/* 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([5, -1, -7, -1, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([21, 21, w^2 - w - 6]) primes_array = [ [3, 3, -w^3 + 2*w^2 + 5*w - 3],\ [5, 5, w],\ [7, 7, -w^3 + 2*w^2 + 4*w - 3],\ [13, 13, -w^2 + w + 4],\ [13, 13, -w^3 + 2*w^2 + 5*w - 2],\ [16, 2, 2],\ [17, 17, -w + 2],\ [19, 19, -w^3 + 2*w^2 + 3*w - 2],\ [27, 3, w^3 - 3*w^2 - 4*w + 7],\ [31, 31, w^3 - 3*w^2 - 3*w + 7],\ [31, 31, -w^3 + w^2 + 6*w + 1],\ [41, 41, w^2 - w - 1],\ [43, 43, 2*w^3 - 5*w^2 - 6*w + 4],\ [47, 47, -w^3 + 2*w^2 + 5*w - 1],\ [53, 53, -w - 3],\ [53, 53, -w^3 + 3*w^2 + 3*w - 6],\ [59, 59, 2*w^2 - 3*w - 6],\ [59, 59, w^3 - w^2 - 7*w - 3],\ [79, 79, 2*w^3 - 4*w^2 - 8*w + 3],\ [79, 79, w^2 - 2*w - 1],\ [101, 101, 2*w^3 - 4*w^2 - 9*w + 4],\ [103, 103, -w^3 + 3*w^2 + 4*w - 2],\ [103, 103, w^2 - 3*w - 2],\ [107, 107, -w^3 + w^2 + 5*w - 1],\ [109, 109, -w^3 + 2*w^2 + 4*w - 6],\ [121, 11, -w^3 + 2*w^2 + 6*w - 3],\ [121, 11, w^2 - 3*w - 3],\ [125, 5, -w^3 + w^2 + 7*w + 1],\ [127, 127, 2*w^3 - 5*w^2 - 5*w + 7],\ [127, 127, 2*w^3 - 5*w^2 - 7*w + 6],\ [167, 167, -2*w^3 + 5*w^2 + 7*w - 11],\ [169, 13, -w^2 + w + 8],\ [181, 181, -2*w^3 + 5*w^2 + 8*w - 7],\ [193, 193, -w^3 + 3*w^2 + 3*w - 1],\ [197, 197, 2*w^3 - 4*w^2 - 7*w + 8],\ [197, 197, 3*w^3 - 7*w^2 - 10*w + 9],\ [199, 199, w^3 - 2*w^2 - 3*w - 3],\ [211, 211, -w^3 + w^2 + 7*w - 2],\ [211, 211, w^3 - w^2 - 6*w + 1],\ [227, 227, 2*w^3 - 3*w^2 - 9*w + 3],\ [229, 229, 3*w^3 - 6*w^2 - 13*w + 8],\ [239, 239, 2*w^3 - 4*w^2 - 9*w + 3],\ [239, 239, 2*w^3 - 4*w^2 - 6*w + 1],\ [241, 241, 3*w^3 - 7*w^2 - 9*w + 6],\ [241, 241, -2*w^3 + 4*w^2 + 6*w - 3],\ [251, 251, -4*w^3 + 10*w^2 + 15*w - 16],\ [251, 251, -w^3 + 3*w^2 - 3],\ [271, 271, -w^3 + 4*w^2 + 4*w - 11],\ [271, 271, -3*w^3 + 9*w^2 + 6*w - 13],\ [277, 277, -w^3 + 3*w^2 + 2*w - 9],\ [281, 281, -w^3 + 4*w^2 + w - 8],\ [293, 293, 3*w^3 - 7*w^2 - 11*w + 13],\ [293, 293, w^2 - 6],\ [293, 293, -4*w^3 + 11*w^2 + 9*w - 14],\ [293, 293, 3*w^3 - 5*w^2 - 15*w + 6],\ [311, 311, -2*w^3 + 4*w^2 + 7*w - 4],\ [313, 313, -w^3 + 4*w^2 + 3*w - 13],\ [313, 313, -w^3 + w^2 + 6*w - 2],\ [317, 317, w^3 - 4*w^2 - w + 9],\ [337, 337, 3*w^3 - 7*w^2 - 8*w + 7],\ [343, 7, -2*w^3 + 3*w^2 + 9*w + 1],\ [359, 359, -w^3 + 2*w^2 + 4*w - 7],\ [359, 359, 2*w^3 - 3*w^2 - 10*w + 4],\ [367, 367, 2*w^3 - 4*w^2 - 5*w + 2],\ [367, 367, 2*w^3 - 3*w^2 - 11*w - 2],\ [367, 367, 2*w - 7],\ [367, 367, w^3 - 7*w - 4],\ [373, 373, -w^3 + 4*w^2 + w - 14],\ [373, 373, -2*w^3 + 4*w^2 + 10*w - 11],\ [379, 379, w^3 - 2*w^2 - 2*w - 1],\ [383, 383, w^3 - 8*w - 4],\ [397, 397, 4*w^3 - 10*w^2 - 12*w + 7],\ [397, 397, -2*w^3 + 3*w^2 + 9*w - 2],\ [401, 401, -w^3 + 6*w + 8],\ [409, 409, 2*w^3 - 3*w^2 - 11*w + 4],\ [419, 419, 3*w^3 - 6*w^2 - 15*w + 7],\ [419, 419, -w^2 + 4*w + 8],\ [431, 431, 5*w^3 - 15*w^2 - 10*w + 18],\ [431, 431, 3*w^3 - 4*w^2 - 16*w - 2],\ [439, 439, w^3 - w^2 - 4*w - 4],\ [443, 443, -4*w^3 + 9*w^2 + 15*w - 16],\ [461, 461, 2*w^3 - 2*w^2 - 13*w - 1],\ [467, 467, 2*w^3 - 6*w^2 - 2*w + 7],\ [467, 467, 3*w^3 - 3*w^2 - 19*w - 6],\ [479, 479, w^2 - 8],\ [479, 479, -3*w^3 + 7*w^2 + 14*w - 11],\ [491, 491, 3*w^3 - 5*w^2 - 16*w + 2],\ [499, 499, -w^2 + w - 2],\ [509, 509, -w^3 + 5*w^2 - w - 11],\ [509, 509, w^3 - 5*w - 1],\ [521, 521, -w^3 + 4*w^2 + 3*w - 7],\ [521, 521, -w^3 + 2*w^2 + 7*w - 4],\ [541, 541, -2*w^3 + 4*w^2 + 11*w - 6],\ [557, 557, -w^3 + w^2 + 8*w + 3],\ [557, 557, w^3 - 2*w^2 - 2*w - 2],\ [563, 563, 2*w^3 - 2*w^2 - 10*w - 3],\ [563, 563, 3*w^3 - 5*w^2 - 15*w + 1],\ [569, 569, 2*w^2 - 3*w - 4],\ [569, 569, 3*w^3 - 8*w^2 - 9*w + 7],\ [571, 571, -w^3 + 4*w^2 - 11],\ [571, 571, 3*w^3 - 8*w^2 - 11*w + 11],\ [577, 577, 3*w^3 - 8*w^2 - 5*w + 11],\ [577, 577, -4*w^3 + 11*w^2 + 9*w - 9],\ [593, 593, 2*w^3 - 7*w^2 - 2*w + 9],\ [601, 601, -w^3 + 3*w^2 + 5*w - 9],\ [613, 613, 3*w^3 - 5*w^2 - 14*w + 8],\ [613, 613, -w^3 + 2*w^2 + 7*w - 3],\ [617, 617, -2*w^3 + 6*w^2 + 5*w - 4],\ [617, 617, -w^3 + 3*w^2 - w - 6],\ [641, 641, 4*w^3 - 9*w^2 - 17*w + 14],\ [641, 641, -5*w^3 + 12*w^2 + 18*w - 17],\ [643, 643, -3*w^3 + 8*w^2 + 10*w - 13],\ [647, 647, 3*w - 4],\ [647, 647, -w^3 + 4*w^2 + 2*w - 9],\ [653, 653, -w^3 + 2*w^2 + 5*w - 8],\ [661, 661, 2*w^3 - 7*w^2 - 5*w + 21],\ [661, 661, -w^3 + 2*w^2 + 3*w - 8],\ [673, 673, 2*w^3 - 7*w^2 - 5*w + 9],\ [673, 673, w^3 - 2*w^2 - 3*w - 4],\ [677, 677, -w^3 + 4*w^2 + 3*w - 8],\ [683, 683, -3*w^3 + 4*w^2 + 17*w - 1],\ [683, 683, -3*w^3 + 7*w^2 + 8*w - 2],\ [691, 691, w^2 - 4*w - 2],\ [701, 701, 3*w^3 - 6*w^2 - 11*w + 3],\ [709, 709, -w^3 + 4*w^2 + 3*w - 17],\ [709, 709, w^3 + w^2 - 10*w - 8],\ [709, 709, -3*w^3 + 10*w^2 + 2*w - 16],\ [709, 709, w^3 - w^2 - 8*w + 1],\ [727, 727, -w^3 + 5*w^2 - w - 13],\ [727, 727, 2*w^3 - 4*w^2 - 11*w + 2],\ [733, 733, -2*w^3 + 3*w^2 + 10*w - 6],\ [733, 733, 2*w^3 - 5*w^2 - 5*w + 9],\ [739, 739, -3*w^3 + 6*w^2 + 12*w - 8],\ [743, 743, 3*w^3 - 4*w^2 - 17*w - 4],\ [743, 743, 2*w^3 - 3*w^2 - 8*w + 1],\ [743, 743, -3*w^3 + 7*w^2 + 11*w - 7],\ [743, 743, -3*w^3 + 6*w^2 + 11*w - 9],\ [757, 757, 2*w^3 - 5*w^2 - 6*w + 11],\ [757, 757, -w^3 + w^2 + 8*w + 2],\ [761, 761, -5*w^3 + 14*w^2 + 13*w - 12],\ [769, 769, -3*w^2 + 4*w + 13],\ [773, 773, 3*w^3 - 4*w^2 - 18*w + 3],\ [797, 797, 4*w^3 - 7*w^2 - 21*w + 8],\ [797, 797, -3*w^3 + 7*w^2 + 11*w - 16],\ [809, 809, -2*w^2 + 5*w + 11],\ [827, 827, w - 6],\ [829, 829, -w^3 - w^2 + 11*w + 8],\ [841, 29, -3*w^3 + 6*w^2 + 14*w - 6],\ [841, 29, -2*w^3 + 5*w^2 + 8*w - 4],\ [853, 853, 3*w^3 - 6*w^2 - 9*w + 8],\ [857, 857, 2*w^2 - 3*w - 3],\ [859, 859, -w^3 + w^2 + 4*w + 7],\ [863, 863, 2*w^3 - 3*w^2 - 7*w - 2],\ [863, 863, -3*w^3 + 6*w^2 + 10*w - 11],\ [881, 881, -3*w^2 + 4*w + 12],\ [881, 881, w^2 - 2*w + 3],\ [883, 883, 4*w^3 - 13*w^2 - 6*w + 17],\ [883, 883, -2*w^3 + 4*w^2 + 12*w - 3],\ [887, 887, -3*w^3 + 5*w^2 + 16*w - 8],\ [887, 887, 3*w^3 - 5*w^2 - 14*w + 3],\ [911, 911, -w^3 + 10*w + 1],\ [919, 919, 5*w^3 - 13*w^2 - 14*w + 11],\ [929, 929, 2*w^3 - 3*w^2 - 12*w - 4],\ [937, 937, -3*w - 7],\ [941, 941, -w^3 + 3*w^2 + 3*w - 12],\ [947, 947, 2*w^3 - 6*w^2 - 9*w + 8],\ [947, 947, -4*w^3 + 7*w^2 + 19*w - 6],\ [953, 953, 3*w^2 - w - 9],\ [961, 31, 4*w^3 - 12*w^2 - 5*w + 17],\ [971, 971, 3*w^3 - 6*w^2 - 16*w + 12],\ [983, 983, w^3 + w^2 - 7*w - 12],\ [991, 991, -2*w^3 + 7*w^2 + 3*w - 12],\ [991, 991, -2*w^3 + 4*w^2 + 12*w - 9],\ [997, 997, 3*w^3 - 5*w^2 - 14*w + 9],\ [997, 997, -2*w^3 + 3*w^2 + 11*w - 7],\ [1013, 1013, -2*w^3 + 6*w^2 + 3*w - 11],\ [1019, 1019, -w^3 + 5*w^2 + 2*w - 8],\ [1021, 1021, -w^3 + 5*w^2 - w - 17],\ [1021, 1021, 3*w^3 - 6*w^2 - 8*w + 3],\ [1033, 1033, -2*w^3 + 5*w^2 + 6*w - 13],\ [1033, 1033, 3*w^3 - 6*w^2 - 11*w + 4],\ [1051, 1051, 3*w^3 - 7*w^2 - 10*w + 13],\ [1061, 1061, 2*w^2 - 5*w - 13],\ [1063, 1063, -w^3 + w^2 + 5*w + 8],\ [1069, 1069, -5*w^3 + 12*w^2 + 19*w - 18],\ [1069, 1069, w^3 - 5*w^2 - w + 9],\ [1097, 1097, -3*w^3 + 9*w^2 + 8*w - 12],\ [1097, 1097, -2*w^3 + 6*w^2 + 7*w - 12],\ [1103, 1103, -3*w^3 + 4*w^2 + 15*w - 3],\ [1103, 1103, -w^3 + 3*w^2 + 5*w - 11],\ [1109, 1109, 5*w^3 - 11*w^2 - 16*w + 6],\ [1129, 1129, -w^2 + 3*w - 4],\ [1129, 1129, 2*w^3 - 2*w^2 - 10*w - 9],\ [1151, 1151, 4*w^3 - 10*w^2 - 14*w + 13],\ [1153, 1153, -2*w^3 + 5*w^2 + 11*w - 12],\ [1153, 1153, -2*w^3 + 8*w^2 - 13],\ [1153, 1153, -w^3 + 3*w^2 + 5*w - 12],\ [1153, 1153, 2*w^3 - w^2 - 11*w - 4],\ [1163, 1163, -w^3 + w^2 + 9*w - 4],\ [1163, 1163, 2*w^3 - 4*w^2 - 10*w - 1],\ [1171, 1171, 3*w^3 - 4*w^2 - 18*w - 1],\ [1193, 1193, w^3 - 4*w^2 + 2*w + 6],\ [1193, 1193, w^3 - 2*w^2 - 5*w - 4],\ [1201, 1201, 2*w^3 - 4*w^2 - 9*w - 3],\ [1201, 1201, 4*w^3 - 10*w^2 - 13*w + 11],\ [1213, 1213, -4*w^3 + 7*w^2 + 19*w - 11],\ [1217, 1217, -2*w^3 + 8*w^2 + 2*w - 13],\ [1229, 1229, -4*w^3 + 13*w^2 + 6*w - 19],\ [1229, 1229, -4*w^3 + 8*w^2 + 16*w - 13],\ [1229, 1229, -3*w^3 + 9*w^2 + 6*w - 16],\ [1229, 1229, 2*w^3 - 3*w^2 - 6*w - 1],\ [1231, 1231, -3*w^3 + 6*w^2 + 16*w - 8],\ [1231, 1231, -3*w^2 + w + 12],\ [1237, 1237, -2*w^3 + 5*w^2 + 4*w - 9],\ [1237, 1237, -w^2 - 3],\ [1237, 1237, -w^3 + 6*w - 3],\ [1237, 1237, -4*w - 1],\ [1249, 1249, -2*w^3 + 4*w^2 + 5*w - 8],\ [1283, 1283, 3*w - 8],\ [1289, 1289, 2*w^3 - w^2 - 14*w - 6],\ [1289, 1289, 4*w^3 - 9*w^2 - 13*w + 8],\ [1291, 1291, w^3 - w^2 - 3*w - 4],\ [1297, 1297, 4*w^3 - 9*w^2 - 15*w + 9],\ [1303, 1303, -w^3 + 4*w^2 - 13],\ [1321, 1321, 3*w^3 - 6*w^2 - 13*w + 3],\ [1321, 1321, -3*w^3 + 6*w^2 + 11*w - 7],\ [1367, 1367, -2*w^3 + 6*w^2 + 4*w - 13],\ [1369, 37, 2*w^2 - w - 11],\ [1369, 37, w^3 - 2*w^2 - w - 2],\ [1373, 1373, -w^3 + 3*w^2 + 8*w - 6],\ [1381, 1381, -w^3 + 3*w^2 - 9],\ [1381, 1381, 3*w^3 - 8*w^2 - 8*w + 14],\ [1427, 1427, 3*w^3 - 7*w^2 - 9*w + 12],\ [1429, 1429, 4*w^3 - 9*w^2 - 11*w + 8],\ [1433, 1433, -w^3 + 8*w + 12],\ [1447, 1447, -4*w^3 + 10*w^2 + 15*w - 14],\ [1447, 1447, 6*w^3 - 14*w^2 - 21*w + 19],\ [1451, 1451, w^3 - w^2 - 9*w - 4],\ [1459, 1459, 3*w^3 - 7*w^2 - 13*w + 9],\ [1481, 1481, -3*w^2 + 3*w + 19],\ [1481, 1481, -w^3 + 5*w^2 - 3*w - 14],\ [1487, 1487, -2*w^3 + 6*w^2 + w - 6],\ [1489, 1489, -3*w^3 + 3*w^2 + 19*w + 3],\ [1499, 1499, 3*w^2 - 4*w - 7],\ [1511, 1511, 4*w^3 - 6*w^2 - 20*w + 3],\ [1523, 1523, 8*w^3 - 23*w^2 - 17*w + 27],\ [1531, 1531, -w^3 + 2*w^2 + 3*w - 9],\ [1531, 1531, 3*w^3 - 9*w^2 - 8*w + 19],\ [1549, 1549, -5*w^3 + 11*w^2 + 17*w - 18],\ [1553, 1553, 2*w^2 - 9],\ [1567, 1567, 2*w^3 - 7*w^2 - 4*w + 22],\ [1579, 1579, -2*w^3 + 4*w^2 + 8*w + 3],\ [1579, 1579, -3*w^3 + 5*w^2 + 13*w - 1],\ [1607, 1607, w^3 - w^2 - 6*w - 8],\ [1609, 1609, -4*w^3 + 7*w^2 + 19*w - 2],\ [1619, 1619, -5*w^3 + 11*w^2 + 13*w - 9],\ [1627, 1627, 4*w^3 - 9*w^2 - 16*w + 11],\ [1657, 1657, 2*w^3 - 8*w^2 - w + 12],\ [1657, 1657, -4*w^3 + 11*w^2 + 15*w - 17],\ [1667, 1667, -w^3 + 5*w^2 - 6],\ [1667, 1667, -3*w^3 + 6*w^2 + 10*w - 6],\ [1697, 1697, w^3 + w^2 - 9*w - 7],\ [1697, 1697, 3*w^3 - 6*w^2 - 13*w + 2],\ [1697, 1697, w^3 - 6*w + 1],\ [1697, 1697, -5*w^3 + 11*w^2 + 22*w - 17],\ [1709, 1709, w^3 - w^2 - 8*w - 9],\ [1709, 1709, -w^3 + 3*w^2 - 13],\ [1723, 1723, 2*w^3 - 3*w^2 - 7*w - 3],\ [1733, 1733, -w^3 + 4*w^2 + 3*w - 19],\ [1741, 1741, -w^3 + 5*w^2 - 14],\ [1747, 1747, -4*w^3 + 6*w^2 + 22*w + 1],\ [1747, 1747, 2*w^3 - 5*w^2 - 9*w + 2],\ [1753, 1753, -2*w^3 + 5*w^2 + 3*w - 1],\ [1759, 1759, -w^3 + 2*w^2 + 4*w - 9],\ [1783, 1783, -3*w^3 + 8*w^2 + 5*w - 12],\ [1783, 1783, -2*w^3 + 6*w^2 + 3*w - 12],\ [1783, 1783, 2*w^3 - 7*w^2 - 4*w + 17],\ [1783, 1783, 2*w^3 - 6*w^2 - w + 7],\ [1787, 1787, -5*w^3 + 8*w^2 + 25*w - 3],\ [1823, 1823, w^3 - 8*w - 1],\ [1823, 1823, -3*w^3 + 5*w^2 + 14*w + 6],\ [1831, 1831, -3*w^3 + 8*w^2 + 14*w - 18],\ [1847, 1847, w^3 - 3*w^2 - w + 14],\ [1847, 1847, -2*w^3 + 2*w^2 + 11*w + 11],\ [1861, 1861, 2*w^2 - w - 12],\ [1871, 1871, 4*w^3 - 8*w^2 - 19*w + 9],\ [1871, 1871, -w^3 + 5*w^2 - w - 7],\ [1873, 1873, -3*w^3 + 5*w^2 + 13*w - 4],\ [1877, 1877, -4*w^3 + 8*w^2 + 16*w - 7],\ [1877, 1877, 3*w^3 - 5*w^2 - 15*w - 2],\ [1879, 1879, -w^3 + 4*w^2 - w - 12],\ [1907, 1907, w^3 - w^2 - 10*w - 6],\ [1931, 1931, -w^3 + 5*w^2 - w - 19],\ [1931, 1931, -6*w^3 + 17*w^2 + 12*w - 22],\ [1951, 1951, 4*w^3 - 10*w^2 - 14*w + 21],\ [1951, 1951, -2*w^3 + 4*w^2 + 13*w - 13],\ [1993, 1993, 2*w^3 - 2*w^2 - 11*w + 1],\ [1993, 1993, -w^3 + 6*w + 14]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x K = QQ e = 1 hecke_eigenvalues_array = [-1, 3, -1, 4, 2, -5, -6, 4, -1, -4, 1, -6, -5, -3, 0, -3, 3, 9, 17, -10, -3, 14, 4, -12, -7, 13, 17, 21, -10, 20, 0, 10, -16, 5, 0, 3, 20, 11, -13, 12, 13, -6, -27, -1, -17, 21, 0, 8, -26, 26, 0, -18, 6, 12, 0, 0, 26, 13, -3, 34, 34, 36, -9, -20, -2, -8, 28, 31, 37, 20, 21, -8, -5, 18, 32, 12, 30, -39, 18, -41, 24, 15, 12, -36, 6, -15, -12, -5, -18, 24, -24, 15, 1, -27, -45, -36, 6, 18, 0, -22, 31, 22, -16, -15, 19, -46, 44, -36, 30, 0, 12, -14, -27, 24, 36, 5, -34, -4, 46, 9, -30, 39, -25, -33, -32, 4, -47, 46, 28, 28, 20, 4, 47, 24, 48, -12, 6, -28, 46, -30, 14, 24, 0, 0, 39, -54, -52, 7, -22, -19, -18, -32, -6, -24, 27, 12, -37, -1, 21, -54, -57, -23, -18, 47, -42, -42, -27, 39, 10, 15, 24, 16, 50, 16, 38, 51, 39, -10, 34, 22, -1, -40, 24, -16, -35, -8, -15, -30, -15, -36, 15, 40, -25, 30, 38, 43, 43, 47, -15, -6, -58, 24, -15, 28, -50, 62, -36, 57, 30, -6, 30, 56, 44, 56, 5, -67, 14, -35, -3, -69, -39, 8, -50, -5, 50, 14, -27, 35, -34, -48, 46, 58, 36, 46, -21, -22, 8, 18, 16, 12, -30, 24, 10, 9, 12, 30, -41, -52, -38, -12, 43, -49, 67, -69, -38, -18, 5, 29, 4, 54, 45, -63, -30, 0, -18, -18, -48, -38, 45, 22, -28, 50, -41, 32, 38, -82, -16, 11, 39, -48, 6, -26, -6, -78, 10, 18, 39, -65, -18, 72, 34, 60, -6, 57, -35, 29, 4, 74] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([3, 3, -w^3 + 2*w^2 + 5*w - 3])] = 1 AL_eigenvalues[ZF.ideal([7, 7, -w^3 + 2*w^2 + 4*w - 3])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]