/* 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([9, 3, 1/6*w^3 - 7/3*w + 13/6]) 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^7 - 4*x^6 - 11*x^5 + 44*x^4 + 42*x^3 - 105*x^2 - 110*x - 20 K. = NumberField(heckePol) hecke_eigenvalues_array = [7/139*e^6 + 13/139*e^5 - 120/139*e^4 - 236/139*e^3 + 401/139*e^2 + 1018/139*e + 685/139, e, 15/278*e^6 + 4/139*e^5 - 277/278*e^4 - 94/139*e^3 + 658/139*e^2 + 851/278*e - 110/139, -1, -35/278*e^6 + 37/139*e^5 + 461/278*e^4 - 383/139*e^3 - 794/139*e^2 + 1721/278*e + 720/139, 7/139*e^6 + 13/139*e^5 - 120/139*e^4 - 236/139*e^3 + 401/139*e^2 + 1157/139*e + 546/139, 13/278*e^6 + 22/139*e^5 - 203/278*e^4 - 517/139*e^3 + 283/139*e^2 + 5167/278*e + 1341/139, 113/278*e^6 - 322/139*e^5 - 567/278*e^4 + 3397/139*e^3 - 566/139*e^2 - 15199/278*e - 3099/139, 29/278*e^6 + 156/139*e^5 - 1073/278*e^4 - 1998/139*e^3 + 2866/139*e^2 + 13451/278*e + 1826/139, -18/139*e^6 + 185/139*e^5 - 168/139*e^4 - 1915/139*e^3 + 2424/139*e^2 + 4372/139*e - 570/139, 187/278*e^6 - 293/139*e^5 - 1637/278*e^4 + 2368/139*e^3 + 799/139*e^2 - 3087/278*e + 1455/139, 57/278*e^6 - 235/139*e^5 + 115/278*e^4 + 2534/139*e^3 - 1753/139*e^2 - 12779/278*e - 1669/139, 53/139*e^6 - 120/139*e^5 - 710/139*e^4 + 1152/139*e^3 + 2361/139*e^2 - 1645/139*e - 314/139, 112/139*e^6 - 348/139*e^5 - 1225/139*e^4 + 3174/139*e^3 + 3497/139*e^2 - 4562/139*e - 2384/139, 31/278*e^6 + 138/139*e^5 - 1147/278*e^4 - 1714/139*e^3 + 3241/139*e^2 + 10803/278*e + 931/139, 15/139*e^6 - 131/139*e^5 - 138/139*e^4 + 1897/139*e^3 + 621/139*e^2 - 6794/139*e - 3834/139, -97/139*e^6 + 356/139*e^5 + 948/139*e^4 - 3362/139*e^3 - 2320/139*e^2 + 5413/139*e + 3276/139, 13/278*e^6 - 256/139*e^5 + 909/278*e^4 + 2958/139*e^3 - 3748/139*e^2 - 15405/278*e - 605/139, 227/278*e^6 - 375/139*e^5 - 2283/278*e^4 + 3461/139*e^3 + 2600/139*e^2 - 10177/278*e - 1850/139, -38/139*e^6 + 267/139*e^5 + 155/139*e^4 - 2869/139*e^3 + 345/139*e^2 + 6249/139*e + 3430/139, 53/278*e^6 + 79/139*e^5 - 1127/278*e^4 - 1231/139*e^3 + 2640/139*e^2 + 9475/278*e + 1650/139, 25/278*e^6 + 53/139*e^5 - 647/278*e^4 - 759/139*e^3 + 1838/139*e^2 + 4847/278*e + 141/139, -126/139*e^6 + 461/139*e^5 + 1187/139*e^4 - 4509/139*e^3 - 2353/139*e^2 + 8364/139*e + 3516/139, -31/278*e^6 - 138/139*e^5 + 869/278*e^4 + 1853/139*e^3 - 2268/139*e^2 - 11637/278*e - 375/139, -135/139*e^6 + 484/139*e^5 + 1242/139*e^4 - 4424/139*e^3 - 2114/139*e^2 + 5963/139*e + 1980/139, 69/278*e^6 - 65/139*e^5 - 885/278*e^4 + 207/139*e^3 + 1470/139*e^2 + 4415/278*e + 328/139, -145/139*e^6 + 525/139*e^5 + 1334/139*e^4 - 4762/139*e^3 - 2389/139*e^2 + 6137/139*e + 1756/139, -17/278*e^6 - 125/139*e^5 + 629/278*e^4 + 1617/139*e^3 - 1450/139*e^2 - 10435/278*e - 1914/139, -53/278*e^6 - 218/139*e^5 + 1683/278*e^4 + 2899/139*e^3 - 4308/139*e^2 - 20317/278*e - 2345/139, -255/278*e^6 + 349/139*e^5 + 3041/278*e^4 - 3267/139*e^3 - 4653/139*e^2 + 8607/278*e + 2565/139, -55/139*e^6 + 573/139*e^5 - 189/139*e^4 - 6585/139*e^3 + 3700/139*e^2 + 17637/139*e + 5440/139, 5/139*e^6 + 49/139*e^5 - 46/139*e^4 - 1082/139*e^3 - 210/139*e^2 + 4917/139*e + 3726/139, -57/278*e^6 - 43/139*e^5 + 997/278*e^4 + 663/139*e^3 - 1861/139*e^2 - 4457/278*e - 694/139, 56/139*e^6 - 313/139*e^5 - 404/139*e^4 + 3394/139*e^3 + 706/139*e^2 - 8119/139*e - 4250/139, -26/139*e^6 - 88/139*e^5 + 545/139*e^4 + 1373/139*e^3 - 2383/139*e^2 - 5747/139*e - 360/139, -135/139*e^6 + 206/139*e^5 + 1937/139*e^4 - 1644/139*e^3 - 7118/139*e^2 + 264/139*e + 2536/139, 38/139*e^6 - 128/139*e^5 - 294/139*e^4 + 1062/139*e^3 - 67/139*e^2 - 1523/139*e + 1296/139, -10/139*e^6 + 41/139*e^5 + 231/139*e^4 - 477/139*e^3 - 1526/139*e^2 + 452/139*e + 2556/139, 108/139*e^6 - 276/139*e^5 - 1355/139*e^4 + 2316/139*e^3 + 4777/139*e^2 - 1768/139*e - 2974/139, 93/139*e^6 - 562/139*e^5 - 522/139*e^4 + 6118/139*e^3 - 570/139*e^2 - 14017/139*e - 5534/139, -143/278*e^6 + 175/139*e^5 + 1955/278*e^4 - 1680/139*e^3 - 3947/139*e^2 + 5435/278*e + 3736/139, 131/278*e^6 - 345/139*e^5 - 677/278*e^4 + 3590/139*e^3 - 666/139*e^2 - 16513/278*e - 1980/139, -255/278*e^6 + 349/139*e^5 + 2763/278*e^4 - 3128/139*e^3 - 3402/139*e^2 + 8051/278*e + 1870/139, -78/139*e^6 + 292/139*e^5 + 523/139*e^4 - 2275/139*e^3 + 774/139*e^2 + 829/139*e - 3304/139, 183/278*e^6 - 118/139*e^5 - 2879/278*e^4 + 827/139*e^3 + 5748/139*e^2 + 2765/278*e - 369/139, -42/139*e^6 + 61/139*e^5 + 442/139*e^4 - 252/139*e^3 - 877/139*e^2 - 1104/139*e + 60/139, 83/139*e^6 - 243/139*e^5 - 986/139*e^4 + 2305/139*e^3 + 3186/139*e^2 - 3835/139*e + 80/139, -163/139*e^6 + 432/139*e^5 + 1861/139*e^4 - 3758/139*e^3 - 5247/139*e^2 + 4115/139*e + 2020/139, -2/139*e^6 - 103/139*e^5 + 74/139*e^4 + 1239/139*e^3 - 55/139*e^2 - 2217/139*e - 2380/139, -90/139*e^6 + 369/139*e^5 + 828/139*e^4 - 3459/139*e^3 - 1502/139*e^2 + 4346/139*e + 1320/139, 163/278*e^6 - 216/139*e^5 - 1583/278*e^4 + 1462/139*e^3 + 1442/139*e^2 + 2001/278*e + 1770/139, 64/139*e^6 - 40/139*e^5 - 978/139*e^4 - 311/139*e^3 + 3289/139*e^2 + 5475/139*e + 2768/139, -61/278*e^6 - 285/139*e^5 + 1701/278*e^4 + 4265/139*e^3 - 4001/139*e^2 - 31965/278*e - 5437/139, -119/278*e^6 - 41/139*e^5 + 1901/278*e^4 + 1589/139*e^3 - 3339/139*e^2 - 20503/278*e - 4919/139, 69/278*e^6 - 482/139*e^5 + 783/278*e^4 + 5767/139*e^3 - 4229/139*e^2 - 33115/278*e - 4954/139, -116/139*e^6 + 420/139*e^5 + 1234/139*e^4 - 4588/139*e^3 - 3607/139*e^2 + 10970/139*e + 7076/139, -57/139*e^6 - 225/139*e^5 + 1414/139*e^4 + 3411/139*e^3 - 7058/139*e^2 - 13631/139*e - 3334/139, -125/278*e^6 + 430/139*e^5 + 177/278*e^4 - 4267/139*e^3 + 1930/139*e^2 + 17465/278*e + 3048/139, -15/278*e^6 + 274/139*e^5 - 835/278*e^4 - 2964/139*e^3 + 3234/139*e^2 + 13605/278*e + 1083/139, 14/139*e^6 + 26/139*e^5 - 101/139*e^4 - 889/139*e^3 + 246/139*e^2 + 4260/139*e + 258/139, -53/278*e^6 + 60/139*e^5 + 571/278*e^4 - 437/139*e^3 - 833/139*e^2 + 1645/278*e + 713/139, 30/139*e^6 - 401/139*e^5 + 280/139*e^4 + 4211/139*e^3 - 3067/139*e^2 - 9140/139*e - 2942/139, -229/278*e^6 + 393/139*e^5 + 1801/278*e^4 - 3328/139*e^3 - 751/139*e^2 + 7265/278*e + 1633/139, 369/278*e^6 - 680/139*e^5 - 3367/278*e^4 + 6806/139*e^3 + 3093/139*e^2 - 27493/278*e - 5069/139, -64/139*e^6 + 318/139*e^5 + 283/139*e^4 - 3164/139*e^3 + 1993/139*e^2 + 6062/139*e - 544/139, 125/278*e^6 - 13/139*e^5 - 1567/278*e^4 - 876/139*e^3 + 2379/139*e^2 + 13393/278*e + 705/139, -15/278*e^6 - 282/139*e^5 + 1111/278*e^4 + 4264/139*e^3 - 3438/139*e^2 - 31709/278*e - 5450/139, -395/278*e^6 + 497/139*e^5 + 4329/278*e^4 - 3965/139*e^3 - 5605/139*e^2 + 5205/278*e + 163/139, 23/139*e^6 - 414/139*e^5 + 539/139*e^4 + 4447/139*e^3 - 4997/139*e^2 - 10853/139*e - 2654/139, 92/139*e^6 - 544/139*e^5 - 763/139*e^4 + 6112/139*e^3 + 2252/139*e^2 - 15890/139*e - 7280/139, -164/139*e^6 + 450/139*e^5 + 1759/139*e^4 - 3208/139*e^3 - 4927/139*e^2 - 260/139*e + 2220/139, -59/139*e^6 + 228/139*e^5 + 376/139*e^4 - 2161/139*e^3 + 254/139*e^2 + 3890/139*e + 3460/139, -209/278*e^6 + 491/139*e^5 + 1339/278*e^4 - 4936/139*e^3 + 358/139*e^2 + 21095/278*e + 4220/139, -130/139*e^6 + 255/139*e^5 + 1474/139*e^4 - 1614/139*e^3 - 4409/139*e^2 - 1491/139*e + 3760/139, 128/139*e^6 - 636/139*e^5 - 1122/139*e^4 + 7162/139*e^3 + 2408/139*e^2 - 17406/139*e - 8920/139, -215/278*e^6 - 11/139*e^5 + 3507/278*e^4 + 1162/139*e^3 - 6744/139*e^2 - 17665/278*e - 3242/139, 280/139*e^6 - 870/139*e^5 - 2715/139*e^4 + 7518/139*e^3 + 5476/139*e^2 - 8625/139*e - 2902/139, -65/139*e^6 + 475/139*e^5 + 320/139*e^4 - 5811/139*e^3 + 645/139*e^2 + 17255/139*e + 4938/139, 74/139*e^6 - 359/139*e^5 - 375/139*e^4 + 3780/139*e^3 - 884/139*e^2 - 9850/139*e - 3402/139, 216/139*e^6 - 552/139*e^5 - 2432/139*e^4 + 4771/139*e^3 + 6913/139*e^2 - 5899/139*e - 2890/139, 1/2*e^6 - 5*e^5 + 3/2*e^4 + 53*e^3 - 31*e^2 - 237/2*e - 30, -52/139*e^6 - 315/139*e^5 + 1229/139*e^4 + 4970/139*e^3 - 5183/139*e^2 - 20390/139*e - 9060/139, 127/278*e^6 - 448/139*e^5 - 251/278*e^4 + 5107/139*e^3 - 2528/139*e^2 - 25395/278*e - 2970/139, -308/139*e^6 + 957/139*e^5 + 3473/139*e^4 - 9354/139*e^3 - 10138/139*e^2 + 16368/139*e + 8780/139, 373/139*e^6 - 1154/139*e^5 - 3932/139*e^4 + 10856/139*e^3 + 10049/139*e^2 - 17499/139*e - 9270/139, -217/139*e^6 + 987/139*e^5 + 1774/139*e^4 - 10059/139*e^3 - 2423/139*e^2 + 20150/139*e + 7538/139, 211/278*e^6 - 231/139*e^5 - 2247/278*e^4 + 1606/139*e^3 + 3075/139*e^2 + 165/278*e - 2752/139, -196/139*e^6 + 609/139*e^5 + 2109/139*e^4 - 5485/139*e^3 - 5251/139*e^2 + 6802/139*e + 1670/139, -441/278*e^6 + 633/139*e^5 + 4363/278*e^4 - 5076/139*e^3 - 3388/139*e^2 + 4393/278*e - 4550/139, 210/139*e^6 - 1000/139*e^5 - 1376/139*e^4 + 9878/139*e^3 - 341/139*e^2 - 19500/139*e - 5304/139, 213/278*e^6 - 666/139*e^5 - 1209/278*e^4 + 7172/139*e^3 - 581/139*e^2 - 31395/278*e - 7539/139, -269/278*e^6 + 475/139*e^5 + 2447/278*e^4 - 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3066/139*e - 60/139, -399/278*e^6 + 1506/139*e^5 + 1697/278*e^4 - 17877/139*e^3 + 1429/139*e^2 + 103075/278*e + 21135/139, 158/139*e^6 - 898/139*e^5 - 1120/139*e^4 + 9844/139*e^3 + 1009/139*e^2 - 22793/139*e - 10194/139, -171/278*e^6 + 288/139*e^5 + 489/278*e^4 - 2181/139*e^3 + 4981/139*e^2 - 1417/278*e - 10144/139, 337/139*e^6 - 1757/139*e^5 - 1766/139*e^4 + 17034/139*e^3 - 4285/139*e^2 - 31273/139*e - 4850/139, 219/139*e^6 - 467/139*e^5 - 2265/139*e^4 + 2704/139*e^3 + 3590/139*e^2 + 5836/139*e + 7630/139, 857/278*e^6 - 1597/139*e^5 - 8079/278*e^4 + 15220/139*e^3 + 9664/139*e^2 - 55611/278*e - 17590/139, -631/139*e^6 + 1628/139*e^5 + 6806/139*e^4 - 12682/139*e^3 - 17561/139*e^2 + 4780/139*e + 3880/139, 349/278*e^6 + 334/139*e^5 - 6519/278*e^4 - 6876/139*e^3 + 12826/139*e^2 + 70155/278*e + 14445/139, 435/278*e^6 - 996/139*e^5 - 4419/278*e^4 + 11174/139*e^3 + 6989/139*e^2 - 56497/278*e - 17785/139, 87/139*e^6 + 380/139*e^5 - 2385/139*e^4 - 5316/139*e^3 + 12748/139*e^2 + 18947/139*e + 2060/139, 375/278*e^6 - 734/139*e^5 - 3589/278*e^4 + 7936/139*e^3 + 4079/139*e^2 - 34603/278*e - 8310/139, -472/139*e^6 + 1407/139*e^5 + 5232/139*e^4 - 13952/139*e^3 - 14926/139*e^2 + 26533/139*e + 15726/139, -35/139*e^6 - 204/139*e^5 + 1017/139*e^4 + 2709/139*e^3 - 5897/139*e^2 - 8148/139*e + 606/139, -109/139*e^6 + 433/139*e^5 + 1253/139*e^4 - 4546/139*e^3 - 5569/139*e^2 + 10320/139*e + 8178/139, -13/139*e^6 + 373/139*e^5 - 631/139*e^4 - 3553/139*e^3 + 5133/139*e^2 + 6092/139*e - 4350/139, -246/139*e^6 + 1509/139*e^5 + 1596/139*e^4 - 16488/139*e^3 - 1483/139*e^2 + 41171/139*e + 14450/139, 691/278*e^6 - 1215/139*e^5 - 6663/278*e^4 + 11803/139*e^3 + 7312/139*e^2 - 43215/278*e - 11415/139, 473/139*e^6 - 2259/139*e^5 - 3462/139*e^4 + 23410/139*e^3 + 2930/139*e^2 - 49958/139*e - 19540/139, 1075/278*e^6 - 1474/139*e^5 - 11419/278*e^4 + 12677/139*e^3 + 15233/139*e^2 - 31771/278*e - 11729/139, -227/139*e^6 + 333/139*e^5 + 2561/139*e^4 - 1640/139*e^3 - 5895/139*e^2 - 4001/139*e - 2694/139, 1325/278*e^6 - 1639/139*e^5 - 14831/278*e^4 + 13149/139*e^3 + 19713/139*e^2 - 12491/278*e - 6149/139, 327/139*e^6 - 743/139*e^5 - 3898/139*e^4 + 5576/139*e^3 + 11425/139*e^2 + 871/139*e + 1876/139, -563/278*e^6 + 897/139*e^5 + 6653/278*e^4 - 9473/139*e^3 - 10556/139*e^2 + 37485/278*e + 10013/139, 392/139*e^6 - 1357/139*e^5 - 3106/139*e^4 + 11248/139*e^3 + 2162/139*e^2 - 12214/139*e - 4/139, -423/139*e^6 + 1498/139*e^5 + 3975/139*e^4 - 14631/139*e^3 - 7671/139*e^2 + 27960/139*e + 9818/139, 23/139*e^6 + 142/139*e^5 - 990/139*e^4 - 1530/139*e^3 + 7513/139*e^2 + 4020/139*e - 5434/139, 185/139*e^6 - 967/139*e^5 - 1424/139*e^4 + 10423/139*e^3 + 987/139*e^2 - 21984/139*e - 2250/139, 133/139*e^6 - 31/139*e^5 - 2141/139*e^4 - 1009/139*e^3 + 8175/139*e^2 + 11419/139*e - 1580/139, -525/278*e^6 + 138/139*e^5 + 8583/278*e^4 + 1066/139*e^3 - 17609/139*e^2 - 31175/278*e - 6575/139, 83/139*e^6 + 869/139*e^5 - 2793/139*e^4 - 11873/139*e^3 + 14167/139*e^2 + 41896/139*e + 12590/139, 612/139*e^6 - 2537/139*e^5 - 4991/139*e^4 + 24522/139*e^3 + 5571/139*e^2 - 42452/139*e - 13424/139, 233/139*e^6 + 254/139*e^5 - 4312/139*e^4 - 4718/139*e^3 + 18709/139*e^2 + 21772/139*e + 5386/139, -7/139*e^6 - 13/139*e^5 + 120/139*e^4 + 375/139*e^3 - 679/139*e^2 + 94/139*e + 1956/139, -63/278*e^6 + 428/139*e^5 - 449/278*e^4 - 5054/139*e^3 + 3269/139*e^2 + 25171/278*e + 1991/139, 423/139*e^6 - 1220/139*e^5 - 4253/139*e^4 + 9766/139*e^3 + 7671/139*e^2 - 3496/139*e + 7696/139, 304/139*e^6 - 1441/139*e^5 - 2352/139*e^4 + 14334/139*e^3 + 3495/139*e^2 - 28169/139*e - 12984/139, 303/278*e^6 - 920/139*e^5 - 925/278*e^4 + 9805/139*e^3 - 4556/139*e^2 - 44915/278*e - 10840/139, 32/139*e^6 - 159/139*e^5 + 623/139*e^4 + 470/139*e^3 - 8572/139*e^2 + 4892/139*e + 11670/139, -270/139*e^6 + 2080/139*e^5 + 816/139*e^4 - 23582/139*e^3 + 5363/139*e^2 + 61410/139*e + 25088/139, -180/139*e^6 + 1155/139*e^5 + 683/139*e^4 - 12478/139*e^3 + 4502/139*e^2 + 29820/139*e + 7088/139, 22/139*e^6 - 118/139*e^5 + 159/139*e^4 - 7/139*e^3 - 2592/139*e^2 + 7012/139*e + 3940/139, -9/139*e^6 - 811/139*e^5 + 2140/139*e^4 + 8147/139*e^3 - 16163/139*e^2 - 15606/139*e + 8750/139, 403/278*e^6 - 291/139*e^5 - 6571/278*e^4 + 2460/139*e^3 + 15167/139*e^2 - 6067/278*e - 10415/139, 7/278*e^6 - 341/139*e^5 + 853/278*e^4 + 5164/139*e^3 - 1954/139*e^2 - 42767/278*e - 9735/139, -165/278*e^6 + 1068/139*e^5 - 2235/278*e^4 - 11059/139*e^3 + 13612/139*e^2 + 45961/278*e - 5184/139, 21/278*e^6 + 228/139*e^5 - 1611/278*e^4 - 1188/139*e^3 + 5814/139*e^2 - 9595/278*e - 7104/139, -1283/278*e^6 + 2234/139*e^5 + 11609/278*e^4 - 20112/139*e^3 - 9753/139*e^2 + 54183/278*e + 8621/139, 223/278*e^6 - 617/139*e^5 - 745/278*e^4 + 6229/139*e^3 - 3988/139*e^2 - 25453/278*e - 199/139, -275/278*e^6 + 946/139*e^5 - 389/278*e^4 - 9165/139*e^3 + 8138/139*e^2 + 38145/278*e + 4148/139, -317/139*e^6 + 1119/139*e^5 + 3250/139*e^4 - 10381/139*e^3 - 9065/139*e^2 + 14940/139*e + 12248/139, 65/139*e^6 - 614/139*e^5 - 42/139*e^4 + 6784/139*e^3 - 2591/139*e^2 - 14753/139*e - 8274/139, 93/139*e^6 - 6/139*e^5 - 1773/139*e^4 - 693/139*e^3 + 8743/139*e^2 + 5026/139*e - 2754/139, -252/139*e^6 - 190/139*e^5 + 4181/139*e^4 + 5577/139*e^3 - 14714/139*e^2 - 34424/139*e - 17710/139, 73/139*e^6 - 341/139*e^5 - 199/139*e^4 + 2523/139*e^3 - 2371/139*e^2 + 92/139*e + 690/139, -226/139*e^6 - 519/139*e^5 + 4748/139*e^4 + 8374/139*e^3 - 22200/139*e^2 - 34932/139*e - 4284/139, 421/278*e^6 - 1009/139*e^5 - 2789/278*e^4 + 11132/139*e^3 - 640/139*e^2 - 55753/278*e - 12354/139, -412/139*e^6 + 605/139*e^5 + 5931/139*e^4 - 3445/139*e^3 - 23701/139*e^2 - 7732/139*e + 10120/139, -497/139*e^6 + 2274/139*e^5 + 3655/139*e^4 - 23137/139*e^3 - 2617/139*e^2 + 50320/139*e + 18780/139] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([9, 3, 1/6*w^3 - 7/3*w + 13/6])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]