/* 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, 2, -5, -2, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([8, 4, w^2 - 2*w - 5]) primes_array = [ [2, 2, w],\ [2, 2, -w - 1],\ [5, 5, -w^3 + 2*w^2 + 5*w - 1],\ [7, 7, -w^3 + 2*w^2 + 5*w - 3],\ [29, 29, -w^2 + w + 3],\ [29, 29, 2*w^3 - 5*w^2 - 7*w + 9],\ [31, 31, -w^3 + 2*w^2 + 5*w + 1],\ [41, 41, -w^3 + 4*w^2 - w - 3],\ [43, 43, -w^3 + 3*w^2 + 2*w - 5],\ [47, 47, w^2 - 3*w - 3],\ [53, 53, w^3 - 2*w^2 - 3*w + 1],\ [59, 59, w^2 - w - 5],\ [61, 61, 5*w^3 - 12*w^2 - 19*w + 17],\ [67, 67, 2*w^3 - 5*w^2 - 7*w + 5],\ [67, 67, w^3 - 4*w^2 + w + 5],\ [67, 67, 3*w^3 - 7*w^2 - 12*w + 11],\ [67, 67, -2*w^3 + 4*w^2 + 8*w - 5],\ [71, 71, -3*w^3 + 8*w^2 + 9*w - 9],\ [71, 71, -w^3 + 3*w^2 + 2*w - 7],\ [79, 79, 3*w^3 - 7*w^2 - 14*w + 15],\ [81, 3, -3],\ [83, 83, -2*w^3 + 6*w^2 + 6*w - 7],\ [83, 83, w^3 - w^2 - 6*w - 1],\ [89, 89, -4*w^3 + 10*w^2 + 14*w - 13],\ [101, 101, -w^3 + w^2 + 8*w + 3],\ [107, 107, -w^3 + 3*w^2 + 1],\ [125, 5, w^3 - 4*w^2 + 3*w + 1],\ [127, 127, -2*w^2 + 6*w + 1],\ [127, 127, 2*w - 3],\ [131, 131, w^2 - 5*w + 5],\ [131, 131, -2*w^3 + 3*w^2 + 9*w - 1],\ [137, 137, 4*w^3 - 8*w^2 - 18*w + 5],\ [139, 139, -3*w^3 + 8*w^2 + 11*w - 15],\ [149, 149, 2*w^3 - 4*w^2 - 10*w + 3],\ [157, 157, -2*w^3 + 6*w^2 + 6*w - 11],\ [163, 163, -w^3 + 3*w^2 - 3],\ [173, 173, -2*w^3 + 4*w^2 + 8*w - 3],\ [173, 173, w^3 - 2*w^2 - 5*w - 3],\ [179, 179, -w^3 + 4*w^2 + 3*w - 9],\ [181, 181, 2*w^3 - 2*w^2 - 14*w - 5],\ [193, 193, -2*w + 7],\ [193, 193, -2*w^3 + 5*w^2 + 5*w - 5],\ [223, 223, 3*w^3 - 9*w^2 - 4*w + 3],\ [223, 223, w^3 - 3*w^2 - 4*w + 9],\ [233, 233, -7*w^3 + 17*w^2 + 26*w - 25],\ [239, 239, w^3 - 3*w^2 - 4*w + 1],\ [241, 241, w^2 - 3*w + 3],\ [263, 263, 4*w^3 - 10*w^2 - 16*w + 13],\ [263, 263, -2*w + 5],\ [269, 269, 3*w^3 - 6*w^2 - 13*w + 3],\ [269, 269, -4*w^3 + 11*w^2 + 9*w - 7],\ [271, 271, 2*w^3 - 5*w^2 - 9*w + 11],\ [271, 271, w^3 - w^2 - 8*w + 3],\ [277, 277, 2*w^3 - 3*w^2 - 9*w - 1],\ [277, 277, w^2 + 3*w - 1],\ [277, 277, w^3 - 4*w^2 - w + 9],\ [277, 277, w^3 - 4*w^2 + 3*w - 3],\ [281, 281, -2*w^2 + 2*w + 3],\ [281, 281, w^3 - w^2 - 8*w - 1],\ [283, 283, -w^3 + 9*w + 3],\ [313, 313, 5*w^3 - 13*w^2 - 20*w + 19],\ [313, 313, -4*w - 1],\ [317, 317, -3*w^3 + 11*w^2 - 4*w - 3],\ [337, 337, 2*w^2 - 4*w - 5],\ [337, 337, -2*w^3 + 5*w^2 + 5*w - 3],\ [343, 7, 11*w^3 - 28*w^2 - 41*w + 45],\ [349, 349, w^3 - 4*w^2 - w + 13],\ [349, 349, -4*w^3 + 15*w^2 - 3*w - 7],\ [359, 359, w^3 - 3*w^2 - 6*w - 1],\ [359, 359, -4*w^3 + 9*w^2 + 17*w - 13],\ [367, 367, -6*w^3 + 15*w^2 + 23*w - 21],\ [379, 379, -2*w^3 + 5*w^2 + 7*w - 3],\ [389, 389, 3*w^3 - 7*w^2 - 10*w + 9],\ [397, 397, -w^3 + 3*w^2 + 2*w - 9],\ [397, 397, -w^3 + 4*w^2 + w - 5],\ [409, 409, w^3 - w^2 - 6*w - 7],\ [433, 433, -5*w^3 + 11*w^2 + 20*w - 13],\ [439, 439, 4*w^3 - 10*w^2 - 14*w + 11],\ [439, 439, -2*w^3 + 7*w^2 - w - 5],\ [443, 443, -2*w^3 + 6*w^2 + 4*w - 3],\ [443, 443, -9*w^3 + 23*w^2 + 34*w - 37],\ [461, 461, 2*w^3 - 6*w^2 - 6*w + 15],\ [461, 461, 3*w^3 - 8*w^2 - 13*w + 11],\ [463, 463, 4*w^3 - 8*w^2 - 18*w + 7],\ [467, 467, 2*w^3 - 6*w^2 - 8*w + 11],\ [467, 467, 3*w^3 - 8*w^2 - 9*w + 15],\ [479, 479, 2*w^2 - 6*w + 1],\ [487, 487, 2*w^2 - 2*w - 9],\ [487, 487, w^3 - 7*w - 9],\ [491, 491, 3*w^3 - 7*w^2 - 10*w + 1],\ [491, 491, -w^3 + w^2 + 6*w - 3],\ [499, 499, 2*w^3 - 7*w^2 - w + 9],\ [499, 499, -w^3 + 5*w^2 - 17],\ [503, 503, -w^3 + 3*w^2 + 6*w - 7],\ [521, 521, 2*w^2 - 8*w + 3],\ [521, 521, 2*w^2 - 2*w - 5],\ [523, 523, 5*w^3 - 10*w^2 - 23*w + 5],\ [547, 547, 2*w^3 - 9*w^2 + 5*w + 1],\ [557, 557, -w^3 + 2*w^2 + 7*w - 1],\ [569, 569, w^3 - 5*w^2 + 6*w + 1],\ [571, 571, -6*w^3 + 16*w^2 + 22*w - 27],\ [571, 571, -2*w^3 + 5*w^2 + 7*w + 1],\ [577, 577, -2*w - 5],\ [587, 587, 2*w^3 - 4*w^2 - 6*w - 3],\ [593, 593, -8*w^3 + 21*w^2 + 29*w - 33],\ [593, 593, -w^3 + 2*w^2 + 5*w - 7],\ [601, 601, w^3 - 4*w^2 - 3*w - 1],\ [607, 607, -w^3 + 4*w^2 - w - 7],\ [613, 613, -w^3 + 5*w^2 - 2*w - 11],\ [617, 617, -w^3 + 2*w^2 + 7*w - 3],\ [617, 617, -3*w^3 + 8*w^2 + 7*w - 5],\ [631, 631, -w^3 + 5*w^2 - 4*w - 7],\ [631, 631, -4*w^3 + 11*w^2 + 15*w - 19],\ [641, 641, 5*w^3 - 16*w^2 - 3*w + 7],\ [641, 641, -6*w^3 + 19*w^2 + 3*w - 5],\ [647, 647, -4*w^3 + 9*w^2 + 17*w - 15],\ [661, 661, -2*w^3 + 2*w^2 + 12*w + 11],\ [673, 673, -2*w^2 - 4*w + 1],\ [677, 677, -w^3 + 3*w^2 + 6*w + 3],\ [677, 677, -4*w^3 + 7*w^2 + 21*w - 1],\ [683, 683, 3*w^3 - 9*w^2 - 6*w + 7],\ [683, 683, -3*w^2 + 11*w - 5],\ [701, 701, -w^3 - w^2 + 8*w + 5],\ [709, 709, -2*w^3 + 4*w^2 + 12*w - 9],\ [709, 709, 6*w^3 - 13*w^2 - 23*w + 17],\ [709, 709, -3*w^2 + 7*w + 7],\ [709, 709, 5*w^3 - 9*w^2 - 24*w + 1],\ [719, 719, 4*w^3 - 10*w^2 - 12*w + 5],\ [719, 719, w^2 + w - 5],\ [733, 733, 2*w^3 - 4*w^2 - 6*w + 3],\ [739, 739, 2*w^3 - 5*w^2 - 7*w + 1],\ [739, 739, -3*w^3 + 8*w^2 + 9*w - 7],\ [743, 743, 4*w^3 - 12*w^2 - 8*w + 15],\ [751, 751, w^3 - 5*w^2 + 2*w + 13],\ [751, 751, 2*w^3 - 9*w^2 + 7*w + 7],\ [757, 757, 4*w^3 - 9*w^2 - 15*w + 11],\ [757, 757, -2*w^3 + 4*w^2 + 6*w - 9],\ [769, 769, 3*w^3 - 6*w^2 - 15*w + 5],\ [769, 769, 3*w^2 - 5*w - 13],\ [773, 773, -w^2 + w - 3],\ [773, 773, -2*w^3 + 4*w^2 + 4*w - 3],\ [809, 809, 2*w^3 - 7*w^2 + w - 1],\ [809, 809, 2*w^3 - 3*w^2 - 7*w + 1],\ [821, 821, 2*w^3 - 4*w^2 - 6*w + 1],\ [821, 821, -2*w^3 + 2*w^2 + 6*w + 1],\ [823, 823, 4*w^3 - 11*w^2 - 11*w + 11],\ [827, 827, -3*w^3 + 9*w^2 + 2*w + 1],\ [841, 29, 5*w^3 - 9*w^2 - 26*w + 5],\ [853, 853, w^2 - 5*w + 7],\ [853, 853, 3*w^2 - 3*w - 11],\ [857, 857, 3*w^3 - 9*w^2 - 8*w + 15],\ [857, 857, -w^3 + 2*w^2 + 7*w + 5],\ [859, 859, -w^3 + 6*w^2 - 3*w - 21],\ [863, 863, w^3 - w^2 - 4*w - 5],\ [877, 877, 3*w^3 - 7*w^2 - 10*w + 5],\ [881, 881, -w^3 + 11*w + 1],\ [907, 907, 2*w^3 - 3*w^2 - 9*w + 3],\ [911, 911, 3*w^3 - 7*w^2 - 10*w - 1],\ [937, 937, 6*w^3 - 13*w^2 - 25*w + 13],\ [967, 967, 3*w^3 - 7*w^2 - 12*w + 5],\ [971, 971, w^2 - 5*w - 5],\ [977, 977, w^3 + w^2 - 6*w - 5],\ [983, 983, -4*w^3 + 7*w^2 + 19*w - 3],\ [991, 991, -w^3 + w^2 + 2*w + 3],\ [997, 997, w^3 - 6*w^2 + 9*w + 1],\ [1019, 1019, -2*w^3 + 6*w^2 + 4*w - 11],\ [1031, 1031, w^3 - 6*w^2 + 7*w + 3],\ [1049, 1049, -4*w^3 + 10*w^2 + 12*w - 7],\ [1051, 1051, 4*w^3 - 13*w^2 + w + 3],\ [1051, 1051, w^3 - 2*w^2 - w - 3],\ [1063, 1063, 2*w^3 - 7*w^2 - 3*w + 17],\ [1069, 1069, -w^3 + 11*w + 3],\ [1069, 1069, -w^3 - w^2 + 10*w + 7],\ [1087, 1087, 2*w^3 - w^2 - 13*w - 5],\ [1091, 1091, w^3 - 4*w^2 - 3*w + 17],\ [1091, 1091, 3*w^3 - 6*w^2 - 13*w + 11],\ [1093, 1093, -5*w^3 + 10*w^2 + 21*w - 9],\ [1097, 1097, -w^3 + 2*w^2 + 3*w - 7],\ [1109, 1109, 2*w^3 - 4*w^2 - 10*w - 5],\ [1117, 1117, 5*w^3 - 12*w^2 - 21*w + 17],\ [1151, 1151, -w^3 + 9*w + 1],\ [1151, 1151, 16*w^3 - 40*w^2 - 58*w + 63],\ [1171, 1171, w^3 - 4*w^2 - 5*w + 1],\ [1187, 1187, 2*w^3 - 6*w^2 - 1],\ [1223, 1223, 4*w - 5],\ [1229, 1229, 9*w^3 - 17*w^2 - 44*w + 9],\ [1231, 1231, 4*w^3 - 12*w^2 + 2*w + 3],\ [1231, 1231, 2*w^2 - 7],\ [1237, 1237, -4*w^3 + 9*w^2 + 15*w - 7],\ [1237, 1237, 3*w^3 - 6*w^2 - 11*w - 1],\ [1249, 1249, 3*w^2 - 5*w - 5],\ [1249, 1249, 4*w^3 - 8*w^2 - 16*w - 1],\ [1249, 1249, -3*w^3 + 5*w^2 + 12*w - 3],\ [1249, 1249, w^2 - 3*w - 9],\ [1259, 1259, 3*w^3 - 9*w^2 - 10*w + 23],\ [1279, 1279, -8*w^3 + 19*w^2 + 31*w - 29],\ [1279, 1279, 2*w^2 - 6*w - 7],\ [1289, 1289, 2*w^2 - 8*w + 7],\ [1289, 1289, w^3 - 2*w^2 - 5*w - 5],\ [1291, 1291, w^3 - 7*w^2 + 10*w + 5],\ [1291, 1291, -2*w^3 + 6*w^2 + 2*w - 7],\ [1297, 1297, -4*w^3 + 6*w^2 + 20*w + 5],\ [1297, 1297, 4*w^3 - 10*w^2 - 10*w + 1],\ [1303, 1303, 3*w^3 - 6*w^2 - 13*w + 9],\ [1307, 1307, -w^3 + 3*w^2 - 7],\ [1307, 1307, 4*w^3 - 7*w^2 - 21*w - 3],\ [1319, 1319, w^3 + w^2 - 6*w - 9],\ [1319, 1319, -7*w^3 + 18*w^2 + 23*w - 21],\ [1327, 1327, 3*w^3 - 5*w^2 - 16*w + 3],\ [1327, 1327, -w^3 + 3*w^2 - 9],\ [1373, 1373, -4*w^3 + 9*w^2 + 15*w - 9],\ [1381, 1381, 3*w^3 - 7*w^2 - 14*w + 17],\ [1399, 1399, -2*w^3 + 3*w^2 + 11*w - 3],\ [1409, 1409, -16*w^3 + 41*w^2 + 61*w - 63],\ [1427, 1427, -5*w^3 + 11*w^2 + 24*w - 7],\ [1429, 1429, 5*w^3 - 11*w^2 - 20*w + 11],\ [1429, 1429, -2*w^3 + 8*w^2 - 6*w - 1],\ [1447, 1447, 5*w^3 - 16*w^2 - w + 3],\ [1451, 1451, -13*w^3 + 33*w^2 + 48*w - 49],\ [1451, 1451, -6*w^3 + 15*w^2 + 21*w - 17],\ [1459, 1459, -7*w^3 + 18*w^2 + 21*w - 15],\ [1481, 1481, -w^3 + 5*w^2 - 2*w - 9],\ [1483, 1483, -4*w^3 + 11*w^2 + 13*w - 13],\ [1483, 1483, w^2 + 3*w - 5],\ [1483, 1483, w^3 - 4*w^2 + w + 9],\ [1483, 1483, 4*w^3 - 11*w^2 - 9*w + 5],\ [1487, 1487, -3*w^2 + 5*w + 1],\ [1489, 1489, w^3 - 3*w^2 + 4*w + 1],\ [1489, 1489, -7*w^3 + 19*w^2 + 22*w - 27],\ [1493, 1493, 5*w^3 - 10*w^2 - 21*w + 1],\ [1493, 1493, -2*w^3 + 4*w^2 + 10*w - 11],\ [1511, 1511, -4*w^3 + 9*w^2 + 17*w - 7],\ [1523, 1523, -2*w^3 + 8*w^2 + 2*w - 3],\ [1531, 1531, w^3 - 3*w^2 + 2*w - 3],\ [1531, 1531, -5*w^3 + 11*w^2 + 22*w - 9],\ [1549, 1549, 6*w^3 - 17*w^2 - 19*w + 31],\ [1553, 1553, 13*w^3 - 34*w^2 - 47*w + 55],\ [1559, 1559, -2*w^2 + 2*w + 13],\ [1559, 1559, -w^3 + 11*w - 3],\ [1571, 1571, -2*w^3 + 5*w^2 + 3*w + 3],\ [1583, 1583, 3*w^3 - 10*w^2 + 3*w + 1],\ [1597, 1597, -3*w^3 + 12*w^2 - 3*w - 7],\ [1597, 1597, 3*w^3 - 8*w^2 - 3*w + 3],\ [1601, 1601, -5*w^3 + 9*w^2 + 26*w - 1],\ [1601, 1601, -4*w^3 + 12*w^2 + 2*w - 5],\ [1607, 1607, -4*w^3 + 12*w^2 + 6*w - 5],\ [1613, 1613, 6*w^3 - 17*w^2 - 19*w + 35],\ [1637, 1637, 3*w^3 - 7*w^2 - 8*w + 7],\ [1637, 1637, -2*w^3 + 4*w^2 + 12*w - 1],\ [1657, 1657, -3*w^3 + 6*w^2 + 11*w - 5],\ [1657, 1657, w^3 - 2*w^2 - 7*w + 11],\ [1657, 1657, -2*w^3 + 7*w^2 + 3*w - 9],\ [1657, 1657, 3*w^3 - 8*w^2 - 5*w + 5],\ [1667, 1667, w^3 - w^2 - 4*w - 7],\ [1667, 1667, -w^3 + 2*w^2 + 9*w - 9],\ [1667, 1667, -w^3 + 6*w^2 - 5*w - 1],\ [1667, 1667, 6*w^3 - 14*w^2 - 24*w + 15],\ [1669, 1669, 2*w^3 - 16*w - 9],\ [1693, 1693, -4*w^3 + 13*w^2 + 7*w - 21],\ [1693, 1693, 3*w^3 - 6*w^2 - 9*w + 7],\ [1723, 1723, 3*w^3 - 8*w^2 - 7*w + 11],\ [1723, 1723, 2*w^3 - 6*w^2 - 8*w + 13],\ [1741, 1741, -3*w^2 + 3*w + 5],\ [1741, 1741, -w^3 + 5*w^2 - 2*w + 1],\ [1747, 1747, -w^3 + 2*w^2 - w - 9],\ [1759, 1759, -w^3 + 6*w^2 - 7*w + 1],\ [1759, 1759, 5*w^3 - 14*w^2 - 13*w + 19],\ [1777, 1777, -9*w^3 + 22*w^2 + 33*w - 31],\ [1783, 1783, 4*w^3 - 14*w^2 + 11],\ [1783, 1783, 4*w^3 - 12*w^2 - 10*w + 27],\ [1787, 1787, 2*w^3 - 9*w^2 + 5*w + 11],\ [1787, 1787, -3*w^3 + 4*w^2 + 15*w + 9],\ [1789, 1789, w^2 - 5*w - 11],\ [1801, 1801, 4*w^3 - 12*w^2 - 10*w + 19],\ [1811, 1811, w^3 + 2*w^2 - 7*w - 3],\ [1861, 1861, -5*w^3 + 11*w^2 + 24*w - 19],\ [1861, 1861, w^3 - 6*w^2 + 3*w + 5],\ [1871, 1871, 4*w^3 - 10*w^2 - 16*w + 11],\ [1873, 1873, 3*w^2 - 5*w - 11],\ [1873, 1873, -4*w^3 + 10*w^2 + 12*w - 9],\ [1879, 1879, -4*w^3 + 11*w^2 + 11*w - 17],\ [1889, 1889, -w^2 - 5*w + 1],\ [1901, 1901, w^3 - 5*w^2 - 2*w + 7],\ [1907, 1907, -3*w^3 + 11*w^2 + 2*w - 5],\ [1907, 1907, -w^3 + 7*w - 1],\ [1907, 1907, -9*w^3 + 19*w^2 + 40*w - 19],\ [1907, 1907, -2*w^3 + 3*w^2 + 13*w - 9],\ [1913, 1913, w^3 - 2*w^2 - 9*w - 1],\ [1913, 1913, 4*w^3 - 13*w^2 - 5*w + 13],\ [1931, 1931, -w^3 + 7*w^2 - 6*w - 21],\ [1931, 1931, -2*w^3 + 5*w^2 + 3*w - 7],\ [1951, 1951, -11*w^3 + 28*w^2 + 41*w - 41],\ [1951, 1951, -w^3 + 7*w - 3],\ [1979, 1979, 2*w^3 - 16*w - 7],\ [1993, 1993, -2*w^2 + 8*w + 1],\ [1993, 1993, -6*w - 1],\ [1999, 1999, -2*w^3 + 5*w^2 + w - 5]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x K = QQ e = 1 hecke_eigenvalues_array = [1, 0, 0, -2, 2, 6, 8, -4, -2, 8, -4, -6, -8, 6, 8, 8, 12, -10, -2, 16, -10, -12, -6, -8, 2, 4, -2, 8, 0, 0, 8, -12, -4, -2, 18, -8, 18, -2, 12, 18, 6, -20, 2, 0, 26, 18, 8, 6, 20, -6, -18, 2, 20, -14, -4, -8, 18, -2, 6, -4, -30, 10, 8, 4, 2, 4, -10, 2, -16, -4, -4, -12, -30, -8, 2, 14, 6, -40, 20, 12, 4, -32, 14, -6, 28, -28, 30, 16, -2, 20, 20, -4, 32, 40, 10, 30, -44, -4, 2, -30, -16, 12, 34, -36, -6, -18, 10, 32, 22, 42, -4, 20, -16, 2, 2, 16, 22, 44, 42, 6, 16, 36, 0, 22, -30, 20, 10, 36, 36, 42, -40, -20, 0, 32, 16, 44, -28, -34, 26, -30, -18, -30, -42, -10, 32, -16, 44, 2, 10, 12, 48, 22, -44, -34, -36, -32, -44, 56, 8, 10, 16, -10, -36, -40, 14, -4, 16, 10, 12, -44, -32, 34, -14, 12, 8, 54, 46, 42, -56, -42, -66, -58, -60, -48, 28, -34, 0, 8, -46, 6, -22, -38, -30, 40, -40, 64, -46, 30, -26, 50, 14, 66, -20, 8, -44, 48, 16, 44, -40, -56, 6, 18, 0, 30, -2, 2, -54, -24, -66, -44, 36, 62, 4, 44, 46, 28, 40, 4, -46, -10, -14, -30, -36, -12, -44, 74, -40, 56, 56, 28, -28, 2, 62, 54, 66, -36, 22, 6, -2, -38, 58, 56, 10, -62, 72, -32, 58, 26, 38, 14, -4, -10, -18, -60, -56, -30, 20, -76, 8, -24, 12, 66, 26, -42, 78, 20, 34, -38, -10, 14, 8, -26, -10, -52, 18, 52, -30, 6, 10, 40, 60, -8, -36, 36, -56, -38, 20] 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 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]