/*
  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<x> := PolynomialRing(Rationals());
g := P![4, 6, -7, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {12, 6, -w - 2}>;

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<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x;
K := Rationals(); e := 1;

heckeEigenvaluesArray := [1, -1, -1, -3, -7, 2, 5, 10, 5, -6, -8, 0, -1, -2, -2, 0, -1, 9, 3, -4, 2, -16, -3, 10, -18, 5, -14, -10, -11, -10, 10, 25, 2, 21, -18, 13, 11, 24, -12, 6, 18, 3, 0, 3, 12, -13, -21, 15, 12, -26, -12, 19, -5, 7, -10, -12, 18, 30, -26, -24, -18, 20, -21, 25, 7, 4, -32, -20, -10, -15, -19, 8, -10, 16, -6, -7, -14, 22, -38, 33, 24, -34, 15, -39, 16, -24, -30, 6, -28, 7, 32, 9, 4, -14, -10, 14, 9, -33, -4, -25, -41, -21, -30, -13, 20, 42, 28, -35, 36, -18, -27, -10, -28, 4, 11, 34, -33, 24, -9, -25, -22, 11, 39, 0, -12, 8, 4, -25, -14, -46, 35, 34, -36, 41, 28, -3, -28, -7, -44, 17, -7, 22, 0, 12, -33, -32, 17, 12, 46, -49, -24, 18, 4, 13, -8, -36, 35, -29, -37, -26, 6, -22, -6, -24, 21, -9, 57, 31, -37, 22, 23, -55, -24, 32, 14, 18, -6, -51, 4, 56, -6, 14, 16, 11, -52, -16, -36, -12, 6, -24, -12, 38, -30, -16, -49, 32, 9, -19, -25, -36, -31, -3, 37, -26, -20, 5, -46, -54, -20, -24, -42, 0, 22, 25, -58, 0, 52, -34, -15, 28, 34, -50, -1, 52, 18, 72, -34, -24, -42, 36, -37, -74, -32, -75, 37, -75, -24, 18, 8, 16, 40, 52, -73, 19, -70, 59, 74, -71, -6, 56, 20, 9, 12, 70, -10, 55, -39, -50, -18, 76, -26, 66, 25, -74, 64, -34, -49, -42, -38, 10, 0, 19, 13, -18, -34, -36, -46, -69, 26, 42, 5, -33, 54, 74, -11, 76, -12, -35, -9, 49];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {3, 3, -w + 1}>] := -1;
ALEigenvalues[ideal<ZF | {4, 2, w}>] := 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;