, -11/8*e^4 + 177/8*e^2 - 165/4, -11/8*e^4 + 177/8*e^2 - 165/4, 1/8*e^4 - 11/8*e^2 - 5/4, 1/8*e^4 - 11/8*e^2 - 5/4, 29/8*e^5 - 487/8*e^3 + 755/4*e, 29/8*e^5 - 487/8*e^3 + 755/4*e, -3/8*e^5 + 65/8*e^3 - 177/4*e, -3/8*e^5 + 65/8*e^3 - 177/4*e, 13/4*e^5 - 215/4*e^3 + 311/2*e, 13/4*e^5 - 215/4*e^3 + 311/2*e, -3/4*e^4 + 33/4*e^2 - 89/2, -3/4*e^4 + 33/4*e^2 - 89/2, -3/8*e^4 + 89/8*e^2 - 37/4, -3/8*e^4 + 89/8*e^2 - 37/4, 9/8*e^4 - 91/8*e^2 + 103/4, 5/8*e^4 - 55/8*e^2 - 33/4, 5/8*e^4 - 55/8*e^2 - 33/4, 9/8*e^4 - 91/8*e^2 + 103/4, 1/4*e^4 - 11/4*e^2 + 15/2, 1/4*e^4 - 11/4*e^2 + 15/2, 2*e^5 - 29*e^3 + 67*e, 2*e^5 - 29*e^3 + 67*e, -7/4*e^5 + 117/4*e^3 - 193/2*e, 19/8*e^5 - 313/8*e^3 + 429/4*e, -7/4*e^5 + 117/4*e^3 - 193/2*e, 19/8*e^5 - 313/8*e^3 + 429/4*e, -2*e^5 + 36*e^3 - 142*e, -2*e^5 + 36*e^3 - 142*e, -21/8*e^5 + 383/8*e^3 - 723/4*e, -37/8*e^5 + 599/8*e^3 - 919/4*e, -21/8*e^5 + 383/8*e^3 - 723/4*e, -37/8*e^5 + 599/8*e^3 - 919/4*e, -5/8*e^4 + 79/8*e^2 - 163/4, -5/8*e^4 + 79/8*e^2 - 163/4, 2*e^4 - 28*e^2 + 46, 2*e^4 - 28*e^2 + 46, 11/4*e^5 - 201/4*e^3 + 397/2*e, 11/4*e^5 - 201/4*e^3 + 397/2*e, 3/2*e^5 - 49/2*e^3 + 69*e, 3/2*e^5 - 49/2*e^3 + 69*e, 9/8*e^4 - 179/8*e^2 + 139/4, 9/8*e^4 - 179/8*e^2 + 139/4, -13/8*e^5 + 215/8*e^3 - 379/4*e, 5*e^5 - 81*e^3 + 236*e, -13/8*e^5 + 215/8*e^3 - 379/4*e, 5*e^5 - 81*e^3 + 236*e, -23/8*e^5 + 405/8*e^3 - 713/4*e, -23/8*e^5 + 405/8*e^3 - 713/4*e, -21/8*e^5 + 319/8*e^3 - 419/4*e, -21/8*e^5 + 319/8*e^3 - 419/4*e, -9/8*e^4 + 67/8*e^2 + 117/4, -9/8*e^4 + 67/8*e^2 + 117/4, 7/4*e^4 - 85/4*e^2 + 133/2, 19/8*e^5 - 289/8*e^3 + 361/4*e, 19/8*e^5 - 289/8*e^3 + 361/4*e, 11/2*e^5 - 183/2*e^3 + 284*e, 11/2*e^5 - 183/2*e^3 + 284*e, -11/8*e^4 + 161/8*e^2 - 189/4, 19/8*e^4 - 193/8*e^2 + 9/4, 19/8*e^4 - 193/8*e^2 + 9/4, -11/8*e^4 + 161/8*e^2 - 189/4, 1/8*e^4 - 51/8*e^2 + 39/4, -11/8*e^4 + 169/8*e^2 - 209/4, 1/8*e^4 - 51/8*e^2 + 39/4, -11/8*e^4 + 169/8*e^2 - 209/4, -11/4*e^5 + 193/4*e^3 - 365/2*e, e^5 - 21*e^3 + 102*e, -11/4*e^5 + 193/4*e^3 - 365/2*e, e^5 - 21*e^3 + 102*e, 3/8*e^4 - 25/8*e^2 - 203/4, 3/8*e^4 - 25/8*e^2 - 203/4, -9/8*e^4 + 155/8*e^2 - 223/4, -9/8*e^4 + 155/8*e^2 - 223/4, -19/8*e^4 + 281/8*e^2 - 309/4, -19/8*e^4 + 281/8*e^2 - 309/4, -17/8*e^5 + 275/8*e^3 - 311/4*e, -17/8*e^5 + 275/8*e^3 - 311/4*e, -19/8*e^4 + 241/8*e^2 - 273/4, -19/8*e^4 + 241/8*e^2 - 273/4, 21/8*e^5 - 367/8*e^3 + 699/4*e, 21/8*e^5 - 367/8*e^3 + 699/4*e, -15/8*e^5 + 237/8*e^3 - 273/4*e, -15/8*e^5 + 237/8*e^3 - 273/4*e, -1/8*e^4 + 11/8*e^2 + 61/4, 9/8*e^4 - 155/8*e^2 + 103/4, 9/8*e^4 - 155/8*e^2 + 103/4, -1/8*e^4 + 11/8*e^2 + 61/4, -5/8*e^4 + 55/8*e^2 - 55/4, -5/8*e^4 + 55/8*e^2 - 55/4, -5/8*e^4 + 31/8*e^2 + 245/4, -5/8*e^4 + 31/8*e^2 + 245/4, -1/8*e^5 + 51/8*e^3 - 247/4*e, -1/8*e^5 + 51/8*e^3 - 247/4*e, 11/8*e^4 - 217/8*e^2 + 297/4, 11/8*e^4 - 217/8*e^2 + 297/4, -1/2*e^5 + 15/2*e^3 - 21*e, -1/2*e^5 + 15/2*e^3 - 21*e] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]