/* 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![3, 6, -8, -1, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [3, 3, w], [7, 7, 1/3*w^3 - 2/3*w^2 - 2*w + 2], [9, 3, w + 1], [13, 13, 2/3*w^3 - 1/3*w^2 - 5*w], [16, 2, 2], [17, 17, 1/3*w^3 - 2/3*w^2 - 3*w + 5], [19, 19, -1/3*w^3 + 2/3*w^2 + 2*w - 5], [23, 23, -1/3*w^3 + 2/3*w^2 + 3*w - 2], [25, 5, 1/3*w^3 + 1/3*w^2 - 3*w], [25, 5, -2/3*w^3 + 1/3*w^2 + 5*w - 3], [29, 29, -2/3*w^3 + 1/3*w^2 + 4*w - 3], [29, 29, -1/3*w^3 + 2/3*w^2 + 2*w], [37, 37, 1/3*w^3 + 1/3*w^2 - 2*w - 3], [41, 41, -2/3*w^3 + 1/3*w^2 + 5*w - 4], [43, 43, 2/3*w^3 - 1/3*w^2 - 6*w], [47, 47, 2/3*w^3 - 1/3*w^2 - 4*w], [59, 59, 1/3*w^3 + 1/3*w^2 - 2*w - 4], [59, 59, 4/3*w^3 - 5/3*w^2 - 10*w + 9], [67, 67, -1/3*w^3 + 2/3*w^2 + 3*w], [71, 71, 2/3*w^3 - 1/3*w^2 - 4*w + 1], [73, 73, -1/3*w^3 + 2/3*w^2 + w - 3], [83, 83, -w^3 + w^2 + 8*w - 4], [83, 83, -2/3*w^3 + 1/3*w^2 + 6*w - 1], [97, 97, 5/3*w^3 - 4/3*w^2 - 13*w + 6], [101, 101, -1/3*w^3 + 2/3*w^2 + 4*w - 2], [127, 127, w^2 - w - 4], [131, 131, 7/3*w^3 - 8/3*w^2 - 18*w + 15], [131, 131, 4/3*w^3 - 8/3*w^2 - 10*w + 17], [137, 137, 1/3*w^3 + 1/3*w^2 - 4*w - 3], [139, 139, w^2 - 5], [151, 151, 1/3*w^3 - 2/3*w^2 - 4*w + 6], [163, 163, -2/3*w^3 + 4/3*w^2 + 6*w - 9], [163, 163, -2/3*w^3 + 1/3*w^2 + 3*w - 4], [179, 179, w^2 - 2*w - 4], [179, 179, w^2 - w - 7], [191, 191, 1/3*w^3 + 1/3*w^2 - w - 3], [191, 191, -2/3*w^3 + 1/3*w^2 + 4*w - 6], [193, 193, -w^3 + 9*w + 1], [193, 193, 4/3*w^3 - 2/3*w^2 - 9*w], [197, 197, 2/3*w^3 - 1/3*w^2 - 5*w - 3], [197, 197, w^3 - 8*w + 1], [197, 197, w^3 - 7*w + 2], [197, 197, -4/3*w^3 + 2/3*w^2 + 11*w - 3], [223, 223, 4/3*w^3 - 5/3*w^2 - 10*w + 8], [233, 233, -4/3*w^3 + 2/3*w^2 + 9*w - 5], [241, 241, 1/3*w^3 + 1/3*w^2 - 4*w - 4], [257, 257, -w^3 - w^2 + 6*w + 5], [269, 269, -1/3*w^3 - 1/3*w^2 + 5*w], [269, 269, -2/3*w^3 + 4/3*w^2 + 4*w - 3], [271, 271, -2/3*w^3 + 4/3*w^2 + 3*w - 6], [271, 271, w^3 - 8*w - 5], [277, 277, 2*w^3 - 2*w^2 - 16*w + 13], [281, 281, 2/3*w^3 + 2/3*w^2 - 5*w], [281, 281, -w^3 + w^2 + 7*w - 2], [331, 331, 2/3*w^3 - 4/3*w^2 - 5*w + 4], [331, 331, 8/3*w^3 - 7/3*w^2 - 20*w + 15], [343, 7, -4/3*w^3 + 2/3*w^2 + 11*w - 6], [347, 347, 2/3*w^3 + 2/3*w^2 - 4*w - 5], [349, 349, -1/3*w^3 + 2/3*w^2 + 3*w - 8], [349, 349, -1/3*w^3 + 2/3*w^2 - 3], [353, 353, -5/3*w^3 + 1/3*w^2 + 12*w], [353, 353, 1/3*w^3 - 5/3*w^2 - 2*w + 14], [367, 367, -1/3*w^3 + 5/3*w^2 + 4*w - 9], [367, 367, 1/3*w^3 + 1/3*w^2 + w - 3], [379, 379, 4/3*w^3 + 1/3*w^2 - 12*w], [379, 379, -5/3*w^3 + 7/3*w^2 + 13*w - 12], [383, 383, -5/3*w^3 + 10/3*w^2 + 12*w - 21], [397, 397, 1/3*w^3 + 4/3*w^2 - 5*w - 6], [397, 397, 4/3*w^3 + 1/3*w^2 - 8*w], [397, 397, -1/3*w^3 + 2/3*w^2 + w - 6], [397, 397, w^3 + w^2 - 8*w - 7], [401, 401, 5/3*w^3 - 1/3*w^2 - 14*w + 1], [419, 419, 1/3*w^3 + 4/3*w^2 - 4*w - 3], [419, 419, 1/3*w^3 - 5/3*w^2 - 3*w + 12], [433, 433, -w^3 + w^2 + 9*w - 4], [433, 433, 1/3*w^3 + 4/3*w^2 - 4*w - 9], [443, 443, 4/3*w^3 - 2/3*w^2 - 8*w - 1], [443, 443, -4/3*w^3 + 8/3*w^2 + 9*w - 17], [449, 449, -w^3 + w^2 + 8*w - 2], [449, 449, 5/3*w^3 - 7/3*w^2 - 13*w + 18], [457, 457, -w^3 + 3*w^2 + 7*w - 20], [457, 457, 2/3*w^3 + 2/3*w^2 - 5*w - 9], [461, 461, 4/3*w^3 - 2/3*w^2 - 9*w + 2], [461, 461, -2/3*w^3 + 7/3*w^2 + 4*w - 16], [463, 463, 4/3*w^3 + 1/3*w^2 - 11*w - 1], [467, 467, w^3 - 2*w^2 - 9*w + 8], [479, 479, 4/3*w^3 - 8/3*w^2 - 10*w + 15], [479, 479, 5/3*w^3 - 1/3*w^2 - 14*w], [491, 491, -1/3*w^3 + 2/3*w^2 + 2*w - 8], [509, 509, -11/3*w^3 + 13/3*w^2 + 27*w - 27], [523, 523, -w - 5], [523, 523, 4/3*w^3 - 2/3*w^2 - 10*w + 9], [541, 541, 8/3*w^3 - 7/3*w^2 - 21*w + 13], [547, 547, -2/3*w^3 + 1/3*w^2 + 5*w - 7], [557, 557, -2/3*w^3 + 1/3*w^2 + 7*w - 3], [563, 563, w^2 + w - 8], [563, 563, 1/3*w^3 + 1/3*w^2 - w - 6], [569, 569, 1/3*w^3 + 4/3*w^2 - 4*w - 6], [569, 569, -1/3*w^3 + 5/3*w^2 + 2*w - 6], [571, 571, 5/3*w^3 - 4/3*w^2 - 11*w + 1], [593, 593, -2/3*w^3 + 1/3*w^2 + 4*w - 7], [593, 593, -2*w^3 + w^2 + 15*w - 1], [599, 599, -1/3*w^3 + 2/3*w^2 + 5*w - 2], [601, 601, w^3 - 10*w - 2], [607, 607, 1/3*w^3 + 4/3*w^2 - 4*w - 4], [607, 607, 7/3*w^3 - 8/3*w^2 - 18*w + 14], [617, 617, -5/3*w^3 + 10/3*w^2 + 13*w - 24], [617, 617, 1/3*w^3 + 4/3*w^2 - 2*w - 6], [617, 617, w^2 + w - 10], [617, 617, 1/3*w^3 + 1/3*w^2 - 6*w - 1], [631, 631, 7/3*w^3 - 2/3*w^2 - 18*w + 3], [631, 631, -1/3*w^3 + 2/3*w^2 + 5*w - 3], [641, 641, 5/3*w^3 - 1/3*w^2 - 11*w - 2], [643, 643, -5/3*w^3 + 7/3*w^2 + 14*w - 13], [643, 643, -2/3*w^3 + 4/3*w^2 + 6*w - 3], [653, 653, w^3 - w^2 - 8*w + 1], [653, 653, 1/3*w^3 + 4/3*w^2 - 3*w - 13], [659, 659, -1/3*w^3 + 5/3*w^2 + 2*w - 8], [661, 661, 5/3*w^3 - 4/3*w^2 - 11*w + 9], [683, 683, 4/3*w^3 - 2/3*w^2 - 7*w + 8], [709, 709, -2*w^3 + 17*w + 2], [709, 709, 1/3*w^3 + 1/3*w^2 - 4], [733, 733, 1/3*w^3 - 2/3*w^2 - 2*w - 3], [733, 733, 5/3*w^3 - 4/3*w^2 - 14*w + 9], [733, 733, 2/3*w^3 - 7/3*w^2 - 2*w + 10], [733, 733, 4/3*w^3 - 8/3*w^2 - 9*w + 18], [743, 743, w^3 - w^2 - 8*w - 1], [751, 751, -4/3*w^3 - 1/3*w^2 + 10*w], [757, 757, 5/3*w^3 - 7/3*w^2 - 11*w + 15], [757, 757, 2/3*w^3 + 5/3*w^2 - 3*w - 5], [761, 761, 7/3*w^3 - 5/3*w^2 - 18*w + 11], [761, 761, 1/3*w^3 + 4/3*w^2 - 4*w - 13], [773, 773, 1/3*w^3 + 4/3*w^2 - 3*w - 7], [773, 773, 1/3*w^3 + 1/3*w^2 - 3*w - 7], [809, 809, w^3 - 8*w + 4], [811, 811, -2/3*w^3 + 4/3*w^2 + 5*w], [823, 823, 7/3*w^3 - 5/3*w^2 - 17*w + 9], [827, 827, -5/3*w^3 + 1/3*w^2 + 12*w - 4], [827, 827, 1/3*w^3 + 4/3*w^2 - 4*w - 10], [829, 829, -1/3*w^3 - 1/3*w^2 - w - 3], [841, 29, -1/3*w^3 - 4/3*w^2 + 2*w + 12], [857, 857, 4*w^3 - 4*w^2 - 31*w + 26], [859, 859, 11/3*w^3 - 10/3*w^2 - 27*w + 21], [863, 863, 7/3*w^3 - 5/3*w^2 - 17*w + 8], [877, 877, 1/3*w^3 + 4/3*w^2 - 3*w - 6], [877, 877, -4/3*w^3 + 5/3*w^2 + 11*w - 6], [881, 881, 5/3*w^3 - 4/3*w^2 - 11*w + 4], [881, 881, 4/3*w^3 - 11/3*w^2 - 10*w + 24], [883, 883, 10/3*w^3 - 17/3*w^2 - 25*w + 36], [883, 883, -4/3*w^3 + 2/3*w^2 + 11*w - 8], [887, 887, -1/3*w^3 + 2/3*w^2 + 3*w - 9], [887, 887, 3*w^3 - 2*w^2 - 23*w + 8], [911, 911, 4/3*w^3 + 1/3*w^2 - 7*w], [919, 919, -4/3*w^3 + 5/3*w^2 + 7*w - 6], [929, 929, 2*w^3 + w^2 - 14*w - 8], [937, 937, -7/3*w^3 + 5/3*w^2 + 15*w - 14], [941, 941, -1/3*w^3 - 1/3*w^2 + 6*w], [941, 941, -7/3*w^3 - 1/3*w^2 + 18*w + 3], [971, 971, -4/3*w^3 + 5/3*w^2 + 10*w - 6], [977, 977, -7/3*w^3 + 11/3*w^2 + 18*w - 27], [977, 977, 7/3*w^3 - 2/3*w^2 - 17*w], [997, 997, 7/3*w^3 - 2/3*w^2 - 17*w + 2], [1009, 1009, 4/3*w^3 + 1/3*w^2 - 13*w + 3], [1009, 1009, 2/3*w^3 - 7/3*w^2 - 6*w + 15], [1013, 1013, 4/3*w^3 + 4/3*w^2 - 12*w - 7], [1013, 1013, 4/3*w^3 + 1/3*w^2 - 9*w + 2], [1019, 1019, -1/3*w^3 + 2/3*w^2 + 6*w], [1021, 1021, w^2 - 11], [1021, 1021, 10/3*w^3 - 8/3*w^2 - 25*w + 17], [1021, 1021, -7/3*w^3 + 2/3*w^2 + 19*w + 4], [1021, 1021, -5/3*w^3 + 7/3*w^2 + 10*w - 13], [1031, 1031, -w^3 + 2*w^2 + 9*w - 14], [1031, 1031, -2/3*w^3 + 4/3*w^2 + 4*w - 13], [1031, 1031, -w^3 + 9*w - 5], [1031, 1031, w^2 + 2*w - 7], [1033, 1033, 4/3*w^3 - 11/3*w^2 - 9*w + 26], [1033, 1033, -4/3*w^3 + 2/3*w^2 + 7*w - 9], [1061, 1061, -1/3*w^3 + 2/3*w^2 - 5], [1061, 1061, -5/3*w^3 + 4/3*w^2 + 14*w - 7], [1069, 1069, 4/3*w^3 + 4/3*w^2 - 11*w - 3], [1069, 1069, -7/3*w^3 + 14/3*w^2 + 17*w - 30], [1087, 1087, 2*w^3 - w^2 - 14*w + 2], [1091, 1091, -w^3 + 10*w - 7], [1091, 1091, -4/3*w^3 + 5/3*w^2 + 8*w - 2], [1093, 1093, 2/3*w^3 + 2/3*w^2 - 7*w + 3], [1103, 1103, 1/3*w^3 - 2/3*w^2 - w - 4], [1109, 1109, -1/3*w^3 + 2/3*w^2 + 5*w - 9], [1109, 1109, w^3 + w^2 - 7*w - 11], [1117, 1117, -w^3 + 10*w - 1], [1129, 1129, -4/3*w^3 + 8/3*w^2 + 10*w - 11], [1163, 1163, -4/3*w^3 + 2/3*w^2 + 12*w - 5], [1163, 1163, -1/3*w^3 + 5/3*w^2 + 2*w - 15], [1181, 1181, -5/3*w^3 + 4/3*w^2 + 11*w - 7], [1187, 1187, 10/3*w^3 - 8/3*w^2 - 26*w + 17], [1193, 1193, -2/3*w^3 + 7/3*w^2 + 5*w - 13], [1193, 1193, 5/3*w^3 - 1/3*w^2 - 11*w], [1193, 1193, 2/3*w^3 - 1/3*w^2 - 6*w - 5], [1193, 1193, 4/3*w^3 + 1/3*w^2 - 10*w - 9], [1201, 1201, 2/3*w^3 + 5/3*w^2 - 6*w - 11], [1229, 1229, -2/3*w^3 + 7/3*w^2 + 7*w - 13], [1231, 1231, 5/3*w^3 - 4/3*w^2 - 11*w - 2], [1249, 1249, -1/3*w^3 - 1/3*w^2 + 4*w - 6], [1249, 1249, -5/3*w^3 + 4/3*w^2 + 11*w - 6], [1259, 1259, 5/3*w^3 - 7/3*w^2 - 13*w + 19], [1277, 1277, 4/3*w^3 + 1/3*w^2 - 13*w], [1291, 1291, 5/3*w^3 - 1/3*w^2 - 9*w - 2], [1297, 1297, -1/3*w^3 + 5/3*w^2 + w - 12], [1327, 1327, w^3 + 2*w^2 - 7*w - 13], [1327, 1327, 2/3*w^3 - 1/3*w^2 - 5*w - 5], [1361, 1361, -1/3*w^3 + 5/3*w^2 + 4*w - 11], [1367, 1367, 2/3*w^3 - 1/3*w^2 - 8*w], [1367, 1367, -3*w - 5], [1399, 1399, -2/3*w^3 + 4/3*w^2 + 3*w - 9], [1399, 1399, -5/3*w^3 + 4/3*w^2 + 13*w - 3], [1433, 1433, -5/3*w^3 + 4/3*w^2 + 13*w - 15], [1433, 1433, -7/3*w^3 + 8/3*w^2 + 19*w - 15], [1439, 1439, -w^3 + 2*w^2 + 6*w - 14], [1439, 1439, -1/3*w^3 + 2/3*w^2 - w + 7], [1447, 1447, 1/3*w^3 + 1/3*w^2 - 5*w - 6], [1447, 1447, -4/3*w^3 + 2/3*w^2 + 12*w - 3], [1451, 1451, 4/3*w^3 - 8/3*w^2 - 10*w + 21], [1451, 1451, w^2 - 3*w - 5], [1451, 1451, -2/3*w^3 + 4/3*w^2 + 5*w - 13], [1451, 1451, 10/3*w^3 - 14/3*w^2 - 25*w + 33], [1453, 1453, 2/3*w^3 - 7/3*w^2 - 5*w + 12], [1453, 1453, 5/3*w^3 - 1/3*w^2 - 15*w - 2], [1453, 1453, -4/3*w^3 - 1/3*w^2 + 9*w - 3], [1453, 1453, 4/3*w^3 + 1/3*w^2 - 7*w - 4], [1471, 1471, -1/3*w^3 + 2/3*w^2 - 6], [1489, 1489, -5/3*w^3 + 1/3*w^2 + 13*w - 4], [1489, 1489, -2/3*w^3 + 1/3*w^2 + 2*w - 6], [1493, 1493, 7/3*w^3 - 11/3*w^2 - 17*w + 27], [1493, 1493, -2*w^3 + 2*w^2 + 15*w - 8], [1511, 1511, 17/3*w^3 - 19/3*w^2 - 43*w + 37], [1523, 1523, -w^3 + 3*w^2 + 9*w - 11], [1523, 1523, 1/3*w^3 + 1/3*w^2 - 5*w - 7], [1523, 1523, -w^3 + 3*w^2 + 8*w - 22], [1523, 1523, 2/3*w^3 + 5/3*w^2 - 4*w - 5], [1543, 1543, 1/3*w^3 + 1/3*w^2 - 6*w - 4], [1543, 1543, -4/3*w^3 + 5/3*w^2 + 10*w - 5], [1553, 1553, -4/3*w^3 + 5/3*w^2 + 11*w - 5], [1559, 1559, 2*w^3 - 15*w + 1], [1559, 1559, -4/3*w^3 + 2/3*w^2 + 5*w - 3], [1567, 1567, 1/3*w^3 + 1/3*w^2 - 6], [1571, 1571, 1/3*w^3 + 1/3*w^2 + 3*w - 1], [1571, 1571, -4/3*w^3 + 5/3*w^2 + 8*w - 14], [1579, 1579, -10/3*w^3 + 11/3*w^2 + 24*w - 20], [1583, 1583, 1/3*w^3 - 2/3*w^2 - 2*w - 4], [1597, 1597, 11/3*w^3 - 19/3*w^2 - 28*w + 42], [1601, 1601, 10/3*w^3 - 14/3*w^2 - 25*w + 26], [1613, 1613, 8/3*w^3 - 16/3*w^2 - 19*w + 36], [1627, 1627, -2/3*w^3 + 1/3*w^2 + 3*w - 9], [1627, 1627, -2/3*w^3 + 1/3*w^2 + 6*w - 9], [1637, 1637, 2*w^3 + w^2 - 16*w - 4], [1657, 1657, -2/3*w^3 + 7/3*w^2 + 6*w - 16], [1657, 1657, 2/3*w^3 + 2/3*w^2 - 3*w - 6], [1663, 1663, -3*w + 10], [1663, 1663, 1/3*w^3 + 4/3*w^2 - w - 9], [1667, 1667, -2/3*w^3 + 1/3*w^2 + 2*w - 7], [1667, 1667, 8/3*w^3 - 4/3*w^2 - 22*w + 3], [1669, 1669, 3*w^3 - 2*w^2 - 22*w + 11], [1693, 1693, 1/3*w^3 - 2/3*w^2 - 3*w - 4], [1697, 1697, -4/3*w^3 + 2/3*w^2 + 7*w - 3], [1709, 1709, 3*w^3 - 3*w^2 - 23*w + 23], [1741, 1741, -3*w^3 + 2*w^2 + 24*w - 10], [1747, 1747, -2/3*w^3 + 7/3*w^2 + 4*w - 10], [1747, 1747, 1/3*w^3 - 8/3*w^2 - 4*w + 9], [1753, 1753, 4/3*w^3 - 2/3*w^2 - 7*w + 2], [1753, 1753, 8/3*w^3 - 16/3*w^2 - 19*w + 37], [1759, 1759, 2*w^2 - w - 5], [1759, 1759, -5/3*w^3 + 10/3*w^2 + 14*w - 21], [1777, 1777, -1/3*w^3 + 2/3*w^2 + 6*w - 2], [1783, 1783, 7/3*w^3 - 8/3*w^2 - 17*w + 12], [1783, 1783, -5/3*w^3 - 2/3*w^2 + 10*w - 3], [1787, 1787, -1/3*w^3 - 1/3*w^2 - 2*w - 3], [1787, 1787, 4/3*w^3 + 1/3*w^2 - 12*w + 2], [1811, 1811, 5/3*w^3 - 1/3*w^2 - 10*w - 6], [1811, 1811, -w^3 + w^2 + 5*w - 10], [1847, 1847, w^2 - 2*w - 10], [1861, 1861, 2*w^2 - w - 11], [1867, 1867, w^3 + 2*w^2 - 7*w - 4], [1871, 1871, -1/3*w^3 + 5/3*w^2 - 9], [1873, 1873, 2/3*w^3 + 5/3*w^2 - 7*w - 6], [1877, 1877, -2/3*w^3 + 1/3*w^2 + 8*w - 3], [1877, 1877, -1/3*w^3 + 2/3*w^2 + 6*w - 3], [1879, 1879, w^3 + 2*w^2 - 11*w - 8], [1879, 1879, 8/3*w^3 - 7/3*w^2 - 20*w + 18], [1901, 1901, -2/3*w^3 + 1/3*w^2 + 2*w + 9], [1901, 1901, 10/3*w^3 - 11/3*w^2 - 24*w + 21], [1907, 1907, w^3 + w^2 - 10*w - 8], [1907, 1907, 5*w^3 - 6*w^2 - 38*w + 35], [1933, 1933, 1/3*w^3 + 1/3*w^2 - 7], [1933, 1933, 3*w^3 - w^2 - 22*w + 4], [1949, 1949, 5/3*w^3 + 2/3*w^2 - 15*w - 9], [1951, 1951, w^3 + w^2 - 12*w - 1], [1979, 1979, -7/3*w^3 + 5/3*w^2 + 17*w - 15], [1993, 1993, -5/3*w^3 + 1/3*w^2 + 12*w - 6], [1999, 1999, -4/3*w^3 - 1/3*w^2 + 13*w + 7], [1999, 1999, 2/3*w^3 + 5/3*w^2 - 6*w - 5]]; primes := [ideal : I in primesArray]; heckePol := x; K := Rationals(); e := 1; heckeEigenvaluesArray := [-1, -1, -4, 1, 5, 3, 1, 1, 1, 6, 9, -2, -7, -8, -5, 11, 3, -6, -3, -11, 0, -16, -6, 12, -5, -7, 12, -6, -22, 18, -20, 16, 4, -15, 14, 18, 7, 6, -8, -2, -2, -17, 2, -4, 0, 0, -2, 30, 18, -14, -2, 1, -8, 15, -23, -1, -16, -30, -16, -34, -21, 22, -7, 24, -20, 18, -36, -18, 15, -34, 10, -13, -16, 3, -27, 34, -8, 19, -4, -35, 26, -35, 13, 26, 6, -20, -5, 24, -14, -20, -13, -32, 2, 19, 22, 9, 15, 45, 12, 42, -10, -6, -26, 34, 1, -8, -6, 22, -35, 26, 8, 24, 9, -34, 36, 42, -36, 42, -38, 0, -5, -43, -21, -30, -18, -32, -7, 42, -1, -44, -14, -8, 39, 24, -48, -34, -22, -23, -22, -25, -19, -18, 0, -12, -18, -8, -26, -54, -12, -32, 2, 6, 17, 55, 51, 7, 24, 43, -23, 45, 33, 7, 28, 0, -54, -45, 38, 5, -36, 12, -18, 48, -41, -62, 52, -23, -35, 25, 18, 45, 40, 34, -57, -58, 31, 9, -18, 10, 19, -48, -44, -15, -9, 41, 16, -31, -27, 11, 32, -24, 12, 11, 18, 55, -46, -61, 29, -14, 8, -3, -48, 66, 1, 57, -2, -29, -69, 31, 56, 52, -36, -56, -30, 57, -42, 47, -33, -54, 10, -31, 38, 28, -66, 0, 24, -64, -42, 36, -34, 5, 13, -58, 0, 4, -72, -6, 46, 1, 47, -33, 68, -26, -4, 30, 74, -77, 71, -14, -3, -23, -38, -36, 30, -27, -10, 62, -64, -23, 28, -40, 56, -34, -6, -23, 28, -28, 54, 20, -48, 43, -62, -42, 57, 18, 84, 40, 40, 30, -9, -12, 73, 66, 44, -39, 46, -18, 64, 9, 75]; heckeEigenvalues := AssociativeArray(); for i := 1 to #heckeEigenvaluesArray do heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i]; end for; ALEigenvalues := AssociativeArray(); ALEigenvalues[ideal] := 1; 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;