/* 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![4, 6, -7, -1, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [3, 3, -w + 1], [4, 2, w], [4, 2, -1/2*w^3 + 1/2*w^2 + 7/2*w - 3], [7, 7, -1/2*w^3 + 1/2*w^2 + 5/2*w - 2], [13, 13, 1/2*w^3 + 1/2*w^2 - 7/2*w - 1], [17, 17, 1/2*w^3 - 1/2*w^2 - 5/2*w], [27, 3, -w^3 + 7*w + 1], [31, 31, w + 3], [31, 31, -w^2 + 5], [37, 37, w^2 - 3], [41, 41, 1/2*w^3 + 1/2*w^2 - 3/2*w - 2], [47, 47, w^3 - 5*w - 3], [53, 53, 1/2*w^3 - 1/2*w^2 - 7/2*w], [61, 61, w^3 - w^2 - 5*w + 3], [83, 83, w^3 + w^2 - 6*w - 7], [83, 83, -w^3 + 5*w + 1], [83, 83, 2*w - 1], [83, 83, -3/2*w^3 - 1/2*w^2 + 19/2*w + 2], [89, 89, 1/2*w^3 - 1/2*w^2 - 7/2*w - 2], [89, 89, w^3 - 5*w + 5], [97, 97, -1/2*w^3 + 1/2*w^2 + 7/2*w - 6], [97, 97, -1/2*w^3 + 1/2*w^2 + 9/2*w - 1], [103, 103, 3/2*w^3 + 3/2*w^2 - 19/2*w - 9], [109, 109, 1/2*w^3 - 1/2*w^2 - 9/2*w + 4], [127, 127, 1/2*w^3 + 1/2*w^2 - 3/2*w - 3], [127, 127, 1/2*w^3 - 1/2*w^2 - 5/2*w + 6], [139, 139, 1/2*w^3 + 3/2*w^2 - 9/2*w - 3], [139, 139, 3/2*w^3 - 1/2*w^2 - 21/2*w + 1], [149, 149, -1/2*w^3 + 1/2*w^2 + 9/2*w - 2], [151, 151, -1/2*w^3 - 1/2*w^2 + 9/2*w - 3], [157, 157, 3/2*w^3 - 1/2*w^2 - 21/2*w + 4], [163, 163, 1/2*w^3 + 3/2*w^2 - 5/2*w - 3], [167, 167, -1/2*w^3 + 3/2*w^2 + 9/2*w - 9], [179, 179, w^3 - w^2 - 5*w + 1], [179, 179, -1/2*w^3 + 1/2*w^2 + 3/2*w - 4], [181, 181, 5/2*w^3 + 3/2*w^2 - 31/2*w - 10], [191, 191, -1/2*w^3 + 5/2*w^2 + 7/2*w - 14], [191, 191, 2*w^3 - 2*w^2 - 12*w + 11], [193, 193, w^2 + 1], [193, 193, 1/2*w^3 + 3/2*w^2 - 9/2*w - 6], [197, 197, 1/2*w^3 + 1/2*w^2 - 11/2*w + 1], [197, 197, -3/2*w^3 + 3/2*w^2 + 17/2*w - 8], [199, 199, 3/2*w^3 - 3/2*w^2 - 19/2*w + 7], [227, 227, 1/2*w^3 + 1/2*w^2 - 7/2*w - 6], [227, 227, -2*w^3 - w^2 + 14*w + 9], [229, 229, w^3 - w^2 - 4*w + 3], [229, 229, 3/2*w^3 - 1/2*w^2 - 17/2*w + 1], [233, 233, -w^3 - w^2 + 7*w + 9], [233, 233, 1/2*w^3 + 3/2*w^2 - 5/2*w - 9], [251, 251, -5/2*w^3 - 1/2*w^2 + 31/2*w + 2], [257, 257, -3/2*w^3 - 1/2*w^2 + 19/2*w + 1], [257, 257, 1/2*w^3 + 1/2*w^2 - 11/2*w - 1], [269, 269, 2*w^3 - 2*w^2 - 13*w + 15], [269, 269, 1/2*w^3 + 3/2*w^2 - 9/2*w - 4], [271, 271, 1/2*w^3 + 3/2*w^2 - 5/2*w - 6], [277, 277, -1/2*w^3 + 5/2*w^2 + 5/2*w - 16], [281, 281, -2*w^3 + 12*w + 1], [281, 281, -3/2*w^3 - 1/2*w^2 + 15/2*w - 3], [283, 283, 5/2*w^3 + 3/2*w^2 - 29/2*w - 8], [283, 283, 3/2*w^3 + 1/2*w^2 - 15/2*w], [293, 293, -w^3 - w^2 + 6*w + 1], [307, 307, -3/2*w^3 + 1/2*w^2 + 13/2*w + 4], [307, 307, -3/2*w^3 - 1/2*w^2 + 13/2*w - 2], [311, 311, 2*w^2 + w - 11], [311, 311, 1/2*w^3 + 3/2*w^2 - 5/2*w - 5], [313, 313, 3/2*w^3 - 1/2*w^2 - 17/2*w + 2], [317, 317, w^3 + 2*w^2 - 7*w - 3], [331, 331, 2*w^2 + 2*w - 9], [343, 7, -w^3 + 2*w^2 + 5*w - 11], [349, 349, 5/2*w^3 + 1/2*w^2 - 35/2*w - 6], [353, 353, 1/2*w^3 + 3/2*w^2 - 7/2*w - 4], [359, 359, w - 5], [361, 19, -5/2*w^3 + 1/2*w^2 + 33/2*w], [361, 19, w^3 - w^2 - 8*w + 3], [367, 367, -3/2*w^3 + 3/2*w^2 + 15/2*w - 8], [383, 383, -1/2*w^3 + 1/2*w^2 + 1/2*w - 3], [397, 397, -3/2*w^3 + 5/2*w^2 + 17/2*w - 16], [397, 397, 1/2*w^3 - 1/2*w^2 - 5/2*w - 3], [401, 401, -w^3 + 3*w - 3], [401, 401, 3/2*w^3 + 3/2*w^2 - 21/2*w - 4], [409, 409, 3/2*w^3 - 1/2*w^2 - 19/2*w - 3], [419, 419, -5/2*w^3 - 1/2*w^2 + 29/2*w + 1], [421, 421, -1/2*w^3 + 3/2*w^2 + 9/2*w - 11], [421, 421, -3*w + 1], [431, 431, 1/2*w^3 - 1/2*w^2 + 1/2*w + 2], [431, 431, w^3 - 7*w - 5], [433, 433, 1/2*w^3 + 1/2*w^2 - 3/2*w - 6], [433, 433, w^3 + w^2 - 6*w + 3], [467, 467, 1/2*w^3 + 3/2*w^2 - 11/2*w - 6], [487, 487, -2*w^3 + w^2 + 12*w - 9], [503, 503, 3/2*w^3 - 3/2*w^2 - 19/2*w + 6], [503, 503, w^2 + 2*w - 7], [523, 523, w^3 - w^2 - 6*w + 1], [523, 523, -w^3 + 3*w^2 + 7*w - 19], [541, 541, -3/2*w^3 + 3/2*w^2 + 15/2*w - 7], [541, 541, w^3 - w^2 - 8*w + 7], [547, 547, -1/2*w^3 + 3/2*w^2 + 3/2*w - 8], [557, 557, 2*w^2 - w - 9], [563, 563, -3*w^3 - w^2 + 19*w + 5], [563, 563, 7/2*w^3 + 5/2*w^2 - 45/2*w - 16], [563, 563, -w^3 + 3*w + 3], [563, 563, -w^2 + 2*w - 3], [569, 569, w^3 - w^2 - 6*w - 1], [571, 571, -5/2*w^3 + 3/2*w^2 + 31/2*w - 11], [577, 577, 2*w^2 - 2*w - 5], [587, 587, 3/2*w^3 + 1/2*w^2 - 23/2*w - 3], [587, 587, -2*w^3 + 14*w + 1], [599, 599, -3/2*w^3 - 1/2*w^2 + 13/2*w + 4], [607, 607, -3/2*w^3 - 3/2*w^2 + 21/2*w + 13], [613, 613, -1/2*w^3 + 3/2*w^2 + 11/2*w - 6], [613, 613, w^3 + w^2 - 8*w - 7], [617, 617, 2*w^2 - w - 5], [625, 5, -5], [653, 653, -1/2*w^3 + 3/2*w^2 + 9/2*w - 3], [653, 653, -2*w^3 - w^2 + 14*w + 3], [659, 659, -7/2*w^3 - 1/2*w^2 + 43/2*w + 6], [661, 661, 5/2*w^3 + 1/2*w^2 - 25/2*w - 5], [661, 661, w^3 + w^2 - 5*w - 7], [677, 677, -11/2*w^3 - 3/2*w^2 + 73/2*w + 13], [683, 683, -7/2*w^3 + 5/2*w^2 + 45/2*w - 16], [683, 683, -w^3 + 3*w^2 + 5*w - 11], [691, 691, 2*w^2 - 15], [691, 691, w^3 + w^2 - 4*w - 5], [701, 701, -7/2*w^3 - 1/2*w^2 + 47/2*w + 5], [701, 701, -w^3 + w^2 + 3*w - 5], [719, 719, 5/2*w^3 + 1/2*w^2 - 27/2*w - 5], [727, 727, -1/2*w^3 - 1/2*w^2 + 3/2*w - 4], [733, 733, 3/2*w^3 - 1/2*w^2 - 15/2*w + 1], [733, 733, -w^3 - 2*w^2 + 5*w + 5], [733, 733, 1/2*w^3 - 1/2*w^2 + 3/2*w + 2], [733, 733, 1/2*w^3 + 5/2*w^2 - 5/2*w - 13], [739, 739, 2*w^2 - 7], [743, 743, 3/2*w^3 - 1/2*w^2 - 9/2*w + 4], [743, 743, 2*w^3 - 12*w + 3], [751, 751, 1/2*w^3 + 1/2*w^2 - 11/2*w - 5], [751, 751, -1/2*w^3 + 1/2*w^2 + 11/2*w - 2], [761, 761, -3/2*w^3 + 3/2*w^2 + 23/2*w - 5], [769, 769, w^3 - 2*w^2 - 5*w + 13], [787, 787, -3/2*w^3 + 1/2*w^2 + 15/2*w - 3], [787, 787, 2*w^3 - 10*w - 3], [797, 797, 3/2*w^3 - 1/2*w^2 - 21/2*w + 7], [797, 797, -2*w^3 + 2*w^2 + 14*w - 13], [809, 809, -4*w^3 - 2*w^2 + 25*w + 15], [821, 821, 2*w^3 + 2*w^2 - 13*w - 11], [839, 839, -1/2*w^3 + 3/2*w^2 + 3/2*w - 9], [839, 839, -7/2*w^3 - 5/2*w^2 + 47/2*w + 18], [841, 29, 3/2*w^3 - 1/2*w^2 - 17/2*w + 10], [841, 29, 2*w^3 - 15*w + 5], [853, 853, 2*w^3 - 9*w - 3], [853, 853, -1/2*w^3 - 3/2*w^2 - 3/2*w + 7], [857, 857, 3/2*w^3 + 3/2*w^2 - 9/2*w - 4], [857, 857, w^3 - 3*w^2 - 5*w + 9], [857, 857, -w^3 + 3*w^2 + 6*w - 13], [857, 857, 1/2*w^3 + 5/2*w^2 - 7/2*w - 14], [859, 859, -5/2*w^3 - 1/2*w^2 + 35/2*w + 3], [863, 863, 2*w^3 - 2*w^2 - 13*w + 9], [881, 881, 2*w^2 + w - 13], [881, 881, 5/2*w^3 - 1/2*w^2 - 33/2*w - 1], [907, 907, -w^3 + 3*w^2 + 6*w - 15], [919, 919, 3/2*w^3 + 5/2*w^2 - 19/2*w - 4], [929, 929, 3/2*w^3 + 1/2*w^2 - 19/2*w + 1], [941, 941, 3/2*w^3 - 1/2*w^2 - 13/2*w], [947, 947, -1/2*w^3 + 5/2*w^2 + 13/2*w - 11], [947, 947, -7/2*w^3 - 1/2*w^2 + 43/2*w + 4], [947, 947, 1/2*w^3 + 1/2*w^2 - 13/2*w - 1], [947, 947, -4*w - 3], [953, 953, 3/2*w^3 - 7/2*w^2 - 21/2*w + 22], [953, 953, 1/2*w^3 + 5/2*w^2 - 3/2*w - 10], [961, 31, 3/2*w^3 + 3/2*w^2 - 17/2*w - 10], [967, 967, w^3 - 2*w^2 - 7*w + 7], [967, 967, 1/2*w^3 - 1/2*w^2 - 5/2*w - 4], [977, 977, -2*w^3 + 11*w + 1], [977, 977, 3/2*w^3 + 1/2*w^2 - 25/2*w + 4], [991, 991, -1/2*w^3 + 1/2*w^2 + 5/2*w - 8], [997, 997, 1/2*w^3 + 3/2*w^2 - 1/2*w - 7], [1013, 1013, 2*w^3 - w^2 - 14*w + 5], [1013, 1013, 3/2*w^3 + 3/2*w^2 - 23/2*w - 3], [1019, 1019, -1/2*w^3 + 3/2*w^2 + 3/2*w - 12], [1019, 1019, -3/2*w^3 + 7/2*w^2 + 17/2*w - 22], [1049, 1049, -5/2*w^3 - 5/2*w^2 + 33/2*w + 15], [1049, 1049, -3*w^3 - w^2 + 18*w + 3], [1051, 1051, 3*w^3 - 3*w^2 - 18*w + 17], [1061, 1061, 3/2*w^3 + 1/2*w^2 - 15/2*w - 7], [1061, 1061, -3*w - 5], [1087, 1087, w^2 - 2*w - 7], [1087, 1087, -w^3 - w^2 + 7*w - 1], [1091, 1091, 3/2*w^3 - 3/2*w^2 - 23/2*w + 3], [1093, 1093, 2*w^2 + w + 1], [1103, 1103, -2*w^3 + w^2 + 12*w - 1], [1103, 1103, 11/2*w^3 + 3/2*w^2 - 69/2*w - 11], [1103, 1103, -3/2*w^3 - 3/2*w^2 + 23/2*w + 9], [1103, 1103, 3/2*w^3 - 3/2*w^2 - 15/2*w + 1], [1109, 1109, -9/2*w^3 - 1/2*w^2 + 57/2*w + 3], [1151, 1151, 5/2*w^3 + 5/2*w^2 - 31/2*w - 14], [1151, 1151, -1/2*w^3 - 1/2*w^2 + 7/2*w - 5], [1153, 1153, -1/2*w^3 + 5/2*w^2 + 9/2*w - 15], [1163, 1163, -1/2*w^3 + 3/2*w^2 + 5/2*w - 13], [1163, 1163, -5/2*w^3 - 1/2*w^2 + 33/2*w + 1], [1181, 1181, -5/2*w^3 + 1/2*w^2 + 33/2*w - 5], [1181, 1181, -w^3 + w^2 + 4*w - 9], [1181, 1181, -5/2*w^3 + 1/2*w^2 + 35/2*w], [1181, 1181, -3/2*w^3 - 3/2*w^2 + 13/2*w + 6], [1187, 1187, 7/2*w^3 - 5/2*w^2 - 43/2*w + 15], [1193, 1193, -3/2*w^3 + 1/2*w^2 + 13/2*w - 2], [1201, 1201, -2*w^3 - 2*w^2 + 13*w + 5], [1201, 1201, 3/2*w^3 + 1/2*w^2 - 25/2*w + 2], [1213, 1213, 3*w^3 - w^2 - 18*w + 3], [1213, 1213, 3/2*w^3 + 5/2*w^2 - 21/2*w - 6], [1217, 1217, -1/2*w^3 - 1/2*w^2 + 11/2*w - 7], [1223, 1223, -1/2*w^3 - 1/2*w^2 - 5/2*w], [1237, 1237, -w^3 - 2*w^2 + 7*w + 9], [1259, 1259, -1/2*w^3 + 3/2*w^2 + 11/2*w - 12], [1279, 1279, 1/2*w^3 + 5/2*w^2 - 3/2*w - 13], [1279, 1279, 3/2*w^3 - 1/2*w^2 - 11/2*w + 1], [1297, 1297, -4*w^3 + 25*w - 1], [1301, 1301, 5/2*w^3 - 1/2*w^2 - 29/2*w + 4], [1303, 1303, -3/2*w^3 - 1/2*w^2 + 25/2*w + 4], [1303, 1303, w^3 + w^2 - 7*w + 3], [1319, 1319, 9/2*w^3 - 9/2*w^2 - 57/2*w + 29], [1321, 1321, 1/2*w^3 + 5/2*w^2 - 3/2*w - 8], [1321, 1321, w^2 - 2*w - 9], [1381, 1381, 1/2*w^3 + 1/2*w^2 - 1/2*w - 7], [1399, 1399, -1/2*w^3 + 1/2*w^2 + 1/2*w - 7], [1399, 1399, -w^2 - 3], [1409, 1409, -5/2*w^3 - 3/2*w^2 + 25/2*w], [1423, 1423, 5/2*w^3 - 3/2*w^2 - 31/2*w + 5], [1423, 1423, -w^3 + 9*w - 3], [1439, 1439, w^3 + 2*w^2 - 7*w - 7], [1439, 1439, -5/2*w^3 - 5/2*w^2 + 33/2*w + 7], [1447, 1447, 3/2*w^3 - 5/2*w^2 - 17/2*w + 9], [1451, 1451, 1/2*w^3 + 5/2*w^2 - 7/2*w - 5], [1459, 1459, -1/2*w^3 + 1/2*w^2 + 13/2*w], [1459, 1459, 1/2*w^3 + 3/2*w^2 - 17/2*w - 3], [1481, 1481, 3/2*w^3 - 3/2*w^2 - 21/2*w + 4], [1487, 1487, 7/2*w^3 - 1/2*w^2 - 43/2*w + 1], [1499, 1499, -3/2*w^3 - 5/2*w^2 + 11/2*w + 8], [1523, 1523, -5/2*w^3 + 1/2*w^2 + 29/2*w - 1], [1523, 1523, -3/2*w^3 + 1/2*w^2 + 11/2*w + 5], [1543, 1543, -2*w^3 + 10*w - 3], [1567, 1567, -1/2*w^3 - 5/2*w^2 + 9/2*w + 5], [1567, 1567, 2*w^3 - 15*w - 3], [1579, 1579, 1/2*w^3 + 3/2*w^2 - 3/2*w - 10], [1579, 1579, -3/2*w^3 + 3/2*w^2 + 13/2*w - 10], [1597, 1597, -1/2*w^3 - 3/2*w^2 + 11/2*w + 8], [1601, 1601, 2*w^3 - w^2 - 10*w + 1], [1601, 1601, 3/2*w^3 + 1/2*w^2 - 21/2*w + 2], [1613, 1613, -1/2*w^3 - 1/2*w^2 + 5/2*w - 5], [1619, 1619, 5/2*w^3 - 1/2*w^2 - 29/2*w + 3], [1619, 1619, 3*w^3 - 3*w^2 - 17*w + 19], [1621, 1621, w^3 + w^2 - w - 5], [1621, 1621, 3/2*w^3 + 5/2*w^2 - 23/2*w - 7], [1637, 1637, -7/2*w^3 + 1/2*w^2 + 45/2*w - 4], [1657, 1657, -7/2*w^3 - 5/2*w^2 + 45/2*w + 14], [1669, 1669, -w^3 - w^2 + 2*w + 5], [1693, 1693, 11/2*w^3 + 1/2*w^2 - 71/2*w - 4], [1699, 1699, 7/2*w^3 - 9/2*w^2 - 47/2*w + 27], [1699, 1699, -1/2*w^3 + 5/2*w^2 + 11/2*w - 14], [1733, 1733, w^3 - 9*w + 1], [1741, 1741, 5/2*w^3 - 1/2*w^2 - 31/2*w - 4], [1747, 1747, 7/2*w^3 - 5/2*w^2 - 45/2*w + 12], [1777, 1777, -3/2*w^3 - 1/2*w^2 + 23/2*w - 3], [1777, 1777, -1/2*w^3 - 1/2*w^2 - 3/2*w - 3], [1787, 1787, -2*w^3 - 2*w^2 + 15*w + 5], [1801, 1801, -1/2*w^3 - 5/2*w^2 + 11/2*w + 6], [1801, 1801, 1/2*w^3 - 1/2*w^2 - 9/2*w - 5], [1823, 1823, w^3 + 3*w^2 - 7*w - 17], [1823, 1823, 5/2*w^3 + 5/2*w^2 - 33/2*w - 9], [1847, 1847, -w^3 - 3*w^2 + 7*w + 5], [1861, 1861, -1/2*w^3 + 1/2*w^2 + 9/2*w - 10], [1861, 1861, -3/2*w^3 + 3/2*w^2 + 17/2*w - 2], [1871, 1871, -w^3 + w^2 + 10*w - 11], [1873, 1873, -5*w^3 - w^2 + 31*w + 7], [1879, 1879, -2*w^3 - w^2 + 8*w + 5], [1889, 1889, w^3 + w^2 - 7*w - 11], [1889, 1889, -3/2*w^3 - 1/2*w^2 + 9/2*w + 4], [1901, 1901, 4*w^3 - 4*w^2 - 26*w + 27], [1901, 1901, 3/2*w^3 + 3/2*w^2 - 21/2*w - 2], [1901, 1901, -1/2*w^3 + 1/2*w^2 + 13/2*w - 1], [1901, 1901, w^2 + 4*w - 3], [1907, 1907, 1/2*w^3 - 5/2*w^2 - 5/2*w + 19], [1913, 1913, -2*w^3 + 2*w^2 + 12*w - 7], [1931, 1931, -w^3 - 3*w^2 + 6*w + 5], [1973, 1973, 4*w^3 - 4*w^2 - 25*w + 21], [1973, 1973, 1/2*w^3 + 3/2*w^2 - 3/2*w - 12], [1979, 1979, -w^3 + 2*w^2 + 9*w - 9], [1979, 1979, -3/2*w^3 + 3/2*w^2 + 23/2*w - 8], [1987, 1987, 1/2*w^3 + 5/2*w^2 - 5/2*w - 7], [1993, 1993, -3*w^3 + 17*w - 3], [1993, 1993, -1/2*w^3 - 5/2*w^2 + 9/2*w + 11], [1999, 1999, -2*w^3 + 12*w - 5]]; primes := [ideal : I in primesArray]; heckePol := x; K := Rationals(); e := 1; heckeEigenvaluesArray := [1, -3, 1, 2, 4, 0, 4, -4, -2, 4, 2, 0, 4, 12, -6, -10, 4, 0, 18, 8, 16, -10, 4, -18, -8, -6, 0, 20, 6, -8, 2, 4, 0, -20, -8, 6, 0, -12, 14, -6, 6, 22, -24, -4, 20, 2, -20, -6, -24, 14, 20, 26, -20, 24, -8, -22, -2, 18, -2, 12, 14, -14, 16, 16, -26, 16, -6, 20, 12, -2, 14, -10, -26, -20, 0, 4, -38, 2, 6, -2, 20, 40, -32, 2, 40, -12, 6, 30, 8, 0, -16, 2, -4, 24, 2, -32, 44, 2, 0, -8, -36, -20, 24, -4, 24, -18, -16, -18, 32, -42, -2, 6, -6, -18, -34, -30, 30, 26, 0, -12, 12, -20, -16, -30, 6, -30, 24, -14, -22, -20, 32, -22, 0, -36, -4, 40, -12, 26, -42, 28, -6, 32, -6, -18, -8, -8, 46, -12, 50, 34, 48, -2, -48, 10, -14, -16, 46, -42, 48, 24, -6, 54, 34, -32, -4, -4, 22, 42, -10, 16, -8, 26, -50, -32, -6, 30, -46, -36, 54, 42, -30, 20, 10, -4, -40, -18, 52, 54, -44, -36, 16, 20, 6, -40, 32, 24, -36, 12, -20, 18, 48, 2, 32, -44, -20, 46, -26, 10, 44, -2, 10, -34, -16, -36, -64, -42, 8, 4, -30, 28, -14, 58, 24, 32, -58, -10, 24, 24, -20, -60, -36, -38, 52, 10, -24, -40, -62, -32, -48, -48, 40, 40, 32, 22, -10, -18, 46, 20, -74, 58, 58, -6, 38, 2, -14, 28, -44, 54, -70, -68, -32, 50, -32, 38, -14, 72, 18, 16, 38, 6, 32, 66, -2, -36, -12, -50, -18, -30, -46, 72, -74, -68, 54, 14, 60, 28, 20, -18, 62, -32]; 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;