/* 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([2, 1, -5, -2, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([4, 4, -w^2 + 2*w + 4]) primes_array = [ [2, 2, -w],\ [3, 3, w - 1],\ [8, 2, w^3 - 2*w^2 - 5*w + 1],\ [13, 13, -w^2 + 2*w + 3],\ [17, 17, -w^2 + 3*w + 3],\ [17, 17, -w^2 + 2*w + 1],\ [27, 3, w^3 - w^2 - 6*w - 5],\ [29, 29, w^3 - 2*w^2 - 2*w + 1],\ [47, 47, w^3 - 2*w^2 - 4*w - 3],\ [49, 7, w^3 - 3*w^2 - 3*w + 1],\ [49, 7, -2*w^3 + 5*w^2 + 7*w - 3],\ [59, 59, -2*w^3 + 6*w^2 + 5*w - 7],\ [61, 61, 2*w^3 - 4*w^2 - 9*w - 3],\ [71, 71, w^3 - 3*w^2 - 3*w + 7],\ [73, 73, 2*w^3 - 6*w^2 - 3*w + 5],\ [73, 73, w^3 - 3*w^2 - 4*w + 5],\ [83, 83, -2*w^3 + 5*w^2 + 8*w - 3],\ [89, 89, -2*w^3 + 6*w^2 + 4*w - 7],\ [89, 89, w^3 - 2*w^2 - 6*w + 3],\ [97, 97, -3*w - 1],\ [101, 101, w^2 - w - 5],\ [101, 101, -w^3 + w^2 + 6*w + 7],\ [103, 103, w^3 - 3*w^2 - 2*w - 1],\ [107, 107, -w^3 + w^2 + 8*w - 1],\ [109, 109, -3*w^3 + 7*w^2 + 11*w - 7],\ [131, 131, w^2 - 4*w - 1],\ [131, 131, -2*w^2 + 2*w + 1],\ [131, 131, w^3 - w^2 - 7*w - 1],\ [131, 131, -w^3 + 4*w^2 - 5],\ [137, 137, 2*w^3 - 5*w^2 - 9*w + 7],\ [139, 139, w^3 - 8*w - 7],\ [139, 139, -3*w + 1],\ [149, 149, -3*w^3 + 8*w^2 + 8*w - 3],\ [149, 149, -2*w^3 + 3*w^2 + 12*w + 3],\ [151, 151, 2*w^3 - 4*w^2 - 10*w + 1],\ [151, 151, -w^3 + 3*w^2 + w - 5],\ [151, 151, -2*w^3 + 6*w^2 + 5*w - 11],\ [151, 151, 2*w - 3],\ [157, 157, 2*w^3 - 7*w^2 + 1],\ [163, 163, -w^3 + w^2 + 6*w + 1],\ [173, 173, 3*w^3 - 9*w^2 - 7*w + 9],\ [173, 173, -2*w^3 + 7*w^2 + 2*w - 5],\ [181, 181, -w^2 + 5*w + 3],\ [191, 191, -3*w^3 + 8*w^2 + 10*w - 7],\ [193, 193, -2*w^3 + 5*w^2 + 5*w - 3],\ [193, 193, 2*w^2 - 4*w - 5],\ [223, 223, 2*w^3 - 4*w^2 - 9*w + 3],\ [227, 227, -3*w^3 + 7*w^2 + 12*w - 9],\ [227, 227, 2*w^3 - 4*w^2 - 7*w - 1],\ [241, 241, w^2 - 2*w - 7],\ [257, 257, -2*w^3 + 7*w^2 + 2*w - 9],\ [257, 257, w^2 - 4*w - 5],\ [263, 263, -3*w^3 + 7*w^2 + 12*w - 3],\ [269, 269, w^3 - 3*w^2 - 4*w + 7],\ [271, 271, 3*w^3 - 10*w^2 - 4*w + 7],\ [271, 271, -w^3 + 3*w^2 - 2*w - 1],\ [277, 277, -2*w^3 + 6*w^2 + 4*w - 9],\ [277, 277, -6*w^3 + 14*w^2 + 22*w - 5],\ [281, 281, -2*w^3 + 6*w^2 + 2*w - 5],\ [289, 17, w^3 - 2*w^2 + 3],\ [293, 293, w^3 - 3*w^2 + w - 1],\ [293, 293, 3*w^3 - 8*w^2 - 8*w + 11],\ [311, 311, -w^3 + 4*w^2 - 2*w - 5],\ [313, 313, 2*w^2 - 4*w - 11],\ [313, 313, 3*w^3 - 7*w^2 - 11*w + 1],\ [317, 317, -w^3 + w^2 + 5*w - 1],\ [317, 317, -2*w^3 + 4*w^2 + 10*w - 7],\ [331, 331, w^3 - 5*w^2 - 3*w + 3],\ [331, 331, -3*w^3 + 6*w^2 + 14*w - 3],\ [337, 337, 3*w^3 - 7*w^2 - 10*w + 1],\ [337, 337, w^3 - w^2 - 5*w - 9],\ [353, 353, -w^3 + 2*w^2 + 1],\ [359, 359, -2*w^2 + 4*w - 1],\ [359, 359, 4*w^3 - 10*w^2 - 14*w + 7],\ [367, 367, -4*w^3 + 10*w^2 + 13*w - 9],\ [379, 379, -2*w^3 + 8*w^2 + w - 17],\ [379, 379, w^3 - 4*w^2 - 2*w + 9],\ [397, 397, 2*w^3 - 5*w^2 - 9*w + 1],\ [401, 401, w^2 - 5],\ [401, 401, 2*w^3 - 5*w^2 - 8*w + 1],\ [431, 431, -3*w^3 + 3*w^2 + 21*w + 13],\ [439, 439, -2*w^2 + 3*w + 13],\ [449, 449, 2*w^3 - 4*w^2 - 7*w - 5],\ [449, 449, 5*w^3 - 10*w^2 - 22*w - 1],\ [457, 457, 2*w^3 - 7*w^2 - 5*w + 5],\ [461, 461, 3*w^3 - 5*w^2 - 14*w - 1],\ [461, 461, -2*w^3 + 3*w^2 + 11*w + 1],\ [463, 463, 2*w^3 - 4*w^2 - 6*w + 1],\ [463, 463, 2*w^3 - 5*w^2 - 5*w + 9],\ [479, 479, 2*w^2 - 5*w - 1],\ [509, 509, -6*w^3 + 10*w^2 + 31*w + 11],\ [523, 523, -w^2 + 5*w + 1],\ [523, 523, 3*w^3 - 6*w^2 - 14*w - 3],\ [547, 547, -3*w^3 + 6*w^2 + 12*w - 1],\ [563, 563, 2*w^2 - 6*w - 7],\ [563, 563, -w^3 + 2*w^2 + 4*w + 5],\ [571, 571, w^2 - 2*w + 3],\ [571, 571, 3*w^2 - 8*w - 9],\ [577, 577, -3*w^3 + 7*w^2 + 10*w - 3],\ [577, 577, -2*w^2 + 5*w - 1],\ [601, 601, w^3 - 2*w^2 - 8*w + 7],\ [601, 601, 4*w^3 - 9*w^2 - 17*w + 3],\ [607, 607, -3*w^3 + 6*w^2 + 12*w + 1],\ [613, 613, -2*w^3 + 6*w^2 + 6*w - 3],\ [617, 617, 3*w^3 - 9*w^2 - 6*w + 11],\ [617, 617, -2*w^3 + 5*w^2 + 7*w + 3],\ [619, 619, -3*w^3 + 5*w^2 + 17*w + 1],\ [625, 5, -5],\ [631, 631, -3*w^2 + 6*w + 11],\ [643, 643, -2*w^3 + 7*w^2 + 4*w - 7],\ [643, 643, -2*w^3 + 7*w^2 + 3*w - 7],\ [653, 653, 3*w^3 - 7*w^2 - 11*w - 1],\ [661, 661, 2*w^2 - 7*w - 3],\ [661, 661, -w^3 + 5*w^2 - w - 5],\ [673, 673, -3*w^3 + 9*w^2 + 8*w - 7],\ [677, 677, -4*w^3 + 9*w^2 + 15*w - 3],\ [677, 677, -2*w^3 + 7*w^2 - 7],\ [683, 683, w^3 - 4*w^2 - 1],\ [683, 683, w^3 - 4*w^2 - 2*w + 13],\ [709, 709, -3*w^3 + 9*w^2 + 5*w - 9],\ [709, 709, 4*w^3 - 12*w^2 - 7*w + 5],\ [719, 719, -2*w^3 + 7*w^2 + 4*w - 11],\ [719, 719, -5*w^3 + 8*w^2 + 28*w + 9],\ [733, 733, -3*w^3 + 8*w^2 + 6*w - 3],\ [733, 733, -4*w^3 + 8*w^2 + 20*w + 1],\ [739, 739, -4*w^3 + 8*w^2 + 20*w - 7],\ [739, 739, -w^2 - 4*w + 1],\ [751, 751, 3*w^3 - 7*w^2 - 14*w + 1],\ [757, 757, -4*w^3 + 3*w^2 + 29*w + 21],\ [761, 761, -w^3 + 5*w^2 - 2*w - 9],\ [773, 773, -w^2 + 4*w - 5],\ [787, 787, 3*w^3 - 5*w^2 - 17*w - 3],\ [787, 787, -w^3 + 3*w^2 - 4*w + 1],\ [787, 787, -3*w^3 + 8*w^2 + 10*w - 5],\ [787, 787, 4*w^3 - 8*w^2 - 18*w - 3],\ [809, 809, -2*w^3 + 7*w^2 + 3*w - 9],\ [811, 811, -3*w^2 + 5*w + 15],\ [811, 811, -4*w^3 + 11*w^2 + 12*w - 9],\ [821, 821, -7*w^3 + 15*w^2 + 29*w - 3],\ [823, 823, w^2 + 3],\ [827, 827, -2*w^3 + 4*w^2 + 11*w - 3],\ [827, 827, 6*w^3 - 12*w^2 - 26*w - 3],\ [829, 829, -3*w^3 + 7*w^2 + 9*w - 5],\ [859, 859, 3*w^2 - 6*w - 7],\ [859, 859, -w^3 + w^2 + 2*w + 3],\ [863, 863, w^3 - 3*w^2 - 2*w - 3],\ [877, 877, -2*w^3 + w^2 + 14*w + 7],\ [877, 877, 5*w^3 - 11*w^2 - 22*w + 3],\ [877, 877, 4*w^3 - 12*w^2 - 10*w + 13],\ [877, 877, -2*w^3 + 6*w^2 - 5],\ [883, 883, 2*w^3 - 4*w^2 - 12*w + 7],\ [887, 887, -5*w^3 + 13*w^2 + 15*w - 7],\ [947, 947, w^3 + w^2 - 10*w - 9],\ [947, 947, -4*w^3 + 13*w^2 + 8*w - 21],\ [953, 953, 3*w - 5],\ [967, 967, -4*w^3 + 11*w^2 + 11*w - 7],\ [967, 967, 5*w^3 - 15*w^2 - 7*w + 7],\ [971, 971, -w^3 + w^2 + 9*w + 1],\ [977, 977, -6*w^3 + 11*w^2 + 28*w + 7],\ [983, 983, 3*w^3 - 9*w^2 - 3*w + 7],\ [997, 997, -4*w^3 + 14*w^2 + 5*w - 23],\ [1009, 1009, -3*w^3 + 11*w^2 + w - 13],\ [1009, 1009, -2*w^3 + 5*w^2 + 11*w - 1],\ [1013, 1013, w^3 - 5*w^2 + 3*w + 11],\ [1031, 1031, w^3 - 3*w^2 - 3*w - 3],\ [1033, 1033, -3*w^3 + 7*w^2 + 6*w - 5],\ [1061, 1061, -2*w^3 + 9*w^2 - 3*w - 15],\ [1063, 1063, 5*w^3 - 7*w^2 - 30*w - 11],\ [1069, 1069, -2*w^3 + 6*w^2 + 2*w - 7],\ [1091, 1091, 2*w^2 - 6*w - 9],\ [1091, 1091, 4*w^3 - 11*w^2 - 14*w + 17],\ [1093, 1093, -2*w^3 + w^2 + 16*w + 7],\ [1097, 1097, -4*w^3 + 12*w^2 + 11*w - 15],\ [1103, 1103, -2*w^3 + 2*w^2 + 11*w + 9],\ [1103, 1103, 3*w^3 - 7*w^2 - 13*w + 1],\ [1117, 1117, -5*w^3 + 7*w^2 + 27*w + 15],\ [1123, 1123, 2*w^3 - 5*w^2 - 10*w + 5],\ [1129, 1129, -5*w^3 + 9*w^2 + 23*w + 7],\ [1129, 1129, 6*w^3 - 10*w^2 - 32*w - 11],\ [1129, 1129, -3*w^3 + 7*w^2 + 9*w - 3],\ [1129, 1129, -5*w^3 + 11*w^2 + 19*w - 9],\ [1153, 1153, 4*w^2 - 7*w - 19],\ [1163, 1163, -2*w^3 + 3*w^2 + 14*w + 5],\ [1193, 1193, -3*w^3 + 9*w^2 + 9*w - 7],\ [1201, 1201, -w^2 + 6*w - 1],\ [1217, 1217, 2*w^3 - 2*w^2 - 14*w - 3],\ [1223, 1223, -w^3 + 3*w^2 + 2*w - 9],\ [1229, 1229, 4*w^3 - 9*w^2 - 16*w + 1],\ [1231, 1231, 5*w^2 - 10*w - 21],\ [1231, 1231, -2*w^3 + 7*w^2 + 5*w - 3],\ [1249, 1249, -2*w^3 + 5*w^2 + 3*w - 7],\ [1259, 1259, -2*w^3 + 3*w^2 + 10*w - 1],\ [1277, 1277, -4*w^3 + 8*w^2 + 16*w + 3],\ [1277, 1277, 3*w^3 - 7*w^2 - 8*w + 1],\ [1283, 1283, -3*w^3 + 6*w^2 + 16*w + 1],\ [1289, 1289, 3*w^3 - 9*w^2 - 6*w + 13],\ [1289, 1289, -2*w^2 + 8*w - 1],\ [1289, 1289, -5*w^3 + 15*w^2 + 13*w - 21],\ [1289, 1289, 2*w^3 - 6*w^2 - 5*w + 13],\ [1291, 1291, -7*w^3 + 11*w^2 + 39*w + 13],\ [1303, 1303, 4*w^3 - 10*w^2 - 17*w + 13],\ [1307, 1307, 2*w^3 - 8*w^2 + 7],\ [1319, 1319, -w^3 + 5*w^2 + w - 7],\ [1319, 1319, -5*w^3 + 10*w^2 + 24*w - 3],\ [1327, 1327, -2*w^3 + 3*w^2 + 11*w - 1],\ [1367, 1367, -4*w^3 + 11*w^2 + 8*w - 7],\ [1367, 1367, 3*w^3 - 7*w^2 - 12*w - 1],\ [1373, 1373, 4*w^3 - 11*w^2 - 7*w + 7],\ [1373, 1373, 3*w^2 - 4*w - 9],\ [1381, 1381, -w^3 + 5*w^2 - w - 17],\ [1381, 1381, -4*w^3 + 8*w^2 + 20*w - 5],\ [1399, 1399, w^2 - 2*w - 9],\ [1399, 1399, -6*w^3 + 10*w^2 + 30*w + 9],\ [1423, 1423, -6*w^3 + 16*w^2 + 16*w - 15],\ [1427, 1427, 3*w^3 - 3*w^2 - 21*w - 7],\ [1429, 1429, -8*w^3 + 12*w^2 + 43*w + 17],\ [1429, 1429, 5*w^3 - 14*w^2 - 10*w + 5],\ [1433, 1433, -w^3 + 5*w^2 - 9],\ [1433, 1433, -w^3 + w^2 + w + 3],\ [1433, 1433, 4*w^3 - 14*w^2 - 3*w + 9],\ [1433, 1433, -w^3 + 5*w^2 - 5*w + 3],\ [1439, 1439, -2*w^3 + 8*w^2 - w - 13],\ [1451, 1451, 3*w^2 - 5*w - 9],\ [1459, 1459, -3*w^3 + 13*w^2 + w - 7],\ [1459, 1459, -4*w^3 + 11*w^2 + 13*w - 9],\ [1471, 1471, -5*w^3 + 8*w^2 + 28*w + 7],\ [1481, 1481, 3*w^3 - 6*w^2 - 10*w - 1],\ [1483, 1483, w^3 - 8*w - 13],\ [1489, 1489, 2*w^3 - 4*w^2 - 2*w + 5],\ [1499, 1499, -w^3 + 2*w^2 + 8*w - 1],\ [1511, 1511, 3*w^2 - 4*w - 7],\ [1511, 1511, -6*w^3 + 15*w^2 + 19*w - 15],\ [1543, 1543, 4*w^3 - 3*w^2 - 28*w - 17],\ [1543, 1543, 4*w - 5],\ [1549, 1549, 3*w^2 - 7*w - 3],\ [1553, 1553, 3*w^3 - 10*w^2 - 4*w + 15],\ [1553, 1553, -w^3 - 3*w^2 + 14*w + 21],\ [1567, 1567, -6*w^3 + 14*w^2 + 22*w - 7],\ [1571, 1571, -w^3 - w^2 + 10*w + 17],\ [1579, 1579, -w^3 + 6*w + 17],\ [1579, 1579, 3*w^3 - 9*w^2 - 7*w + 3],\ [1597, 1597, 2*w^3 - w^2 - 18*w - 5],\ [1607, 1607, w^3 - 5*w^2 + 17],\ [1609, 1609, -3*w^3 + 7*w^2 + 14*w - 5],\ [1621, 1621, w^2 - 3*w - 9],\ [1627, 1627, 5*w^3 - 13*w^2 - 14*w + 3],\ [1627, 1627, -w^3 + 6*w^2 - 4*w + 1],\ [1663, 1663, 6*w^3 - 11*w^2 - 31*w - 5],\ [1663, 1663, 6*w^3 - 18*w^2 - 15*w + 25],\ [1667, 1667, -w^3 + 2*w^2 + 4*w - 7],\ [1667, 1667, 4*w^3 - 14*w^2 - 7*w + 31],\ [1669, 1669, 3*w^2 - 8*w - 11],\ [1669, 1669, -4*w^3 + 11*w^2 + 10*w - 15],\ [1681, 41, 6*w^3 - 11*w^2 - 31*w + 1],\ [1681, 41, 4*w^3 - 13*w^2 - 8*w + 9],\ [1693, 1693, w^3 - w^2 - 7*w - 9],\ [1693, 1693, w^3 - 2*w^2 - 3],\ [1693, 1693, w^3 + w^2 - 11*w - 5],\ [1693, 1693, -2*w^3 + 4*w^2 + 2*w + 1],\ [1697, 1697, -4*w^3 + 12*w^2 + w - 7],\ [1699, 1699, -7*w^3 + 19*w^2 + 19*w - 17],\ [1699, 1699, 6*w^3 - 14*w^2 - 21*w + 3],\ [1709, 1709, -w^3 + 8*w + 1],\ [1709, 1709, 2*w^3 - 4*w^2 - 6*w - 5],\ [1723, 1723, -2*w^3 + 7*w^2 + w - 11],\ [1723, 1723, 4*w^3 - 9*w^2 - 14*w + 5],\ [1733, 1733, -w^3 + w^2 + 3*w - 5],\ [1733, 1733, -w^2 - 5*w + 1],\ [1741, 1741, -3*w^3 + 10*w^2 + 6*w - 15],\ [1741, 1741, -3*w^3 + 9*w^2 + 4*w - 9],\ [1747, 1747, 2*w^3 - 2*w^2 - 13*w - 3],\ [1753, 1753, 4*w^2 - 8*w - 15],\ [1753, 1753, -w^3 + 7*w^2 - 4*w - 27],\ [1759, 1759, -7*w^3 + 17*w^2 + 27*w - 21],\ [1801, 1801, -3*w^3 + 7*w^2 + 6*w - 3],\ [1811, 1811, -4*w^2 + 7*w + 5],\ [1811, 1811, w^3 - w^2 - 4*w - 7],\ [1823, 1823, -7*w^3 + 21*w^2 + 18*w - 31],\ [1831, 1831, -2*w^2 + 3*w - 3],\ [1847, 1847, 5*w^3 - 15*w^2 - 12*w + 17],\ [1861, 1861, -4*w^3 + 8*w^2 + 14*w - 5],\ [1861, 1861, -4*w^2 + 9*w + 11],\ [1871, 1871, -4*w^3 + 10*w^2 + 11*w - 3],\ [1873, 1873, -2*w^3 + 7*w^2 + 6*w - 3],\ [1879, 1879, w^3 - 7*w^2 + 8*w + 11],\ [1879, 1879, -2*w^3 + 19*w + 15],\ [1889, 1889, -2*w^3 + 6*w^2 + w - 7],\ [1933, 1933, 2*w^3 - 3*w^2 - 12*w + 3],\ [1949, 1949, -4*w^3 + 15*w^2 + 4*w - 7],\ [1951, 1951, 2*w^2 + 4*w - 5],\ [1973, 1973, 6*w^3 - 14*w^2 - 25*w + 7],\ [1993, 1993, -4*w^3 + 11*w^2 + 6*w - 5],\ [1997, 1997, -4*w^3 + 9*w^2 + 14*w - 3],\ [1997, 1997, 5*w^3 - 11*w^2 - 20*w + 9],\ [1999, 1999, -3*w^3 + 11*w^2 + 3*w - 19]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x K = QQ e = 1 hecke_eigenvalues_array = [0, -1, -3, -5, 0, 3, 2, 0, -3, 4, 1, -12, -7, -9, -2, -2, -9, 3, 9, 2, -12, 18, -2, -12, 1, -3, 3, -9, -18, 9, -7, -5, 6, -6, -8, -10, -17, -4, -13, 11, -21, -6, 11, 27, 20, 19, -2, 18, -27, 4, -24, -24, -9, 12, 20, 8, -2, -26, -27, 2, -6, -24, -6, 26, -13, -6, -12, 10, -8, -8, -14, 6, 27, 24, 16, 28, 20, -4, -30, 18, -33, 28, 6, 6, -28, 18, 12, -29, 14, 21, -24, -38, 2, 26, -24, 21, -32, -28, -22, 34, -10, -20, 40, -2, 12, -3, -20, -11, 11, 46, -46, -30, 10, 20, -50, 3, 27, 12, 51, -44, 19, 0, -24, -4, 2, 28, 26, 8, -29, -21, 30, -34, -4, 4, -32, -39, 37, -2, 30, 8, -36, 21, -28, 28, -50, -36, -56, -14, 13, -32, -16, 30, -27, -39, 42, -40, 22, 12, 33, -24, 29, 10, 50, 0, 0, 8, 42, 44, 22, 3, 30, -25, 42, 42, -3, 16, 28, -20, -40, -50, 23, -55, 6, -15, -13, 36, 0, 60, 44, 37, 49, -6, 36, -12, 24, 18, -6, 18, 9, 34, -46, -27, 45, -60, -32, -6, 24, -45, 21, -17, 59, 11, 40, -13, -30, 34, -28, 9, 6, 0, 12, 42, -24, -28, 11, 43, 54, -56, -50, 48, 0, 54, 5, -34, 50, -9, -54, -40, -60, 14, 43, -4, 63, 5, -77, 46, 28, 55, -17, 78, 27, -38, -2, -50, -64, -61, 68, -8, 49, -72, -22, 62, 6, -27, 35, -10, 6, 33, -1, -22, 32, 70, 2, -64, 4, 6, 42, -69, -8, 12, 19, 10, 18, 74, -2, -19, -30, 38, 54, -35, -30, -10, -27, -12, 71] 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 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]