/* 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([8, 1, -7, -1, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([16, 2, 2]) primes_array = [ [2, 2, -w^2 + 2],\ [2, 2, -w^3 + 2*w^2 + 4*w - 5],\ [5, 5, -w^3 + 2*w^2 + 3*w - 1],\ [13, 13, w^3 - 2*w^2 - 3*w + 5],\ [17, 17, -w^2 - w + 3],\ [19, 19, -w^3 + 3*w^2 + 2*w - 7],\ [23, 23, w^3 - 2*w^2 - 3*w + 3],\ [31, 31, -w^2 + w + 1],\ [47, 47, -w^3 + w^2 + 4*w - 3],\ [49, 7, 2*w^3 - 5*w^2 - 7*w + 11],\ [53, 53, -3*w^3 + 9*w^2 + 10*w - 31],\ [53, 53, w^3 - w^2 - 4*w + 1],\ [61, 61, 3*w^3 - 6*w^2 - 13*w + 13],\ [61, 61, 2*w^2 - 7],\ [71, 71, w^2 - 3*w - 5],\ [73, 73, 2*w - 3],\ [73, 73, -2*w^3 + 6*w^2 + 6*w - 19],\ [79, 79, 2*w^2 - 5],\ [81, 3, -3],\ [101, 101, 2*w^2 - 4*w - 9],\ [107, 107, 2*w - 1],\ [109, 109, 2*w^3 - 3*w^2 - 11*w + 7],\ [109, 109, 2*w^2 - 2*w - 7],\ [125, 5, -2*w^3 + 8*w^2 - 17],\ [127, 127, 2*w^2 - 4*w - 7],\ [127, 127, -w^2 + 3*w + 1],\ [131, 131, w^3 - w^2 - 4*w - 3],\ [137, 137, 4*w^3 - 10*w^2 - 14*w + 25],\ [139, 139, -w^3 + 2*w^2 + w - 5],\ [139, 139, 2*w^2 - 2*w - 5],\ [149, 149, -w^3 + 4*w^2 - w - 7],\ [149, 149, -2*w^3 + 5*w^2 + 9*w - 19],\ [151, 151, -w^3 + 4*w^2 + w - 11],\ [151, 151, w^2 + w + 1],\ [157, 157, -2*w + 5],\ [163, 163, 4*w^3 - 8*w^2 - 18*w + 19],\ [163, 163, w^3 - 4*w^2 - w + 9],\ [167, 167, -w^3 + 3*w^2 + 2*w - 9],\ [167, 167, w^3 - 2*w^2 - 5*w + 1],\ [173, 173, w^2 - 3*w - 7],\ [173, 173, w^3 - 3*w^2 - 4*w + 5],\ [181, 181, -w^3 + 4*w^2 + w - 13],\ [191, 191, -w^3 + 2*w^2 + 3*w + 1],\ [191, 191, -2*w^3 + 3*w^2 + 9*w - 3],\ [193, 193, -w^3 + 4*w^2 + 3*w - 13],\ [197, 197, 5*w^3 - 10*w^2 - 21*w + 21],\ [211, 211, -w^3 + 3*w^2 + 4*w - 13],\ [227, 227, w^3 - 4*w^2 + w + 5],\ [229, 229, -w^3 + 5*w^2 - 2*w - 11],\ [229, 229, -w^3 + 3*w^2 + 4*w - 3],\ [239, 239, -2*w^3 + 4*w^2 + 8*w - 13],\ [241, 241, 4*w^3 - 10*w^2 - 16*w + 31],\ [241, 241, -5*w^3 + 12*w^2 + 21*w - 31],\ [251, 251, -w^3 + 3*w^2 + 2*w - 11],\ [263, 263, -2*w^3 + 2*w^2 + 10*w + 1],\ [263, 263, -2*w - 5],\ [269, 269, w^2 + w - 5],\ [269, 269, 4*w^3 - 7*w^2 - 19*w + 15],\ [271, 271, 2*w^3 - 5*w^2 - 5*w + 11],\ [271, 271, -w^3 + 7*w + 3],\ [277, 277, -2*w^3 + 4*w^2 + 6*w - 5],\ [277, 277, 4*w^2 - 13],\ [281, 281, w^3 - 5*w^2 + 11],\ [307, 307, w^3 - 3*w - 3],\ [313, 313, 5*w^3 - 10*w^2 - 23*w + 27],\ [313, 313, -2*w^3 + 3*w^2 + 9*w - 7],\ [317, 317, -3*w^3 + 6*w^2 + 11*w - 15],\ [347, 347, -w^3 - w^2 + 1],\ [347, 347, -2*w^3 + 6*w^2 + 6*w - 17],\ [353, 353, -2*w^3 + 4*w^2 + 8*w - 5],\ [353, 353, w^3 - 5*w - 1],\ [359, 359, -2*w^2 + 1],\ [367, 367, -w^3 + 4*w^2 + 3*w - 17],\ [373, 373, 2*w^2 - 2*w - 13],\ [397, 397, 5*w^3 - 13*w^2 - 18*w + 39],\ [397, 397, 2*w^3 - 5*w^2 - 7*w + 9],\ [401, 401, -4*w^3 + 12*w^2 + 8*w - 25],\ [401, 401, -3*w^3 + 7*w^2 + 8*w - 13],\ [421, 421, -4*w^3 + 11*w^2 + 13*w - 31],\ [433, 433, 2*w^3 - 12*w - 15],\ [433, 433, w^3 - w^2 - 6*w + 5],\ [439, 439, 5*w^3 - 13*w^2 - 12*w + 19],\ [439, 439, -2*w^3 + 3*w^2 + 7*w + 1],\ [449, 449, -w^3 + 4*w^2 + 3*w - 11],\ [461, 461, -4*w^3 + 11*w^2 + 7*w - 19],\ [461, 461, -w^3 + w^2 + 2*w + 1],\ [463, 463, -w^3 + 2*w^2 + 7*w - 1],\ [463, 463, -4*w^3 + 11*w^2 + 7*w - 13],\ [467, 467, 2*w^2 - 2*w - 3],\ [467, 467, 2*w^3 - 4*w^2 - 6*w + 11],\ [487, 487, -3*w^3 + 11*w^2 + 2*w - 23],\ [487, 487, w^3 - w + 5],\ [491, 491, w^3 - 2*w^2 - 3*w + 9],\ [499, 499, w^3 - 3*w^2 - 4*w + 1],\ [503, 503, -6*w^3 + 11*w^2 + 27*w - 21],\ [503, 503, -3*w^3 + 8*w^2 + 5*w - 9],\ [521, 521, 2*w^2 - 4*w - 11],\ [541, 541, -3*w^3 + 7*w^2 + 8*w - 11],\ [557, 557, 5*w^3 - 11*w^2 - 18*w + 21],\ [557, 557, 4*w^3 - 7*w^2 - 17*w + 9],\ [557, 557, 4*w^3 - 8*w^2 - 16*w + 15],\ [557, 557, -3*w^3 + 6*w^2 + 11*w - 9],\ [563, 563, -2*w^3 + 6*w^2 + 8*w - 13],\ [569, 569, -2*w^3 + 3*w^2 + 7*w - 11],\ [587, 587, 3*w^3 - 8*w^2 - 9*w + 15],\ [601, 601, -3*w^2 + 5*w + 9],\ [607, 607, -w^3 + w^2 - 5],\ [619, 619, -2*w^3 + 4*w^2 + 6*w - 13],\ [631, 631, -2*w^3 + 7*w^2 + 5*w - 23],\ [643, 643, 2*w^3 - 2*w^2 - 12*w - 5],\ [659, 659, -w^3 + 6*w^2 + 3*w - 11],\ [659, 659, -3*w^2 + w + 9],\ [661, 661, -2*w^3 + 4*w^2 + 8*w - 3],\ [661, 661, -w^3 + 4*w^2 + 5*w - 7],\ [673, 673, w^2 - w - 9],\ [673, 673, 2*w^3 - 7*w^2 - 3*w + 17],\ [677, 677, 9*w^3 - 20*w^2 - 37*w + 51],\ [701, 701, -7*w^3 + 18*w^2 + 27*w - 57],\ [709, 709, 4*w^3 - 6*w^2 - 14*w + 17],\ [709, 709, w^3 + w^2 - 6*w - 17],\ [719, 719, -2*w^3 + 6*w^2 + 4*w - 15],\ [719, 719, -2*w^3 + 3*w^2 + 7*w + 3],\ [727, 727, -w^3 + 3*w^2 + 6*w - 3],\ [743, 743, -2*w^3 + 2*w^2 + 6*w - 1],\ [757, 757, 6*w^3 - 16*w^2 - 22*w + 51],\ [757, 757, w^3 + w^2 - 10*w - 11],\ [761, 761, w^2 + w - 7],\ [761, 761, 2*w^2 - 17],\ [761, 761, 2*w^3 - 6*w^2 - 2*w + 13],\ [761, 761, 2*w^2 - 4*w - 3],\ [769, 769, -w^3 - w^2 + 4*w + 7],\ [769, 769, w^3 + w^2 - 4*w - 1],\ [787, 787, 3*w^3 - 6*w^2 - 11*w + 3],\ [809, 809, -w^3 + 3*w^2 + 2*w - 1],\ [809, 809, 2*w^3 - w^2 - 9*w - 3],\ [809, 809, 2*w^3 - 7*w^2 - 3*w + 15],\ [809, 809, 2*w^3 - 2*w^2 - 8*w - 1],\ [811, 811, -8*w^3 + 19*w^2 + 33*w - 57],\ [811, 811, -2*w^3 + 2*w^2 + 4*w - 11],\ [823, 823, 2*w^3 - w^2 - 7*w + 1],\ [823, 823, -3*w^3 + 8*w^2 + 7*w - 9],\ [829, 829, -w^3 + 5*w - 1],\ [853, 853, -w^3 + 5*w^2 - 2*w - 15],\ [853, 853, -2*w^3 + 4*w^2 + 4*w - 9],\ [853, 853, 2*w^3 - w^2 - 11*w - 3],\ [853, 853, 2*w^3 - 5*w^2 - 7*w + 5],\ [857, 857, -2*w^3 + 5*w^2 + 7*w - 7],\ [859, 859, w^3 - 2*w^2 - w + 9],\ [859, 859, w^3 - 2*w^2 - w - 1],\ [877, 877, 2*w^3 - 6*w^2 - 8*w + 25],\ [877, 877, -4*w^3 + 8*w^2 + 16*w - 19],\ [881, 881, -4*w^2 + 6*w + 13],\ [881, 881, w^3 + 2*w^2 - 7*w - 9],\ [887, 887, -4*w^2 + 4*w + 11],\ [887, 887, -w^3 - w^2 + 6*w + 11],\ [887, 887, w^3 - 6*w^2 - w + 29],\ [887, 887, 4*w^3 - 9*w^2 - 19*w + 31],\ [907, 907, -3*w^3 + 9*w^2 + 12*w - 25],\ [907, 907, -w^3 + 2*w^2 + 7*w - 7],\ [929, 929, -7*w^3 + 13*w^2 + 32*w - 29],\ [941, 941, -2*w^3 + 10*w^2 - 2*w - 23],\ [941, 941, 4*w^3 - 9*w^2 - 13*w + 13],\ [967, 967, -w^3 + 2*w^2 - w - 1],\ [977, 977, w^3 - 5*w^2 + 15],\ [983, 983, w^3 + 2*w^2 - 9*w - 11],\ [983, 983, 2*w^3 - 5*w^2 - 5*w + 13],\ [997, 997, -4*w^3 + 10*w^2 + 12*w - 23],\ [1009, 1009, 2*w^2 + 2*w + 1],\ [1019, 1019, 2*w^3 - 2*w^2 - 6*w + 3],\ [1031, 1031, w^3 - 9*w - 7],\ [1033, 1033, 2*w^3 - 2*w^2 - 8*w + 1],\ [1063, 1063, -w^3 - w - 7],\ [1063, 1063, 4*w^3 - 8*w^2 - 12*w + 5],\ [1091, 1091, 2*w^3 - 2*w^2 - 4*w + 1],\ [1093, 1093, 3*w^3 - 4*w^2 - 13*w - 1],\ [1093, 1093, w^3 - w^2 - 4*w - 7],\ [1097, 1097, w^3 + 3*w^2 - 2*w - 9],\ [1109, 1109, w^2 + w - 9],\ [1109, 1109, -2*w^3 + 5*w^2 + 3*w - 11],\ [1123, 1123, -5*w^3 + 16*w^2 + 7*w - 27],\ [1123, 1123, 8*w^3 - 17*w^2 - 35*w + 47],\ [1163, 1163, 3*w^3 - 3*w^2 - 14*w - 5],\ [1171, 1171, 2*w^3 - w^2 - 7*w - 1],\ [1171, 1171, -3*w^3 + 6*w^2 + 9*w - 7],\ [1181, 1181, 3*w^3 - 7*w^2 - 10*w + 21],\ [1187, 1187, w^3 - 2*w^2 - 7*w + 3],\ [1201, 1201, -3*w^3 + 5*w^2 + 10*w - 3],\ [1201, 1201, -3*w^2 + 3*w + 19],\ [1213, 1213, w^3 - 5*w^2 + 4*w + 7],\ [1217, 1217, w^3 - 4*w^2 - 5*w + 5],\ [1223, 1223, -w^3 - w^2 + 2*w + 5],\ [1223, 1223, w^3 - 5*w^2 + 4*w + 5],\ [1231, 1231, 7*w^3 - 15*w^2 - 28*w + 33],\ [1237, 1237, w^3 - 5*w^2 + 2*w + 9],\ [1249, 1249, -2*w^3 + 2*w^2 + 10*w - 3],\ [1259, 1259, -4*w^3 + 8*w^2 + 16*w - 13],\ [1283, 1283, w^3 - 2*w^2 - 3*w - 3],\ [1289, 1289, -w^3 + w^2 + 4*w - 7],\ [1291, 1291, 2*w^3 - 5*w^2 - 9*w + 9],\ [1297, 1297, -10*w^3 + 24*w^2 + 38*w - 63],\ [1301, 1301, 3*w^3 - 8*w^2 - 11*w + 17],\ [1301, 1301, 2*w^2 - 4*w - 1],\ [1307, 1307, w^3 - w^2 + 7],\ [1327, 1327, -6*w^3 + 12*w^2 + 24*w - 31],\ [1327, 1327, -w^3 + 2*w^2 - w - 5],\ [1327, 1327, 8*w^3 - 16*w^2 - 34*w + 35],\ [1327, 1327, 3*w^3 - 8*w^2 - 3*w + 15],\ [1367, 1367, -w^3 + 2*w^2 + 7*w - 5],\ [1369, 37, 7*w^3 - 17*w^2 - 18*w + 19],\ [1369, 37, w^3 - 7*w^2 + 6*w + 19],\ [1381, 1381, 6*w^2 - 6*w - 35],\ [1427, 1427, -2*w^3 + 3*w^2 + 11*w + 1],\ [1427, 1427, -2*w^3 + 7*w^2 + 7*w - 13],\ [1427, 1427, 3*w^3 - 8*w^2 - 9*w + 9],\ [1427, 1427, -12*w^3 + 26*w^2 + 50*w - 65],\ [1433, 1433, w^3 - 7*w^2 - 4*w + 11],\ [1439, 1439, -5*w^3 + 12*w^2 + 21*w - 39],\ [1453, 1453, -w^3 - w^2 - 2*w - 7],\ [1453, 1453, 2*w^3 - 4*w^2 - 6*w - 1],\ [1459, 1459, 3*w^2 - 7*w - 15],\ [1483, 1483, 4*w^3 - 15*w^2 - 9*w + 55],\ [1483, 1483, w^3 - 3*w - 5],\ [1493, 1493, -2*w^3 + 8*w^2 + 2*w - 23],\ [1499, 1499, -w^3 + 9*w + 1],\ [1499, 1499, w^3 - 4*w^2 + 3*w + 3],\ [1499, 1499, 5*w^3 - 12*w^2 - 13*w + 17],\ [1499, 1499, w^3 + w^2 - 8*w - 15],\ [1511, 1511, -7*w^3 + 17*w^2 + 26*w - 47],\ [1531, 1531, w^3 - 4*w^2 - 7*w + 13],\ [1543, 1543, 4*w^3 - 7*w^2 - 21*w + 21],\ [1543, 1543, 5*w^2 + 3*w - 7],\ [1579, 1579, 4*w^3 - 9*w^2 - 13*w + 17],\ [1583, 1583, -2*w^3 + 6*w^2 + 6*w - 23],\ [1597, 1597, -4*w^3 + 11*w^2 + 13*w - 25],\ [1609, 1609, 4*w^3 - 11*w^2 - 11*w + 27],\ [1609, 1609, -4*w^3 + 13*w^2 + 5*w - 21],\ [1637, 1637, -6*w^3 + 16*w^2 + 20*w - 45],\ [1667, 1667, 5*w^2 - 3*w - 13],\ [1667, 1667, -5*w^2 - w + 11],\ [1697, 1697, -w^3 + 3*w^2 + 8*w + 1],\ [1699, 1699, -w^3 + 4*w^2 - w - 13],\ [1699, 1699, 5*w^3 - 14*w^2 - 9*w + 25],\ [1721, 1721, 2*w^3 - 7*w^2 - w + 17],\ [1723, 1723, -3*w^3 + 6*w^2 + 9*w - 11],\ [1733, 1733, w^3 - w^2 - 8*w + 3],\ [1747, 1747, w^3 - 2*w^2 - 7*w - 5],\ [1753, 1753, -w^3 + 3*w^2 - 11],\ [1759, 1759, -w^3 + 6*w^2 - w - 15],\ [1759, 1759, -4*w^3 + 8*w^2 + 14*w - 19],\ [1783, 1783, 3*w^3 - 6*w^2 - 7*w + 13],\ [1783, 1783, -5*w^2 - 3*w + 13],\ [1787, 1787, -2*w^3 + 7*w^2 + 5*w - 21],\ [1787, 1787, w^2 - 3*w - 11],\ [1787, 1787, -2*w - 7],\ [1787, 1787, -w^3 + 3*w^2 + 4*w - 15],\ [1789, 1789, w^3 - w^2 - 8*w - 1],\ [1789, 1789, -4*w^2 + 8*w + 13],\ [1801, 1801, -3*w^3 + 9*w^2 + 6*w - 11],\ [1823, 1823, -2*w^3 + 5*w^2 + 5*w - 15],\ [1831, 1831, 3*w^3 - 5*w^2 - 12*w + 9],\ [1831, 1831, 3*w^3 - 3*w^2 - 18*w + 1],\ [1831, 1831, w^3 - 5*w^2 - 2*w + 5],\ [1831, 1831, w^3 - 5*w^2 - 2*w + 13],\ [1847, 1847, 2*w^3 - 2*w^2 - 8*w - 9],\ [1847, 1847, -3*w^3 + 5*w^2 + 14*w - 17],\ [1849, 43, 3*w^3 - 3*w^2 - 12*w + 5],\ [1849, 43, 3*w^3 - 2*w^2 - 19*w - 5],\ [1861, 1861, 4*w^3 - 13*w^2 - 5*w + 27],\ [1867, 1867, w^3 + 2*w^2 - 5*w - 5],\ [1871, 1871, -w^3 + 5*w^2 + 2*w - 15],\ [1871, 1871, 2*w^3 - w^2 - 15*w + 1],\ [1873, 1873, w^3 - 7*w - 11],\ [1873, 1873, -4*w^3 + 9*w^2 + 11*w - 17],\ [1877, 1877, -4*w - 1],\ [1877, 1877, 4*w + 9],\ [1889, 1889, -3*w^3 + 7*w^2 + 6*w - 7],\ [1901, 1901, 2*w^3 - 8*w^2 - 2*w + 17],\ [1901, 1901, -3*w^2 + 7*w + 7],\ [1913, 1913, 4*w^3 - 14*w^2 - 2*w + 27],\ [1931, 1931, w^2 + 3*w - 7],\ [1931, 1931, 3*w^3 - 3*w^2 - 12*w - 5],\ [1933, 1933, -2*w^3 + 7*w^2 + 7*w - 17],\ [1951, 1951, -2*w^3 + 6*w^2 + 4*w - 17],\ [1951, 1951, -5*w^3 + 13*w^2 + 20*w - 43],\ [1973, 1973, -2*w^3 + 9*w^2 - w - 19],\ [1979, 1979, -4*w^3 + 10*w^2 + 18*w - 37],\ [1979, 1979, -2*w^3 + 6*w^2 + 8*w - 11],\ [1987, 1987, -w^3 + 3*w^2 + 6*w - 13],\ [1997, 1997, -4*w^3 + 11*w^2 + 13*w - 29],\ [1997, 1997, 4*w - 1],\ [1997, 1997, 9*w^3 - 23*w^2 - 22*w + 35],\ [1997, 1997, -6*w^3 + 18*w^2 + 8*w - 31],\ [1999, 1999, 6*w^3 - 19*w^2 - 9*w + 37]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^2 - 8 K. = NumberField(heckePol) hecke_eigenvalues_array = [-1, 0, e, 0, 0, 0, -8, -3*e, e, -3*e, 6, -6, 3*e, -2*e, -8, 6*e, -4*e, -8, -2, -2*e, -4, 6, 14, 2*e, -16, -16, -2*e, 6, 0, -6*e, 6, 6, 0, -8, -2*e, 20, -20, -3*e, -5*e, 5*e, -7*e, 7*e, 24, 8, -18, 18, -20, 5*e, 2*e, -6*e, 0, 8*e, 5*e, -5*e, 0, -16, -4*e, 11*e, -3*e, 9*e, -10, -6, 6, -3*e, -e, -10*e, 34, -20, -36, -e, -13*e, 0, -10*e, -22, 7*e, 6*e, -30, -18, 26, -4*e, 9*e, 9*e, -6*e, 6, 0, -e, 8, -40, 8*e, -10*e, 32, -8, 20, -4, 15*e, -e, 3*e, 6, 10, 18, -42, 22, -5*e, -6, 4*e, -9*e, 14*e, 14*e, 8, 15*e, 28, -20, -9*e, -15*e, 14, 2, 10*e, -10, -22, 34, 14*e, -e, -13*e, -8, -10, -42, -e, 6*e, -10*e, -11*e, -12*e, -5*e, -2*e, 26, 22, -30, -26, 3*e, 15*e, 40, 32, 14*e, -5*e, 14*e, -5*e, -19*e, -10*e, 17*e, -6*e, 38, 34, 0, 10*e, 15*e, -2*e, 14*e, -13*e, -12, -12, 10*e, -21*e, 4*e, 8, -58, -4*e, 19*e, 6*e, -46, -44, 8, -10, 11*e, -21*e, 2*e, -26, -26, 5*e, -e, -19*e, -12*e, -19*e, -36, 28, -12, -8*e, -36, 22, -34, -30, e, 7*e, -2*e, -18*e, 0, -12*e, 5*e, 4, -34, -12*e, 50, -3*e, -4*e, -22*e, 40, -16, -48, -16, 56, -58, 70, -66, -2*e, -7*e, -e, 11*e, 10*e, -48, -6, 34, 15*e, -9*e, 8*e, -22, 4, -28, -20, -52, 3*e, -25*e, 12*e, -9*e, -20, 24, 66, -6*e, -10*e, -e, -36, -52, 9*e, 9*e, -11*e, 14*e, 44, -58, -52, -12*e, 8, -8, -25*e, -24*e, -60, -12, -60, 12, 58, -50, 46, -20*e, -48, -48, 32, -16, 0, -6*e, 58, -58, 4*e, 13*e, 80, -48, 50, -2, 22, 10, 26*e, -50, 34, 18, 21*e, -18*e, -14, -2*e, -3*e, 25*e, -22*e, 11*e, 12*e, 18, -82, 66, -50, -72] 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^2 + 2])] = 1 AL_eigenvalues[ZF.ideal([2, 2, -w^3 + 2*w^2 + 4*w - 5])] = -1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]