/* 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^4 - 4*x^2 + 1; K := NumberField(heckePol); heckeEigenvaluesArray := [e, -e^3 + 2*e, -e, 1, -e^2 - 2, e^3 - 2*e, -5*e^3 + 21*e, -5*e^2 + 9, 4*e^3 - 10*e, 3*e^3 - 9*e, -2*e^3 + 9*e, 5*e^3 - 16*e, 5*e^3 - 24*e, e^2 - 8, 3*e^3 - 11*e, e^2 - 6, -3*e^3 + 10*e, -3*e^2 - 2, e^2 - 15, 3, 5*e^2 - 3, 2*e^3 - 8*e, e^3 - 3*e, -8*e^3 + 22*e, -3*e^3 + 7*e, -6*e^2 + 4, -10*e^3 + 35*e, -10*e^3 + 43*e, 14*e^3 - 42*e, -2, -3*e^3 + 9*e, 12*e^3 - 45*e, 8*e^3 - 36*e, 11*e^3 - 38*e, 3*e^2 - 10, e^3 - 5*e, -14*e^3 + 42*e, 3*e^2 - 6, -3*e^2 - 11, -3*e^3 + 17*e, e^3 - 15*e, 8*e^2 - 22, -8*e, -3*e^2 + 7, -5*e^2 + 25, e^2 - 5, 7*e^2 - 16, 12*e^2 - 18, -8*e^3 + 37*e, -4*e^2 - 14, 7*e^2 - 26, 15*e^3 - 47*e, -16*e^2 + 29, 9*e^3 - 25*e, 10*e^3 - 28*e, 9*e^2 - 24, -2*e^2 + 5, -6*e^2 + 22, -13*e^3 + 47*e, 7*e^2 - 22, 6*e^2 - 1, -6*e^3 + 10*e, 10*e^2 - 3, -9*e^3 + 41*e, e^3 + 8*e, -5*e^2 + 15, -2*e^2 + 9, 12*e^3 - 29*e, -e^2 - 16, -e^3 + 18*e, -12*e^3 + 28*e, -8*e^2 + 10, -10*e^3 + 28*e, -15*e^3 + 63*e, 9*e^2 - 8, 17*e^2 - 36, -5*e^2 - 16, -8*e^2 + 8, -11*e^2 + 35, -5*e^2 - 14, e^3 + 4*e, 17*e^2 - 43, -5*e^3 + e, 4*e^3 - 32*e, 9*e^2 - 25, -e^2 + 16, -9*e^3 + 33*e, -11*e^2 + 27, -12, -10*e^2 + 39, -10*e^3 + 35*e, 2*e^3 - 2*e, 16*e^3 - 61*e, 16*e^3 - 73*e, -2*e^2 + 9, -23*e^3 + 73*e, -3*e^2 - 15, 14*e^3 - 65*e, 20*e^2 - 30, -8*e^3 + 26*e, -e^3 + 18*e, -e^2 + 29, 20*e^3 - 82*e, -e^2 - 7, 26*e^3 - 82*e, -19*e^3 + 51*e, 11*e^3 - 30*e, -19*e^3 + 63*e, -6*e^2 - 24, -27*e^3 + 81*e, 3*e^2 - 19, -26*e^3 + 100*e, -6*e^2 + 38, -9*e^3 + 29*e, 13*e^2 - 34, -5*e^2 - 14, -4*e^2 - 2, -17*e^3 + 64*e, -5*e^3 + 5*e, 20*e^2 - 48, 23*e^3 - 92*e, -10*e^2 + 11, -15*e^3 + 33*e, -14*e^3 + 71*e, 8*e^2 - 40, 26*e^2 - 50, 24*e^2 - 52, 16*e^2 - 36, 6*e^3 - 22*e, -6*e^2 - 4, 11*e^3 - 25*e, 19*e^3 - 44*e, -6*e^2 - 22, -13*e^2 + 42, -7*e^2 + 10, 19*e^3 - 61*e, -2*e^3 - 16*e, -4*e^2 + 26, 10*e^2 - 24, 10*e^2 + 16, 4*e^2 - 49, -4*e^3 + 14*e, 3*e^3 - 11*e, -27*e^3 + 97*e, -6*e^2 - 17, e^2 - 25, -28*e^2 + 56, -10*e^2 - 1, 4*e^2 - 28, 16*e^3 - 83*e, 28*e^3 - 115*e, 3*e^3 - 28*e, 11*e^3 - 41*e, 23*e^3 - 87*e, -12*e^3 + 27*e, 13*e^3 - 19*e, -13*e^3 + 70*e, -17*e^3 + 51*e, -13*e^3 + 22*e, 38, -3*e^2 + 7, -18*e^2 + 21, 30*e^3 - 97*e, 4*e^2 + 5, -e^3 + 3*e, 7*e^2 + 4, 15*e^3 - 79*e, -19*e^3 + 59*e, 17*e^3 - 65*e, 8*e^3 - 16*e, 28*e^2 - 65, 10*e^2 - 30, -9*e^2 - 5, -14*e^2 + 8, -11*e^3 + 25*e, -7*e^3 + 11*e, -5*e^2 - 24, 18*e^2 - 52, -10*e^2 + 45, 10*e^3 - 18*e, 14*e^2 - 10, -10*e^2 + 18, -11*e^3 + 43*e, 5*e^2 - 62, 5*e^2 - 19, 15*e^2 - 9, 9*e^3 - 7*e, e^2 - 13, 14*e^3 - 66*e, -4*e^2 + 53, 2*e^2 + 26, -31*e^3 + 119*e, -23*e^2 + 35, 18*e^3 - 66*e, -27*e^2 + 63, 11*e^3 - 57*e, -14*e^2 + 46, -8*e^2 - 14, 16*e^2 - 44, -12*e^2 + 14, -28*e^3 + 126*e, 9*e^3 - 33*e, -10*e^2 - 31, -21*e^2 + 58, -2*e^2 - 40, -9*e^3 + 55*e, -2*e^2 - 1, 7*e^3 - 49*e, -11*e^2 + 40, -9*e^3 + 54*e, -2*e^3 - 18*e, 9*e^3 - 39*e, -27*e^3 + 75*e, 3*e^2 - 9, -30*e^2 + 68, -32*e^2 + 78, 15*e^3 - 61*e, -6*e^2 + 11, 23*e^2 - 73, 38*e^3 - 125*e, 2*e^2 - 9, -38*e^3 + 148*e, -28*e^2 + 60, 22*e^2 - 34, -20*e^2 + 30, 9*e^3 - 54*e, -15*e^3 + 51*e, -33*e^3 + 103*e, 28*e^2 - 55, -29*e^3 + 113*e, -17*e^3 + 27*e, -e^3 - 4*e, 26*e^3 - 113*e, 5*e^3 + 13*e, 2*e^2 - 37, -33*e^3 + 97*e, -29*e^2 + 49, -11*e^3 + 41*e, -4*e^2 + 27, -21*e^3 + 90*e, -13*e^3 + 26*e, -40*e^3 + 157*e, 7*e^2 - 32, 26*e^2 - 60, 4*e^2 + 6, -14*e^2 + 40, 13*e^2 - 57, 3*e^2 + 59, 29*e^2 - 55, -21*e^3 + 102*e, -9*e^3 + 21*e, -29*e^2 + 49, -18*e^3 + 79*e, -21*e^3 + 105*e, 25*e^2 - 20, -7*e, -15*e^3 + 97*e, 21*e^3 - 81*e, 22*e^3 - 50*e, -2*e^3 + e, 8*e^2 - 25, -e^2 - 37, 3*e^2 - 11, -45*e^3 + 165*e, 10*e^2 + 24, 31*e^3 - 86*e, -3*e^3 + 46*e, -32*e^3 + 134*e, -15*e^2 + 7, 22*e^3 - 77*e, -4*e^3 - 23*e, -11*e^2 + 1, 35*e^3 - 122*e, -e^2 + 3, -24*e^3 + 78*e, 29*e^2 - 31, 5*e^2 - 13, 22*e^3 - 60*e, -28*e^3 + 124*e, -7*e^2 + 19, 5*e^3 - 45*e, 20*e^3 - 69*e, -7*e^3 - 13*e, -13*e^3 + 42*e, -30*e^3 + 112*e, 19*e^3 - 79*e, -36*e^3 + 96*e, -23*e^2 + 83, -47*e^3 + 150*e, 13*e^2 + 11]; 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;