/* This code can be loaded, or copied and pasted, into Magma. It will load the data associated to the HMF, including the field, level, and Hecke and Atkin-Lehner eigenvalue data. At the *bottom* of the file, there is code to recreate the Hilbert modular form in Magma, by creating the HMF space and cutting out the corresponding Hecke irreducible subspace. From there, you can ask for more eigenvalues or modify as desired. It is commented out, as this computation may be lengthy. */ P := PolynomialRing(Rationals()); g := P![5, -1, -7, -1, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [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 := [ideal : I in primesArray]; heckePol := x^13 + 3*x^12 - 19*x^11 - 56*x^10 + 121*x^9 + 339*x^8 - 359*x^7 - 853*x^6 + 545*x^5 + 817*x^4 - 389*x^3 - 122*x^2 + 40*x + 4; K := NumberField(heckePol); heckeEigenvaluesArray := [e, -733/6436*e^12 - 453/1609*e^11 + 14471/6436*e^10 + 33759/6436*e^9 - 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445253/3218*e^10 - 2140837/9654*e^9 + 4577750/4827*e^8 + 10633693/9654*e^7 - 4569174/1609*e^6 - 18126881/9654*e^5 + 16430347/4827*e^4 + 6444673/9654*e^3 - 1221909/1609*e^2 - 109766/4827*e + 125048/4827, -418759/19308*e^12 - 126988/4827*e^11 + 2994151/6436*e^10 + 7739117/19308*e^9 - 33202987/9654*e^8 - 28505603/19308*e^7 + 35028239/3218*e^6 + 2243941/19308*e^5 - 125130059/9654*e^4 + 83826043/19308*e^3 + 4954577/3218*e^2 - 2419615/4827*e - 116309/4827, 133577/9654*e^12 + 82579/4827*e^11 - 945137/3218*e^10 - 2449957/9654*e^9 + 10367120/4827*e^8 + 8248423/9654*e^7 - 10887960/1609*e^6 + 3378871/9654*e^5 + 38884642/4827*e^4 - 32853905/9654*e^3 - 1480106/1609*e^2 + 2714602/4827*e - 55216/4827, 60892/1609*e^12 + 147321/3218*e^11 - 2607669/3218*e^10 - 1109430/1609*e^9 + 19269547/3218*e^8 + 7876371/3218*e^7 - 30637683/1609*e^6 + 449689/1609*e^5 + 74208577/3218*e^4 - 26195199/3218*e^3 - 5402068/1609*e^2 + 1550644/1609*e + 164573/1609, -219865/4827*e^12 - 301573/4827*e^11 + 1544486/1609*e^10 + 4693676/4827*e^9 - 33571082/4827*e^8 - 18658220/4827*e^7 + 34991940/1609*e^6 + 10997395/4827*e^5 - 125482642/4827*e^4 + 32945986/4827*e^3 + 5924169/1609*e^2 - 3362884/4827*e - 378119/4827, 86573/6436*e^12 + 46599/3218*e^11 - 1860129/6436*e^10 - 1306883/6436*e^9 + 3448522/1609*e^8 + 3421215/6436*e^7 - 21918895/3218*e^6 + 7780193/6436*e^5 + 12958212/1609*e^4 - 27058003/6436*e^3 - 2266413/3218*e^2 + 798611/1609*e + 35645/1609, 34039/6436*e^12 + 42413/3218*e^11 - 655123/6436*e^10 - 1508413/6436*e^9 + 1048042/1609*e^8 + 8345725/6436*e^7 - 5891801/3218*e^6 - 18155125/6436*e^5 + 3499589/1609*e^4 + 13881987/6436*e^3 - 2188717/3218*e^2 - 464893/1609*e + 2702/1609, -504731/9654*e^12 - 705281/9654*e^11 + 1771745/1609*e^10 + 11098303/9654*e^9 - 76918519/9654*e^8 - 22784594/4827*e^7 + 39977160/1609*e^6 + 34373393/9654*e^5 - 285512627/9654*e^4 + 32487100/4827*e^3 + 6656086/1609*e^2 - 3360001/4827*e - 518468/4827, -114377/4827*e^12 - 173729/4827*e^11 + 788844/1609*e^10 + 2742424/4827*e^9 - 16740847/4827*e^8 - 11411623/4827*e^7 + 17053119/1609*e^6 + 9947615/4827*e^5 - 59322947/4827*e^4 + 12935780/4827*e^3 + 2088812/1609*e^2 - 1354862/4827*e + 104450/4827, 3624/1609*e^12 + 8175/1609*e^11 - 65812/1609*e^10 - 128249/1609*e^9 + 379405/1609*e^8 + 532464/1609*e^7 - 901335/1609*e^6 - 560539/1609*e^5 + 606980/1609*e^4 - 314124/1609*e^3 + 528687/1609*e^2 + 82516/1609*e - 87818/1609, 41632/1609*e^12 + 55212/1609*e^11 - 884933/1609*e^10 - 863982/1609*e^9 + 6475924/1609*e^8 + 3477344/1609*e^7 - 20390520/1609*e^6 - 2127010/1609*e^5 + 24536677/1609*e^4 - 6208884/1609*e^3 - 3737657/1609*e^2 + 863980/1609*e + 124325/1609, 738467/19308*e^12 + 232214/4827*e^11 - 5237187/6436*e^10 - 14037553/19308*e^9 + 57647507/9654*e^8 + 50615203/19308*e^7 - 60882827/3218*e^6 - 908165/19308*e^5 + 221341357/9654*e^4 - 152355383/19308*e^3 - 10999817/3218*e^2 + 4619849/4827*e + 574957/4827, 3569/1609*e^12 + 16291/3218*e^11 - 142129/3218*e^10 - 144811/1609*e^9 + 967147/3218*e^8 + 1601953/3218*e^7 - 1501456/1609*e^6 - 1730497/1609*e^5 + 4306369/3218*e^4 + 2467029/3218*e^3 - 1153023/1609*e^2 + 55969/1609*e + 42772/1609, -292205/19308*e^12 - 87836/4827*e^11 + 2070821/6436*e^10 + 5137879/19308*e^9 - 22759841/9654*e^8 - 16362025/19308*e^7 + 23940593/3218*e^6 - 12540997/19308*e^5 - 85428085/9654*e^4 + 76741397/19308*e^3 + 3164391/3218*e^2 - 2556080/4827*e - 207310/4827, 28087/6436*e^12 + 29349/3218*e^11 - 579939/6436*e^10 - 1070473/6436*e^9 + 1026915/1609*e^8 + 6172677/6436*e^7 - 6404265/3218*e^6 - 13950525/6436*e^5 + 4238057/1609*e^4 + 10916279/6436*e^3 - 3310457/3218*e^2 - 364012/1609*e + 106391/1609, -105808/4827*e^12 - 160927/4827*e^11 + 733656/1609*e^10 + 2580404/4827*e^9 - 15665987/4827*e^8 - 11193851/4827*e^7 + 16025351/1609*e^6 + 11810545/4827*e^5 - 56286106/4827*e^4 + 8556652/4827*e^3 + 2417334/1609*e^2 - 706186/4827*e - 71048/4827, 239179/4827*e^12 + 643787/9654*e^11 - 3373383/3218*e^10 - 5005712/4827*e^9 + 73677223/9654*e^8 + 39631423/9654*e^7 - 38552278/1609*e^6 - 10868359/4827*e^5 + 277543019/9654*e^4 - 75038387/9654*e^3 - 6689018/1609*e^2 + 5116138/4827*e + 382742/4827, 599321/19308*e^12 + 396877/9654*e^11 - 4239511/6436*e^10 - 12308623/19308*e^9 + 23250559/4827*e^8 + 48214651/19308*e^7 - 48942149/3218*e^6 - 22894511/19308*e^5 + 89147924/4827*e^4 - 100198019/19308*e^3 - 9677835/3218*e^2 + 3806417/4827*e + 536467/4827, 2567/9654*e^12 + 17657/9654*e^11 - 4328/1609*e^10 - 330517/9654*e^9 - 67871/9654*e^8 + 1015607/4827*e^7 + 157636/1609*e^6 - 5455007/9654*e^5 - 2300797/9654*e^4 + 3005783/4827*e^3 + 319955/1609*e^2 - 713195/4827*e - 159487/4827, 92579/19308*e^12 - 32723/9654*e^11 - 758697/6436*e^10 + 1950431/19308*e^9 + 4853206/4827*e^8 - 19440215/19308*e^7 - 11326021/3218*e^6 + 77730811/19308*e^5 + 20789384/4827*e^4 - 109727945/19308*e^3 - 191597/3218*e^2 + 3655430/4827*e - 79766/4827, 168745/4827*e^12 + 442553/9654*e^11 - 2383863/3218*e^10 - 3405629/4827*e^9 + 52153291/9654*e^8 + 26110783/9654*e^7 - 27294301/1609*e^6 - 4880440/4827*e^5 + 195383033/9654*e^4 - 58838777/9654*e^3 - 4288696/1609*e^2 + 3586759/4827*e + 190226/4827, -495755/19308*e^12 - 156050/4827*e^11 + 3521907/6436*e^10 + 9547669/19308*e^9 - 38752061/9654*e^8 - 35719015/19308*e^7 + 40619459/3218*e^6 + 6952505/19308*e^5 - 144239881/9654*e^4 + 96008051/19308*e^3 + 5520383/3218*e^2 - 3229046/4827*e - 26602/4827, -205295/19308*e^12 - 192955/9654*e^11 + 1395805/6436*e^10 + 6602509/19308*e^9 - 7286974/4827*e^8 - 33656293/19308*e^7 + 14808667/3218*e^6 + 60949733/19308*e^5 - 27412316/4827*e^4 - 27790615/19308*e^3 + 4772243/3218*e^2 + 384616/4827*e - 376459/4827, -88927/6436*e^12 - 60585/3218*e^11 + 1875063/6436*e^10 + 1873001/6436*e^9 - 3406398/1609*e^8 - 7299657/6436*e^7 + 21514841/3218*e^6 + 3608761/6436*e^5 - 13257399/1609*e^4 + 14378377/6436*e^3 + 5090117/3218*e^2 - 464647/1609*e - 155619/1609, 173221/19308*e^12 + 84256/4827*e^11 - 1158053/6436*e^10 - 5695235/19308*e^9 + 11813437/9654*e^8 + 28379813/19308*e^7 - 11722207/3218*e^6 - 50113435/19308*e^5 + 41698409/9654*e^4 + 23592923/19308*e^3 - 2778399/3218*e^2 - 1304738/4827*e + 98810/4827, -113085/3218*e^12 - 77610/1609*e^11 + 2382115/3218*e^10 + 2415885/3218*e^9 - 8622050/1609*e^8 - 9601967/3218*e^7 + 26912868/1609*e^6 + 5617727/3218*e^5 - 32023270/1609*e^4 + 17236937/3218*e^3 + 4326729/1609*e^2 - 1095028/1609*e - 100068/1609, 58100/1609*e^12 + 77970/1609*e^11 - 1228558/1609*e^10 - 1211032/1609*e^9 + 8930961/1609*e^8 + 4773869/1609*e^7 - 27927850/1609*e^6 - 2494871/1609*e^5 + 33122051/1609*e^4 - 9268103/1609*e^3 - 4167943/1609*e^2 + 1115354/1609*e + 51940/1609, 112025/9654*e^12 + 69427/4827*e^11 - 791311/3218*e^10 - 2069539/9654*e^9 + 8634419/4827*e^8 + 7078309/9654*e^7 - 8942756/1609*e^6 + 2250235/9654*e^5 + 30728035/4827*e^4 - 26091545/9654*e^3 - 529652/1609*e^2 + 1146823/4827*e - 297034/4827, 228593/4827*e^12 + 583459/9654*e^11 - 3234277/3218*e^10 - 4416742/4827*e^9 + 70979705/9654*e^8 + 32073725/9654*e^7 - 37373292/1609*e^6 - 1090745/4827*e^5 + 270675235/9654*e^4 - 93011569/9654*e^3 - 6531206/1609*e^2 + 5909927/4827*e + 790936/4827, -89177/6436*e^12 - 46683/3218*e^11 + 1917157/6436*e^10 + 1284887/6436*e^9 - 3555009/1609*e^8 - 3032431/6436*e^7 + 22565161/3218*e^6 - 9622877/6436*e^5 - 13239714/1609*e^4 + 29627335/6436*e^3 + 1877421/3218*e^2 - 776901/1609*e - 19947/1609, 46063/1609*e^12 + 119975/3218*e^11 - 1950087/3218*e^10 - 916294/1609*e^9 + 14196687/3218*e^8 + 6845429/3218*e^7 - 22230635/1609*e^6 - 759299/1609*e^5 + 52702939/3218*e^4 - 17659511/3218*e^3 - 3187966/1609*e^2 + 1117891/1609*e + 55730/1609, 26429/6436*e^12 + 16127/1609*e^11 - 514627/6436*e^10 - 1154503/6436*e^9 + 1676941/3218*e^8 + 6472381/6436*e^7 - 4808355/3218*e^6 - 14425079/6436*e^5 + 5803611/3218*e^4 + 11644507/6436*e^3 - 1815905/3218*e^2 - 466121/1609*e + 3027/1609, 355319/19308*e^12 + 228625/9654*e^11 - 2508981/6436*e^10 - 6947785/19308*e^9 + 13711204/4827*e^8 + 25496449/19308*e^7 - 28641487/3218*e^6 - 3015005/19308*e^5 + 50863178/4827*e^4 - 71247305/19308*e^3 - 3974187/3218*e^2 + 2022332/4827*e + 57001/4827, 207431/6436*e^12 + 139875/3218*e^11 - 4408203/6436*e^10 - 4400853/6436*e^9 + 8077263/1609*e^8 + 18015669/6436*e^7 - 51219547/3218*e^6 - 12843985/6436*e^5 + 31520174/1609*e^4 - 28069085/6436*e^3 - 11846111/3218*e^2 + 956076/1609*e + 200621/1609, -613063/19308*e^12 - 463445/9654*e^11 + 4264913/6436*e^10 + 14918561/19308*e^9 - 22882904/4827*e^8 - 65413925/19308*e^7 + 47153621/3218*e^6 + 72441289/19308*e^5 - 84139027/4827*e^4 + 42749005/19308*e^3 + 8630157/3218*e^2 - 836965/4827*e - 265196/4827, -58382/4827*e^12 - 103087/9654*e^11 + 849891/3218*e^10 + 657193/4827*e^9 - 19261079/9654*e^8 - 1612553/9654*e^7 + 10348047/1609*e^6 - 9637360/4827*e^5 - 74356639/9654*e^4 + 46116223/9654*e^3 + 1219458/1609*e^2 - 2323325/4827*e - 216124/4827]; heckeEigenvalues := AssociativeArray(); for i := 1 to #heckeEigenvaluesArray do heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i]; end for; ALEigenvalues := AssociativeArray(); ALEigenvalues[ideal] := -1; // EXAMPLE: // pp := Factorization(2*ZF)[1][1]; // heckeEigenvalues[pp]; print "To reconstruct the Hilbert newform f, type f, iso := Explode(make_newform());"; function make_newform(); M := HilbertCuspForms(F, NN); S := NewSubspace(M); // SetVerbose("ModFrmHil", 1); NFD := NewformDecomposition(S); newforms := [* Eigenform(U) : U in NFD *]; if #newforms eq 0 then; print "No Hilbert newforms at this level"; return 0; end if; print "Testing ", #newforms, " possible newforms"; newforms := [* f: f in newforms | IsIsomorphic(BaseField(f), K) *]; print #newforms, " newforms have the correct Hecke field"; if #newforms eq 0 then; print "No Hilbert newform found with the correct Hecke field"; return 0; end if; autos := Automorphisms(K); xnewforms := [* *]; for f in newforms do; if K eq RationalField() then; Append(~xnewforms, [* f, autos[1] *]); else; flag, iso := IsIsomorphic(K,BaseField(f)); for a in autos do; Append(~xnewforms, [* f, a*iso *]); end for; end if; end for; newforms := xnewforms; for P in primes do; xnewforms := [* *]; for f_iso in newforms do; f, iso := Explode(f_iso); if HeckeEigenvalue(f,P) eq iso(heckeEigenvalues[P]) then; Append(~xnewforms, f_iso); end if; end for; newforms := xnewforms; if #newforms eq 0 then; print "No Hilbert newform found which matches the Hecke eigenvalues"; return 0; else if #newforms eq 1 then; print "success: unique match"; return newforms[1]; end if; end if; end for; print #newforms, "Hilbert newforms found which match the Hecke eigenvalues"; return newforms[1]; end function;