/* 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([1, 2, -5, -1, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([22, 22, -w^2 + 3]) primes_array = [ [2, 2, -w + 1],\ [7, 7, -w + 2],\ [8, 2, -w^3 + 5*w + 3],\ [11, 11, w^3 - w^2 - 5*w + 4],\ [11, 11, w^3 - w^2 - 4*w + 1],\ [17, 17, -w^3 + w^2 + 5*w],\ [25, 5, -w^2 + w + 3],\ [25, 5, -w^2 + 2],\ [29, 29, w^3 - 2*w^2 - 4*w + 4],\ [29, 29, -w^3 + 4*w + 2],\ [37, 37, 4*w^3 - 2*w^2 - 21*w - 4],\ [43, 43, 4*w^3 - 2*w^2 - 20*w - 3],\ [47, 47, -w^3 + 6*w],\ [53, 53, w^3 - w^2 - 5*w - 2],\ [81, 3, -3],\ [83, 83, 2*w^3 - w^2 - 10*w - 4],\ [89, 89, w^3 - 2*w^2 - 4*w + 2],\ [89, 89, 2*w - 3],\ [101, 101, 6*w^3 - 3*w^2 - 31*w - 5],\ [107, 107, -w^3 + w^2 + 4*w - 5],\ [107, 107, 3*w^3 - 2*w^2 - 14*w + 2],\ [113, 113, 3*w^3 - 2*w^2 - 16*w - 2],\ [121, 11, w^3 - w^2 - 7*w + 6],\ [127, 127, 2*w^3 + w^2 - 11*w - 11],\ [131, 131, -2*w^3 + w^2 + 8*w - 2],\ [139, 139, w^3 - w^2 - 3*w + 6],\ [149, 149, w^3 - 6*w + 2],\ [149, 149, 2*w^3 - 2*w^2 - 11*w + 2],\ [151, 151, w^2 - 2*w - 4],\ [157, 157, w^3 + w^2 - 7*w - 4],\ [163, 163, 2*w^2 - 7],\ [167, 167, w^2 - 4*w - 2],\ [167, 167, -2*w^3 + 2*w^2 + 10*w - 1],\ [173, 173, w^2 - w - 7],\ [179, 179, 2*w^3 - 2*w^2 - 8*w + 3],\ [181, 181, w^3 + w^2 - 4*w - 5],\ [191, 191, -w^3 + w^2 + 2*w - 3],\ [193, 193, -2*w^3 + 2*w^2 + 7*w - 4],\ [193, 193, -w^3 + w^2 + 6*w + 3],\ [193, 193, 2*w^3 - 3*w^2 - 8*w + 4],\ [193, 193, 2*w^3 - 11*w - 2],\ [197, 197, 3*w^3 - 2*w^2 - 14*w],\ [199, 199, -w^3 + 8*w - 2],\ [199, 199, -5*w^3 + 2*w^2 + 26*w + 4],\ [223, 223, 2*w^3 - w^2 - 9*w + 3],\ [223, 223, 2*w^2 - 3*w - 4],\ [229, 229, w^3 - w^2 - 3*w - 4],\ [233, 233, -w - 4],\ [239, 239, -w^3 - w^2 + 5*w + 4],\ [241, 241, 2*w^3 - 2*w^2 - 7*w],\ [241, 241, -2*w^3 + w^2 + 10*w - 2],\ [251, 251, 2*w^2 - w - 8],\ [251, 251, -3*w^3 + 3*w^2 + 14*w - 3],\ [257, 257, -w^2 - 2],\ [257, 257, -4*w^3 + 5*w^2 + 17*w - 11],\ [257, 257, -3*w^3 + w^2 + 13*w],\ [257, 257, -2*w^3 + 3*w^2 + 7*w - 7],\ [269, 269, -w^3 - w^2 + 6*w + 5],\ [269, 269, w^2 + w - 5],\ [271, 271, 3*w^3 - w^2 - 16*w - 7],\ [283, 283, 4*w^3 - w^2 - 20*w - 8],\ [283, 283, w^2 - 2*w - 6],\ [283, 283, 2*w^3 - 12*w - 7],\ [283, 283, 3*w^3 - 3*w^2 - 13*w + 6],\ [307, 307, w^3 - 6*w - 6],\ [311, 311, 2*w^3 - w^2 - 13*w + 5],\ [313, 313, -2*w^3 + 3*w^2 + 7*w + 1],\ [337, 337, w^3 - 8*w],\ [337, 337, 3*w^3 - 3*w^2 - 13*w + 4],\ [343, 7, -4*w^3 + 3*w^2 + 19*w - 5],\ [349, 349, 2*w^3 - w^2 - 8*w - 4],\ [359, 359, w^3 + w^2 - 2*w - 3],\ [359, 359, 4*w^3 - 4*w^2 - 20*w + 11],\ [373, 373, -8*w^3 + 2*w^2 + 43*w + 16],\ [373, 373, 2*w^3 - 10*w - 1],\ [379, 379, w^3 + w^2 - 5*w - 10],\ [383, 383, 2*w^3 - 3*w^2 - 12*w + 8],\ [389, 389, -2*w^3 + 2*w^2 + 7*w - 2],\ [389, 389, -3*w^3 + w^2 + 18*w - 1],\ [397, 397, 5*w^3 - 26*w - 14],\ [397, 397, 4*w^3 - 3*w^2 - 19*w - 1],\ [409, 409, -5*w^3 + 6*w^2 + 22*w - 12],\ [419, 419, -2*w^3 + w^2 + 12*w],\ [419, 419, -6*w^3 + 4*w^2 + 31*w],\ [431, 431, 2*w^2 - 5],\ [439, 439, 5*w^3 - 3*w^2 - 24*w + 1],\ [443, 443, -3*w^3 + w^2 + 13*w + 2],\ [443, 443, -w^3 + 3*w^2 + 4*w - 9],\ [457, 457, -8*w^3 + 3*w^2 + 43*w + 11],\ [463, 463, -3*w^3 + 3*w^2 + 12*w - 1],\ [467, 467, -7*w^3 + 3*w^2 + 37*w + 6],\ [467, 467, -4*w^3 + 3*w^2 + 20*w],\ [479, 479, -2*w^3 + 4*w^2 + 9*w - 10],\ [479, 479, 4*w^3 - 2*w^2 - 19*w - 2],\ [487, 487, 2*w^3 - 3*w^2 - 12*w + 4],\ [487, 487, w^2 + w - 7],\ [499, 499, 2*w^2 - w - 4],\ [541, 541, -w^3 + 3*w^2 + 3*w - 8],\ [547, 547, -w^3 + 2*w^2 + 6*w - 8],\ [547, 547, -2*w^3 + 2*w^2 + 12*w - 1],\ [569, 569, 4*w^3 - 3*w^2 - 21*w - 1],\ [569, 569, -w^3 + 3*w^2 - 5],\ [587, 587, 7*w^3 - 4*w^2 - 36*w - 4],\ [587, 587, -2*w^3 + 14*w + 3],\ [593, 593, -3*w^3 + w^2 + 13*w - 2],\ [601, 601, 5*w^3 - w^2 - 28*w - 13],\ [601, 601, 8*w^3 - 3*w^2 - 40*w - 8],\ [613, 613, -2*w^3 + 3*w^2 + 11*w - 3],\ [613, 613, 4*w^3 - w^2 - 19*w - 7],\ [617, 617, -2*w^3 + 3*w^2 + 8*w - 2],\ [619, 619, -w^3 + 3*w^2 + 3*w - 6],\ [641, 641, -4*w^3 + 2*w^2 + 22*w + 3],\ [643, 643, -w^3 + 3*w^2 + 4*w - 7],\ [643, 643, 2*w^3 - 3*w^2 - 11*w + 13],\ [653, 653, 2*w^3 - 9*w - 6],\ [659, 659, -3*w^3 + 3*w^2 + 15*w - 2],\ [673, 673, -w^3 + 2*w^2 + 6*w - 10],\ [677, 677, -2*w^3 + 3*w^2 + 9*w - 3],\ [683, 683, -3*w^3 + 3*w^2 + 16*w - 5],\ [691, 691, -w^3 + w^2 + 8*w - 3],\ [691, 691, w^3 - w^2 - 5*w - 4],\ [701, 701, -2*w^3 - w^2 + 13*w + 9],\ [709, 709, -5*w^3 + 2*w^2 + 24*w + 4],\ [709, 709, -2*w^3 + 4*w^2 + 6*w - 3],\ [719, 719, -w^3 + 2*w^2 + 4*w - 10],\ [739, 739, -3*w^3 + 4*w^2 + 12*w - 4],\ [739, 739, 5*w^3 - w^2 - 25*w - 10],\ [743, 743, -3*w^3 + 18*w + 8],\ [743, 743, -2*w^3 + 3*w^2 + 7*w - 9],\ [761, 761, 2*w^3 - 3*w^2 - 12*w + 10],\ [769, 769, -2*w^3 + 14*w - 3],\ [773, 773, -w^3 + 4*w^2 + 2*w - 4],\ [787, 787, -6*w^3 + 4*w^2 + 29*w - 2],\ [787, 787, -3*w^3 + 2*w^2 + 14*w - 10],\ [797, 797, 3*w^3 - w^2 - 15*w],\ [797, 797, -2*w^3 + 5*w^2 + 5*w - 11],\ [809, 809, 6*w^3 - 32*w - 17],\ [811, 811, 5*w^3 - 28*w - 18],\ [811, 811, -2*w^3 + 2*w^2 + 12*w - 5],\ [821, 821, -4*w^3 + 21*w + 8],\ [823, 823, -3*w^3 + 3*w^2 + 14*w - 1],\ [827, 827, 6*w^3 - 2*w^2 - 30*w - 5],\ [827, 827, -4*w^3 + 4*w^2 + 22*w - 13],\ [829, 829, 5*w^3 - w^2 - 27*w - 6],\ [829, 829, w^3 + 2*w^2 - 8*w - 4],\ [841, 29, -2*w^3 + 3*w^2 + 8*w],\ [853, 853, -w^3 + 3*w^2 + 7*w - 2],\ [853, 853, -2*w^3 + 11*w],\ [863, 863, -w^3 + w^2 + 4*w - 7],\ [863, 863, 2*w - 7],\ [877, 877, 2*w^3 + w^2 - 10*w - 10],\ [877, 877, -3*w^3 + w^2 + 18*w + 3],\ [883, 883, 3*w^2 - 3*w - 11],\ [919, 919, 3*w^3 - 3*w^2 - 17*w + 14],\ [919, 919, -2*w^3 + 2*w^2 + 12*w - 3],\ [937, 937, -w^2 + 2*w - 4],\ [941, 941, 4*w^3 - 3*w^2 - 18*w + 4],\ [941, 941, w^3 + w^2 - 4*w - 7],\ [941, 941, 7*w^3 - 5*w^2 - 34*w + 3],\ [941, 941, 4*w^3 - 2*w^2 - 18*w - 3],\ [953, 953, 3*w^3 - 3*w^2 - 10*w - 1],\ [961, 31, 4*w^3 - 3*w^2 - 21*w + 7],\ [961, 31, 2*w^2 + w - 8],\ [967, 967, w^3 + w^2 - 5*w - 12],\ [967, 967, 2*w^3 - 8*w - 5],\ [971, 971, w^2 + 2*w - 6],\ [977, 977, w^2 + 3*w - 5],\ [983, 983, w^3 + 2*w^2 - 4*w - 6],\ [991, 991, -4*w^3 + 4*w^2 + 19*w - 4],\ [1009, 1009, -w^3 - w^2 + 8*w + 7],\ [1013, 1013, -3*w^3 + 4*w^2 + 10*w - 4],\ [1013, 1013, -2*w^3 + w^2 + 13*w + 1],\ [1021, 1021, -4*w^3 + 2*w^2 + 20*w - 1],\ [1033, 1033, w^3 + w^2 - 8*w - 11],\ [1033, 1033, -2*w^3 + w^2 + 13*w - 3],\ [1049, 1049, 3*w^3 - 2*w^2 - 16*w + 6],\ [1061, 1061, 3*w^3 - w^2 - 11*w - 4],\ [1063, 1063, 6*w^3 - w^2 - 32*w - 16],\ [1091, 1091, 6*w^3 - 3*w^2 - 29*w - 5],\ [1091, 1091, -6*w^3 + 3*w^2 + 32*w + 2],\ [1093, 1093, -10*w^3 + 3*w^2 + 53*w + 19],\ [1093, 1093, -4*w^3 + 6*w^2 + 19*w - 12],\ [1097, 1097, 4*w^3 - 2*w^2 - 19*w + 2],\ [1123, 1123, -4*w^3 + 5*w^2 + 18*w - 8],\ [1123, 1123, 4*w^3 - 3*w^2 - 20*w - 2],\ [1129, 1129, 2*w^3 - 10*w + 5],\ [1129, 1129, -w^3 + 2*w^2 + 2*w - 12],\ [1151, 1151, 11*w^3 - 5*w^2 - 57*w - 12],\ [1153, 1153, 6*w^3 - 4*w^2 - 31*w - 2],\ [1163, 1163, w^3 - 4*w^2 - 6*w + 12],\ [1163, 1163, 4*w^3 - 3*w^2 - 20*w - 4],\ [1171, 1171, w^3 - w^2 - w - 6],\ [1181, 1181, -5*w^3 + 5*w^2 + 26*w - 13],\ [1181, 1181, 5*w^3 - 5*w^2 - 22*w + 11],\ [1187, 1187, -w^3 - w^2 + 11*w + 2],\ [1187, 1187, 3*w^3 - w^2 - 18*w + 3],\ [1193, 1193, 12*w^3 - 5*w^2 - 61*w - 13],\ [1193, 1193, 2*w^3 - 9*w - 8],\ [1201, 1201, 10*w^3 - 3*w^2 - 51*w - 17],\ [1213, 1213, -5*w^3 + 7*w^2 + 20*w - 11],\ [1217, 1217, 2*w^3 - 13*w - 8],\ [1229, 1229, 4*w^3 - 4*w^2 - 16*w + 1],\ [1237, 1237, 2*w^3 - 13*w + 2],\ [1237, 1237, -8*w^3 + 4*w^2 + 42*w + 9],\ [1249, 1249, 4*w^3 - 4*w^2 - 21*w + 6],\ [1249, 1249, 3*w^3 - 2*w^2 - 12*w],\ [1259, 1259, 4*w + 5],\ [1259, 1259, -w^3 - 2*w^2 + 8*w + 12],\ [1279, 1279, -w^3 + w^2 + 9*w - 4],\ [1283, 1283, -10*w^3 + 5*w^2 + 52*w + 6],\ [1291, 1291, 3*w^3 + w^2 - 16*w - 11],\ [1301, 1301, 5*w^3 - 26*w - 12],\ [1301, 1301, 4*w^3 - 3*w^2 - 22*w],\ [1321, 1321, -w - 6],\ [1327, 1327, w^3 + 2*w^2 - 8*w - 6],\ [1361, 1361, 2*w^3 - 3*w^2 - 12*w],\ [1367, 1367, 4*w^3 - 22*w - 15],\ [1381, 1381, -5*w^3 + 2*w^2 + 26*w + 2],\ [1381, 1381, w^3 - w^2 - 6*w - 5],\ [1399, 1399, -w^3 + 6*w - 6],\ [1399, 1399, -2*w^3 + 3*w^2 + 11*w - 1],\ [1423, 1423, 2*w^3 + w^2 - 12*w - 14],\ [1423, 1423, 3*w^3 - 14*w - 8],\ [1427, 1427, -4*w^3 + w^2 + 18*w],\ [1447, 1447, -5*w^3 + w^2 + 29*w + 10],\ [1447, 1447, -4*w^3 + 2*w^2 + 22*w - 3],\ [1453, 1453, 6*w^3 - 3*w^2 - 29*w - 3],\ [1453, 1453, 4*w^3 - 4*w^2 - 22*w + 1],\ [1459, 1459, 6*w^3 - 7*w^2 - 26*w + 12],\ [1471, 1471, -w^3 + w^2 + w - 6],\ [1483, 1483, 6*w^3 - 3*w^2 - 34*w - 8],\ [1489, 1489, -2*w^3 - w^2 + 8*w - 2],\ [1493, 1493, 3*w^3 - w^2 - 10*w + 3],\ [1493, 1493, 3*w^3 - 5*w^2 - 12*w + 7],\ [1511, 1511, 8*w^3 - 3*w^2 - 41*w - 7],\ [1523, 1523, -w^3 + 4*w^2 + 4*w - 6],\ [1543, 1543, -w^3 + w^2 + 5*w - 8],\ [1543, 1543, -3*w^3 + 2*w^2 + 16*w - 8],\ [1553, 1553, w^3 + w^2 - 8*w - 9],\ [1559, 1559, 3*w^3 - 3*w^2 - 15*w - 2],\ [1559, 1559, -3*w^3 + 3*w^2 + 11*w - 4],\ [1567, 1567, w^3 - w^2 - 3*w - 6],\ [1567, 1567, 2*w^3 - 11*w - 10],\ [1567, 1567, -w^3 + 2*w^2 + 8*w - 8],\ [1567, 1567, w^3 + w^2 - 2*w - 5],\ [1579, 1579, w^2 + 2*w - 10],\ [1579, 1579, w^3 + w^2 - 9*w - 6],\ [1597, 1597, -2*w^3 + 5*w^2 + 7*w - 5],\ [1597, 1597, 2*w^3 - 4*w^2 - 10*w + 5],\ [1607, 1607, -8*w^3 + 3*w^2 + 40*w + 6],\ [1609, 1609, 3*w^2 - 4*w - 6],\ [1609, 1609, -2*w^3 - w^2 + 14*w + 8],\ [1613, 1613, -3*w^3 + 3*w^2 + 10*w - 5],\ [1613, 1613, 2*w^3 - 10*w + 3],\ [1627, 1627, -5*w^3 + 2*w^2 + 28*w + 2],\ [1627, 1627, -3*w^3 + w^2 + 13*w - 4],\ [1627, 1627, 4*w^3 - 5*w^2 - 16*w + 4],\ [1627, 1627, -3*w^3 - w^2 + 17*w + 4],\ [1663, 1663, -4*w^3 + w^2 + 22*w + 2],\ [1667, 1667, 2*w^3 - 2*w^2 - 8*w - 5],\ [1669, 1669, 2*w^3 - 2*w^2 - 13*w + 6],\ [1669, 1669, -4*w^3 + 2*w^2 + 21*w - 2],\ [1693, 1693, w^3 - w^2 - 2*w + 9],\ [1693, 1693, -w^3 + w^2 + 9*w - 2],\ [1697, 1697, 6*w^3 - 4*w^2 - 29*w - 4],\ [1697, 1697, -2*w^3 + 11*w - 2],\ [1721, 1721, -4*w^3 + 4*w^2 + 14*w - 11],\ [1721, 1721, w^3 - 6*w - 8],\ [1733, 1733, 4*w^3 - 2*w^2 - 17*w + 6],\ [1741, 1741, 5*w^3 - 4*w^2 - 28*w + 14],\ [1747, 1747, -w^3 - 3*w^2 + 6*w + 17],\ [1787, 1787, 7*w^3 - 5*w^2 - 36*w + 3],\ [1789, 1789, -w^3 - w^2 + 11*w],\ [1789, 1789, w^3 + 2*w^2 - 6*w - 8],\ [1811, 1811, 4*w^3 - 2*w^2 - 17*w + 2],\ [1847, 1847, 6*w^3 - 5*w^2 - 32*w + 2],\ [1847, 1847, -2*w^3 + 3*w^2 + 15*w + 1],\ [1861, 1861, 2*w^3 - 3*w^2 - 13*w + 11],\ [1861, 1861, -4*w^3 + 3*w^2 + 17*w - 5],\ [1879, 1879, -8*w^3 + 5*w^2 + 39*w - 1],\ [1933, 1933, 5*w^3 - 5*w^2 - 22*w + 7],\ [1949, 1949, 3*w^3 - 3*w^2 - 17*w + 4],\ [1951, 1951, -4*w^3 + 3*w^2 + 23*w - 13],\ [1973, 1973, 9*w^3 - w^2 - 47*w - 22],\ [1973, 1973, -3*w^3 + 12*w - 2],\ [1973, 1973, w^2 - 2*w - 10],\ [1973, 1973, w^3 - w^2 - 10*w + 5],\ [1979, 1979, 5*w^3 - 3*w^2 - 23*w - 2],\ [1979, 1979, 5*w^3 - w^2 - 27*w - 14],\ [1987, 1987, 7*w^3 - w^2 - 35*w - 14],\ [1993, 1993, -w^3 + 3*w^2 - 7],\ [1993, 1993, 3*w^2 - w - 5],\ [1993, 1993, -10*w^3 + 5*w^2 + 53*w + 9],\ [1993, 1993, -2*w^3 + 14*w - 7],\ [1997, 1997, -7*w^3 + 4*w^2 + 36*w + 6]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x K = QQ e = 1 hecke_eigenvalues_array = [1, -2, 1, -1, 2, -4, -6, 8, -6, -6, -10, -4, -12, -4, -16, 18, 0, 6, 14, -4, -8, -6, 10, -4, -14, 12, -10, 18, 18, 16, 0, -10, -24, -6, 0, 14, -16, 2, 14, 4, 2, -8, 4, 4, -16, 18, 6, 24, -6, 22, -2, -12, 20, -4, -22, 14, 22, -2, -28, 4, -20, -4, 20, -28, 8, 0, 10, 2, -34, -16, -34, 0, -24, 16, 10, 12, 24, 0, 36, 0, -28, -18, 14, 30, -12, -16, -2, -24, -34, -34, -22, -16, -20, 14, -24, -26, -8, 10, -10, -46, -38, -2, 22, -26, 30, 10, -20, -26, -30, 38, 36, -42, 2, 22, 14, -12, -16, -22, 20, 16, -2, 34, -14, 4, 0, -38, -42, 36, 16, 46, 38, 36, 40, -40, 46, -34, -54, -32, 10, -10, 54, -20, -48, -38, 2, -34, -34, -26, 30, 10, 26, 46, -4, -16, -32, -4, 10, 30, -30, 42, 6, -10, 6, -20, 2, -30, 6, 8, 8, 4, 2, 6, 34, -34, 22, 36, -54, -8, -18, 36, -38, -22, 54, 42, -28, 26, 14, -60, -2, -28, 10, -46, 54, -30, 12, 60, 34, 40, 58, 36, -46, 4, 12, 66, -14, 46, 30, 20, 40, 4, 28, -30, 30, -10, 48, 30, -2, 70, 50, 32, 16, 56, -64, -44, -40, 12, 16, -58, 52, -22, 16, 40, 22, 30, 8, 16, 16, -22, 54, -8, 16, 14, 12, 16, -58, -52, -64, -22, 2, 52, -42, -4, -54, -30, -12, -44, 28, -34, -2, -52, 64, 44, -2, 30, -24, 54, -26, -46, 82, -2, -28, -68, 32, -16, 20, 72, 2, 6, -2, 44, -62, 30, 60, 24, 42, -24, -6, -24, -24, 52, 6, 50, -6, 6, 46] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([2, 2, -w + 1])] = -1 AL_eigenvalues[ZF.ideal([11, 11, w^3 - w^2 - 5*w + 4])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]