/*
  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 | {21, 21, 1/2*w^3 + 1/2*w^2 - 9/2*w - 1}>;

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^9 - 3*x^8 - 14*x^7 + 41*x^6 + 57*x^5 - 158*x^4 - 78*x^3 + 171*x^2 + 34*x - 19;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [1, e, -21/554*e^8 + 29/277*e^7 + 321/554*e^6 - 745/554*e^5 - 1625/554*e^4 + 2509/554*e^3 + 3317/554*e^2 - 1150/277*e - 644/277, 1, -147/1108*e^8 + 203/554*e^7 + 985/554*e^6 - 5215/1108*e^5 - 1805/277*e^4 + 8643/554*e^3 + 1996/277*e^2 - 10837/1108*e - 2091/1108, 20/277*e^8 - 108/277*e^7 - 187/277*e^6 + 1435/277*e^5 + 189/277*e^4 - 5054/277*e^3 + 376/277*e^2 + 4235/277*e + 488/277, -107/554*e^8 + 95/277*e^7 + 798/277*e^6 - 2345/554*e^5 - 3421/277*e^4 + 3866/277*e^3 + 4368/277*e^2 - 7353/554*e - 2223/554, -1/1108*e^8 - 25/554*e^7 + 67/554*e^6 + 967/1108*e^5 - 359/277*e^4 - 3073/554*e^3 + 854/277*e^2 + 12461/1108*e + 31/1108, -247/1108*e^8 + 98/277*e^7 + 934/277*e^6 - 4357/1108*e^5 - 4188/277*e^4 + 5489/554*e^3 + 5958/277*e^2 - 2373/1108*e - 3977/1108, 33/277*e^8 - 12/277*e^7 - 544/277*e^6 - 56/277*e^5 + 2791/277*e^4 + 1439/277*e^3 - 4975/277*e^2 - 2915/277*e + 2301/277, 115/1108*e^8 - 86/277*e^7 - 390/277*e^6 + 4581/1108*e^5 + 1397/277*e^4 - 8367/554*e^3 - 983/277*e^2 + 16249/1108*e + 1421/1108, -281/1108*e^8 + 227/277*e^7 + 965/277*e^6 - 12475/1108*e^5 - 3652/277*e^4 + 23441/554*e^3 + 3693/277*e^2 - 41843/1108*e - 4031/1108, 183/1108*e^8 - 67/277*e^7 - 729/277*e^6 + 3089/1108*e^5 + 3372/277*e^4 - 3829/554*e^3 - 3655/277*e^2 - 2315/1108*e - 5119/1108, 32/277*e^8 - 62/277*e^7 - 410/277*e^6 + 634/277*e^5 + 1078/277*e^4 - 829/277*e^3 + 1488/277*e^2 - 2365/277*e - 3208/277, 393/1108*e^8 - 212/277*e^7 - 1393/277*e^6 + 10539/1108*e^5 + 5634/277*e^4 - 17759/554*e^3 - 6823/277*e^2 + 33427/1108*e + 9423/1108, -45/554*e^8 - 17/277*e^7 + 522/277*e^6 + 303/554*e^5 - 3502/277*e^4 + 215/277*e^3 + 6779/277*e^2 - 5443/554*e - 4145/554, -9/277*e^8 + 104/277*e^7 - 179/277*e^6 - 1269/277*e^5 + 3142/277*e^4 + 3964/277*e^3 - 9421/277*e^2 - 3083/277*e + 3603/277, 125/1108*e^8 - 199/554*e^7 - 619/554*e^6 + 4329/1108*e^5 + 1/277*e^4 - 4229/554*e^3 + 4050/277*e^2 - 4209/1108*e - 12739/1108, -79/1108*e^8 + 241/554*e^7 + 307/554*e^6 - 6707/1108*e^5 + 447/277*e^4 + 13181/554*e^3 - 4000/277*e^2 - 27185/1108*e + 10205/1108, -33/554*e^8 + 6/277*e^7 + 272/277*e^6 - 221/554*e^5 - 1257/277*e^4 + 1081/277*e^3 + 1241/277*e^2 - 8165/554*e + 1577/554, -112/277*e^8 + 217/277*e^7 + 1712/277*e^6 - 2773/277*e^5 - 7928/277*e^4 + 9688/277*e^3 + 11689/277*e^2 - 8758/277*e - 1237/277, -73/554*e^8 + 114/277*e^7 + 459/277*e^6 - 3091/554*e^5 - 1446/277*e^4 + 5858/277*e^3 + 1419/277*e^2 - 11649/554*e - 3831/554, -119/1108*e^8 + 36/277*e^7 + 524/277*e^6 - 1821/1108*e^5 - 2833/277*e^4 + 3277/554*e^3 + 4122/277*e^2 - 8509/1108*e + 4243/1108, 49/277*e^8 - 43/277*e^7 - 749/277*e^6 + 538/277*e^5 + 3053/277*e^4 - 2438/277*e^3 - 1738/277*e^2 + 4351/277*e - 2073/277, -159/554*e^8 + 180/277*e^7 + 958/277*e^6 - 4137/554*e^5 - 1977/277*e^4 + 5561/277*e^3 - 4320/277*e^2 - 5345/554*e + 10469/554, -73/554*e^8 + 114/277*e^7 + 459/277*e^6 - 3091/554*e^5 - 1446/277*e^4 + 5858/277*e^3 + 1973/277*e^2 - 12757/554*e - 9371/554, 305/1108*e^8 - 204/277*e^7 - 938/277*e^6 + 9211/1108*e^5 + 2573/277*e^4 - 11183/554*e^3 + 464/277*e^2 + 2051/1108*e - 6685/1108, 61/554*e^8 - 137/277*e^7 - 209/277*e^6 + 3615/554*e^5 - 1076/277*e^4 - 6724/277*e^3 + 6335/277*e^2 + 14371/554*e - 3553/554, -31/554*e^8 + 56/277*e^7 + 138/277*e^6 - 1047/554*e^5 - 98/277*e^4 + 25/277*e^3 + 318/277*e^2 + 9017/554*e - 2917/554, -93/554*e^8 + 168/277*e^7 + 414/277*e^6 - 4249/554*e^5 + 537/277*e^4 + 7554/277*e^3 - 7079/277*e^2 - 16715/554*e + 7869/554, -32/277*e^8 + 62/277*e^7 + 410/277*e^6 - 634/277*e^5 - 1355/277*e^4 + 1106/277*e^3 + 1005/277*e^2 + 1534/277*e + 2654/277, 125/554*e^8 - 199/277*e^7 - 619/277*e^6 + 4883/554*e^5 + 2/277*e^4 - 7830/277*e^3 + 7269/277*e^2 + 13519/554*e - 6645/554, 157/554*e^8 - 230/277*e^7 - 1101/277*e^6 + 6071/554*e^5 + 4419/277*e^4 - 10599/277*e^3 - 5837/277*e^2 + 14755/554*e + 9537/554, 93/554*e^8 - 168/277*e^7 - 691/277*e^6 + 4803/554*e^5 + 3341/277*e^4 - 9493/277*e^3 - 6217/277*e^2 + 15607/554*e + 4873/554, -49/554*e^8 + 160/277*e^7 + 236/277*e^6 - 4139/554*e^5 - 557/277*e^4 + 6482/277*e^3 + 1146/277*e^2 - 4905/554*e - 697/554, 26/277*e^8 - 85/277*e^7 - 437/277*e^6 + 1173/277*e^5 + 2434/277*e^4 - 3634/277*e^3 - 5439/277*e^2 - 1004/277*e + 4180/277, -93/1108*e^8 + 84/277*e^7 + 207/277*e^6 - 3695/1108*e^5 - 147/277*e^4 + 3953/554*e^3 - 77/277*e^2 + 3229/1108*e + 4545/1108, 59/277*e^8 - 97/277*e^7 - 981/277*e^6 + 1394/277*e^5 + 5225/277*e^4 - 6073/277*e^3 - 10137/277*e^2 + 7992/277*e + 3434/277, 263/1108*e^8 - 73/554*e^7 - 2109/554*e^6 + 519/1108*e^5 + 5223/277*e^4 + 2129/554*e^3 - 9096/277*e^2 - 5319/1108*e + 11791/1108, 10/277*e^8 - 54/277*e^7 - 232/277*e^6 + 1133/277*e^5 + 1618/277*e^4 - 7513/277*e^3 - 3136/277*e^2 + 14444/277*e + 2460/277, -73/277*e^8 + 228/277*e^7 + 918/277*e^6 - 2814/277*e^5 - 2892/277*e^4 + 8392/277*e^3 + 899/277*e^2 - 3062/277*e + 4202/277, -269/1108*e^8 - 77/554*e^7 + 2511/554*e^6 + 3067/1108*e^5 - 6823/277*e^4 - 8933/554*e^3 + 10342/277*e^2 + 32441/1108*e - 7173/1108, 409/1108*e^8 - 289/277*e^7 - 1375/277*e^6 + 13903/1108*e^5 + 5284/277*e^4 - 19559/554*e^3 - 7191/277*e^2 + 15763/1108*e + 17791/1108, -545/1108*e^8 + 779/554*e^7 + 3829/554*e^6 - 20337/1108*e^5 - 8126/277*e^4 + 35413/554*e^3 + 12258/277*e^2 - 59519/1108*e - 7481/1108, -161/1108*e^8 + 407/554*e^7 + 815/554*e^6 - 11621/1108*e^5 - 460/277*e^4 + 23237/554*e^3 - 2114/277*e^2 - 45795/1108*e - 549/1108, 303/554*e^8 - 458/277*e^7 - 2019/277*e^6 + 11699/554*e^5 + 7311/277*e^4 - 19822/277*e^3 - 6459/277*e^2 + 35837/554*e - 1637/554, 337/554*e^8 - 439/277*e^7 - 2358/277*e^6 + 11507/554*e^5 + 9286/277*e^4 - 20046/277*e^3 - 11347/277*e^2 + 28771/554*e + 8389/554, -105/554*e^8 + 145/277*e^7 + 664/277*e^6 - 3725/554*e^5 - 2262/277*e^4 + 7242/277*e^3 + 2614/277*e^2 - 19533/554*e - 623/554, -93/554*e^8 + 168/277*e^7 + 691/277*e^6 - 4803/554*e^5 - 3618/277*e^4 + 9770/277*e^3 + 8710/277*e^2 - 20593/554*e - 5981/554, 141/554*e^8 - 76/277*e^7 - 1137/277*e^6 + 1599/554*e^5 + 5673/277*e^4 - 1874/277*e^3 - 10641/277*e^2 + 1949/554*e + 13357/554, 5/554*e^8 + 125/277*e^7 - 335/277*e^6 - 3173/554*e^5 + 3590/277*e^4 + 4562/277*e^3 - 9648/277*e^2 - 1365/554*e + 6493/554, -455/1108*e^8 + 268/277*e^7 + 1531/277*e^6 - 12633/1108*e^5 - 5732/277*e^4 + 16701/554*e^3 + 7141/277*e^2 - 4313/1108*e - 6393/1108, -111/554*e^8 - 5/277*e^7 + 1066/277*e^6 + 415/554*e^5 - 6293/277*e^4 - 1224/277*e^3 + 10923/277*e^2 + 3711/554*e - 1545/554, -99/1108*e^8 + 286/277*e^7 + 131/277*e^6 - 16729/1108*e^5 + 746/277*e^4 + 33159/554*e^3 - 3263/277*e^2 - 57181/1108*e + 11379/1108, -165/554*e^8 + 307/277*e^7 + 1083/277*e^6 - 8307/554*e^5 - 3792/277*e^4 + 14823/277*e^3 + 2327/277*e^2 - 18665/554*e + 6777/554, 309/554*e^8 - 308/277*e^7 - 2144/277*e^6 + 7005/554*e^5 + 8295/277*e^4 - 9140/277*e^3 - 10059/277*e^2 + 4283/554*e + 10919/554, 199/554*e^8 - 288/277*e^7 - 1422/277*e^6 + 7561/554*e^5 + 6321/277*e^4 - 13662/277*e^3 - 10816/277*e^2 + 23787/554*e + 4911/554, -109/1108*e^8 + 45/554*e^7 + 655/554*e^6 + 143/1108*e^5 - 905/277*e^4 - 3665/554*e^3 + 1676/277*e^2 + 8705/1108*e - 18781/1108, 315/1108*e^8 - 79/277*e^7 - 1273/277*e^6 + 2865/1108*e^5 + 6163/277*e^4 - 951/554*e^3 - 9184/277*e^2 - 18407/1108*e + 10733/1108, -147/554*e^8 + 203/277*e^7 + 1262/277*e^6 - 5769/554*e^5 - 7211/277*e^4 + 11413/277*e^3 + 15626/277*e^2 - 24687/554*e - 14279/554, -65/554*e^8 + 37/277*e^7 + 754/277*e^6 - 1409/554*e^5 - 5674/277*e^4 + 4127/277*e^3 + 14624/277*e^2 - 13279/554*e - 16267/554, 199/1108*e^8 + 133/277*e^7 - 988/277*e^6 - 9059/1108*e^5 + 5792/277*e^4 + 21517/554*e^3 - 10117/277*e^2 - 51003/1108*e + 11005/1108, 144/277*e^8 - 279/277*e^7 - 2122/277*e^6 + 3407/277*e^5 + 9006/277*e^4 - 10517/277*e^3 - 11032/277*e^2 + 7501/277*e + 2184/277, 135/1108*e^8 + 164/277*e^7 - 783/277*e^6 - 10327/1108*e^5 + 4976/277*e^4 + 23731/554*e^3 - 8368/277*e^2 - 70095/1108*e + 6341/1108, 41/554*e^8 - 83/277*e^7 - 531/277*e^6 + 3011/554*e^5 + 4231/277*e^4 - 7798/277*e^3 - 11027/277*e^2 + 22047/554*e + 15349/554, -165/554*e^8 + 30/277*e^7 + 1360/277*e^6 + 3/554*e^5 - 6562/277*e^4 - 2351/277*e^3 + 9529/277*e^2 + 11251/554*e - 4857/554, 55/277*e^8 - 297/277*e^7 - 445/277*e^6 + 4154/277*e^5 - 519/277*e^4 - 17084/277*e^3 + 4635/277*e^2 + 21549/277*e - 874/277, 17/277*e^8 + 19/277*e^7 - 339/277*e^6 - 373/277*e^5 + 2252/277*e^4 + 1438/277*e^3 - 5165/277*e^2 - 1040/277*e + 27/277, 117/1108*e^8 - 399/554*e^7 - 637/554*e^6 + 12065/1108*e^5 + 730/277*e^4 - 26597/554*e^3 - 1029/277*e^2 + 64455/1108*e + 17425/1108, 32/277*e^8 - 62/277*e^7 - 410/277*e^6 + 634/277*e^5 + 1078/277*e^4 - 829/277*e^3 + 1488/277*e^2 - 2365/277*e - 3208/277, 211/554*e^8 - 265/277*e^7 - 1672/277*e^6 + 7591/554*e^5 + 8289/277*e^4 - 15843/277*e^3 - 13861/277*e^2 + 36577/554*e + 1769/554, -95/1108*e^8 + 395/554*e^7 + 271/554*e^6 - 11179/1108*e^5 + 520/277*e^4 + 22737/554*e^3 - 1139/277*e^2 - 61597/1108*e - 2595/1108, 323/554*e^8 - 512/277*e^7 - 1974/277*e^6 + 12857/554*e^5 + 5605/277*e^4 - 20964/277*e^3 - 1285/277*e^2 + 34255/554*e - 595/554, 60/277*e^8 - 47/277*e^7 - 1115/277*e^6 + 704/277*e^5 + 6661/277*e^4 - 2974/277*e^3 - 14384/277*e^2 + 1625/277*e + 8112/277, 647/1108*e^8 - 445/554*e^7 - 5123/554*e^6 + 11451/1108*e^5 + 11781/277*e^4 - 20573/554*e^3 - 14881/277*e^2 + 40537/1108*e + 8751/1108, -217/1108*e^8 + 115/554*e^7 + 1797/554*e^6 - 2897/1108*e^5 - 4221/277*e^4 + 4607/554*e^3 + 3329/277*e^2 - 1699/1108*e + 26671/1108, -165/554*e^8 + 307/277*e^7 + 1083/277*e^6 - 8307/554*e^5 - 4346/277*e^4 + 15377/277*e^3 + 8421/277*e^2 - 28637/554*e - 14275/554, 221/277*e^8 - 584/277*e^7 - 3022/277*e^6 + 7616/277*e^5 + 11271/277*e^4 - 26457/277*e^3 - 12022/277*e^2 + 20551/277*e + 3952/277, 667/1108*e^8 - 1053/554*e^7 - 4247/554*e^6 + 27567/1108*e^5 + 6773/277*e^4 - 48307/554*e^3 - 3707/277*e^2 + 71641/1108*e + 1483/1108, 101/277*e^8 - 213/277*e^7 - 1346/277*e^6 + 2330/277*e^5 + 4874/277*e^4 - 5551/277*e^3 - 5968/277*e^2 + 958/277*e + 6841/277, -185/1108*e^8 + 42/277*e^7 + 796/277*e^6 - 3371/1108*e^5 - 3813/277*e^4 + 10425/554*e^3 + 3424/277*e^2 - 31487/1108*e + 11829/1108, 91/554*e^8 + 59/277*e^7 - 834/277*e^6 - 1573/554*e^5 + 4121/277*e^4 + 1535/277*e^3 - 4186/277*e^2 + 9505/554*e - 5037/554, 41/1108*e^8 + 97/277*e^7 - 404/277*e^6 - 4745/1108*e^5 + 3362/277*e^4 + 6883/554*e^3 - 7591/277*e^2 - 8977/1108*e + 21443/1108, 51/277*e^8 + 57/277*e^7 - 1017/277*e^6 - 1119/277*e^5 + 5925/277*e^4 + 7084/277*e^3 - 9955/277*e^2 - 14200/277*e + 2020/277, 148/277*e^8 - 356/277*e^7 - 2104/277*e^6 + 4802/277*e^5 + 8379/277*e^4 - 18896/277*e^3 - 9461/277*e^2 + 22198/277*e + 1506/277, 93/1108*e^8 - 445/554*e^7 + 417/554*e^6 + 12005/1108*e^5 - 5393/277*e^4 - 22789/554*e^3 + 18636/277*e^2 + 48847/1108*e - 36123/1108, 169/554*e^8 - 207/277*e^7 - 1074/277*e^6 + 4993/554*e^5 + 3063/277*e^4 - 7240/277*e^3 + 536/277*e^2 + 7601/554*e - 4131/554, 207/1108*e^8 - 365/554*e^7 - 1127/554*e^6 + 10351/1108*e^5 + 631/277*e^4 - 20379/554*e^3 + 3549/277*e^2 + 36561/1108*e + 231/1108, 49/554*e^8 + 117/277*e^7 - 513/277*e^6 - 3617/554*e^5 + 2496/277*e^4 + 7645/277*e^3 - 315/277*e^2 - 13931/554*e - 10937/554, -126/277*e^8 + 348/277*e^7 + 1372/277*e^6 - 4193/277*e^5 - 1717/277*e^4 + 13392/277*e^3 - 9460/277*e^2 - 12415/277*e + 4183/277, -51/277*e^8 + 220/277*e^7 + 740/277*e^6 - 3036/277*e^5 - 3432/277*e^4 + 11752/277*e^3 + 5246/277*e^2 - 13500/277*e - 635/277, 61/1108*e^8 - 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2180/277*e^6 + 3087/554*e^5 + 9272/277*e^4 - 3462/277*e^3 - 9323/277*e^2 - 767/554*e - 11299/554, 339/554*e^8 - 112/277*e^7 - 2492/277*e^6 + 1263/554*e^5 + 10168/277*e^4 + 1889/277*e^3 - 12824/277*e^2 - 10001/554*e + 16637/554, 310/277*e^8 - 843/277*e^7 - 4422/277*e^6 + 10747/277*e^5 + 18857/277*e^4 - 34848/277*e^3 - 27966/277*e^2 + 20630/277*e + 6733/277, -179/554*e^8 + 511/277*e^7 + 1190/277*e^6 - 15267/554*e^5 - 4426/277*e^4 + 29694/277*e^3 + 4079/277*e^2 - 38111/554*e + 11089/554, -37/554*e^8 - 94/277*e^7 + 540/277*e^6 + 1985/554*e^5 - 3575/277*e^4 - 408/277*e^3 + 4472/277*e^2 - 23693/554*e + 5579/554, 372/277*e^8 - 513/277*e^7 - 6082/277*e^6 + 5916/277*e^5 + 30883/277*e^4 - 16389/277*e^3 - 54168/277*e^2 + 5366/277*e + 24755/277, -102/277*e^8 + 163/277*e^7 + 1480/277*e^6 - 1640/277*e^5 - 5479/277*e^4 + 790/277*e^3 + 3844/277*e^2 + 17043/277*e - 439/277, 369/277*e^8 - 663/277*e^7 - 5680/277*e^6 + 8263/277*e^5 + 25744/277*e^4 - 28456/277*e^3 - 32840/277*e^2 + 27514/277*e - 3129/277, 543/1108*e^8 - 276/277*e^7 - 1709/277*e^6 + 11745/1108*e^5 + 5192/277*e^4 - 12751/554*e^3 - 2240/277*e^2 + 7989/1108*e - 35115/1108, -57/554*e^8 + 514/277*e^7 - 336/277*e^6 - 14131/554*e^5 + 5887/277*e^4 + 25664/277*e^3 - 12059/277*e^2 - 49257/554*e + 105/554];
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 | {7, 7, -1/2*w^3 + 1/2*w^2 + 5/2*w - 2}>] := -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;