/*
  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![45, 15, -14, -2, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {25, 5, 2/3*w^2 - 2/3*w - 5}>;

primesArray := [
[5, 5, 2/3*w^2 + 1/3*w - 5],
[5, 5, 2/3*w^2 - 5/3*w - 4],
[9, 3, -w + 3],
[9, 3, w + 2],
[11, 11, 2/3*w^2 + 1/3*w - 4],
[16, 2, 2],
[19, 19, 1/3*w^3 - 10/3*w - 3],
[19, 19, 1/3*w^3 - w^2 - 7/3*w + 6],
[29, 29, -1/3*w^3 + 2/3*w^2 + 8/3*w - 2],
[29, 29, 1/3*w^3 - 1/3*w^2 - 3*w + 1],
[31, 31, -1/3*w^3 + 1/3*w^2 + 3*w + 1],
[31, 31, -1/3*w^3 + 2/3*w^2 + 8/3*w - 4],
[49, 7, 1/3*w^3 - 10/3*w - 1],
[49, 7, -1/3*w^3 + w^2 + 7/3*w - 4],
[59, 59, -1/3*w^3 + 5/3*w^2 + 5/3*w - 11],
[59, 59, -2/3*w^3 + 5/3*w^2 + 5*w - 13],
[61, 61, -1/3*w^3 + 5/3*w^2 + 2/3*w - 9],
[61, 61, -1/3*w^3 + 1/3*w^2 + 4*w + 2],
[61, 61, 2/3*w^2 - 5/3*w - 1],
[61, 61, -1/3*w^2 + 4/3*w + 6],
[71, 71, -4/3*w^2 + 7/3*w + 11],
[79, 79, -1/3*w^2 + 7/3*w - 3],
[79, 79, -2/3*w^3 + 1/3*w^2 + 16/3*w - 4],
[101, 101, 2/3*w^2 - 5/3*w - 6],
[101, 101, -2/3*w^2 - 1/3*w + 7],
[109, 109, -4/3*w^2 + 1/3*w + 9],
[109, 109, 1/3*w^3 - 1/3*w^2 - 3*w - 4],
[121, 11, -1/3*w^2 + 1/3*w + 6],
[131, 131, -2/3*w^3 + 2/3*w^2 + 5*w - 7],
[131, 131, 2/3*w^3 - 4/3*w^2 - 13/3*w - 2],
[151, 151, -1/3*w^3 - 2/3*w^2 + 5*w + 7],
[151, 151, 1/3*w^3 - 10/3*w + 2],
[169, 13, 5/3*w^2 + 1/3*w - 11],
[169, 13, 5/3*w^2 - 11/3*w - 9],
[181, 181, -1/3*w^3 + 4/3*w^2 + 3*w - 8],
[181, 181, -1/3*w^3 - 1/3*w^2 + 14/3*w + 4],
[191, 191, -2/3*w^3 + w^2 + 17/3*w - 9],
[191, 191, 1/3*w^3 - 2*w^2 - 7/3*w + 13],
[191, 191, 2/3*w^3 - 1/3*w^2 - 22/3*w - 7],
[191, 191, 2/3*w^3 - w^2 - 17/3*w - 3],
[199, 199, 1/3*w^3 + w^2 - 13/3*w - 9],
[199, 199, 2/3*w^2 - 5/3*w - 7],
[199, 199, 1/3*w^3 - w^2 - 4/3*w + 6],
[199, 199, -1/3*w^3 + 2*w^2 + 4/3*w - 12],
[269, 269, -1/3*w^3 - 4/3*w^2 + 11/3*w + 14],
[269, 269, 1/3*w^3 - 7/3*w^2 + 16],
[271, 271, 2/3*w^2 + 1/3*w - 9],
[271, 271, 2/3*w^2 - 5/3*w - 8],
[281, 281, 1/3*w^3 + w^2 - 13/3*w - 11],
[281, 281, -2/3*w^3 + 20/3*w + 3],
[281, 281, -w^2 + 3*w + 2],
[281, 281, -1/3*w^3 + 2*w^2 + 4/3*w - 14],
[289, 17, w^3 - 8/3*w^2 - 16/3*w + 4],
[289, 17, -w^3 + 1/3*w^2 + 23/3*w - 3],
[331, 331, -2/3*w^3 + 2*w^2 + 11/3*w - 7],
[331, 331, 2/3*w^3 - 17/3*w - 2],
[359, 359, -1/3*w^3 - 1/3*w^2 + 11/3*w - 2],
[359, 359, 1/3*w^3 + 4/3*w^2 - 17/3*w - 12],
[361, 19, 4/3*w^2 - 4/3*w - 11],
[389, 389, 5/3*w^2 - 8/3*w - 13],
[389, 389, 5/3*w^2 - 2/3*w - 14],
[401, 401, -1/3*w^3 + 5/3*w^2 + 11/3*w - 12],
[401, 401, 1/3*w^3 + 2/3*w^2 - 6*w - 7],
[409, 409, 1/3*w^3 + 1/3*w^2 - 14/3*w - 3],
[409, 409, -1/3*w^3 + 4/3*w^2 + 3*w - 7],
[421, 421, -1/3*w^3 + 5/3*w^2 + 11/3*w - 13],
[421, 421, 1/3*w^3 + 2/3*w^2 - 6*w - 8],
[431, 431, 2/3*w^2 + 4/3*w - 9],
[431, 431, 2/3*w^2 - 8/3*w - 7],
[449, 449, 2/3*w^3 - 17/3*w + 2],
[449, 449, 1/3*w^3 + 4/3*w^2 - 20/3*w - 11],
[461, 461, 1/3*w^2 + 5/3*w - 8],
[461, 461, 1/3*w^3 + 1/3*w^2 - 17/3*w - 3],
[461, 461, -1/3*w^3 + 4/3*w^2 + 4*w - 8],
[461, 461, 1/3*w^2 - 7/3*w - 6],
[479, 479, -1/3*w^3 + 2/3*w^2 + 5/3*w - 9],
[479, 479, 4/3*w^3 + w^2 - 40/3*w - 18],
[479, 479, 4/3*w^3 - 5*w^2 - 22/3*w + 29],
[479, 479, 1/3*w^3 - 1/3*w^2 - 2*w - 7],
[491, 491, 2/3*w^3 - 5/3*w^2 - 5*w + 8],
[491, 491, -1/3*w^3 + 3*w^2 + 10/3*w - 24],
[491, 491, 1/3*w^3 + 2*w^2 - 25/3*w - 18],
[491, 491, 2/3*w^3 - 1/3*w^2 - 19/3*w - 2],
[499, 499, -1/3*w^3 + 10/3*w^2 - 4*w - 17],
[499, 499, 1/3*w^3 - 7/3*w - 9],
[499, 499, -1/3*w^3 + w^2 + 4/3*w - 11],
[499, 499, -w^3 + 5/3*w^2 + 22/3*w - 14],
[509, 509, w^3 - 14/3*w^2 - 10/3*w + 24],
[509, 509, -2/3*w^2 + 5/3*w + 11],
[509, 509, 2/3*w^2 + 1/3*w - 12],
[509, 509, -2/3*w^3 + 3*w^2 + 8/3*w - 18],
[521, 521, 1/3*w^3 - 13/3*w + 1],
[521, 521, w^3 + 2/3*w^2 - 32/3*w - 12],
[521, 521, 2/3*w^3 + 7/3*w^2 - 13*w - 23],
[521, 521, -1/3*w^3 + w^2 + 10/3*w - 3],
[541, 541, 2/3*w^3 - 17/3*w - 3],
[541, 541, -1/3*w^3 + 4/3*w^2 + 2*w - 14],
[541, 541, -1/3*w^3 + 4/3*w^2 + w - 1],
[541, 541, -1/3*w^3 + 1/3*w^2 + 5*w - 4],
[569, 569, 1/3*w^3 + 11/3*w^2 - 12*w - 26],
[569, 569, -w^3 + 10/3*w^2 + 23/3*w - 21],
[571, 571, w^3 - 7/3*w^2 - 23/3*w + 17],
[571, 571, 2/3*w^3 - 3*w^2 - 11/3*w + 17],
[599, 599, 1/3*w^3 + 2/3*w^2 - 4*w - 1],
[599, 599, 1/3*w^3 - 5/3*w^2 - 5/3*w + 4],
[601, 601, 1/3*w^3 - 7/3*w - 6],
[601, 601, -2/3*w^3 + 2*w^2 + 11/3*w - 9],
[601, 601, 2/3*w^3 - 17/3*w - 4],
[601, 601, -1/3*w^3 + w^2 + 4/3*w - 8],
[619, 619, -2/3*w^3 + 10/3*w^2 + 19/3*w - 26],
[619, 619, 2/3*w^3 + 4/3*w^2 - 11*w - 17],
[631, 631, 1/3*w^3 - 7/3*w^2 - w + 14],
[631, 631, 1/3*w^3 + 4/3*w^2 - 14/3*w - 11],
[659, 659, -2/3*w^3 + 2/3*w^2 + 6*w + 1],
[659, 659, 2/3*w^3 - 4/3*w^2 - 16/3*w + 7],
[661, 661, -2/3*w^3 + 13/3*w^2 + 16/3*w - 32],
[661, 661, -2/3*w^3 - 7/3*w^2 + 12*w + 23],
[691, 691, 1/3*w^3 - 11/3*w^2 - 11/3*w + 29],
[691, 691, -2/3*w^3 + 10/3*w^2 + 10/3*w - 23],
[701, 701, 2/3*w^3 - 4/3*w^2 - 13/3*w - 3],
[701, 701, 2/3*w^3 - 4/3*w^2 - 16/3*w - 1],
[701, 701, -2/3*w^3 + 2/3*w^2 + 6*w - 7],
[701, 701, -2/3*w^3 + 2/3*w^2 + 5*w - 8],
[709, 709, -4/3*w^3 + 31/3*w - 2],
[709, 709, -w^3 + 1/3*w^2 + 32/3*w + 4],
[719, 719, -2/3*w^3 + w^2 + 17/3*w - 2],
[719, 719, 1/3*w^3 - 2*w^2 - 4/3*w + 9],
[719, 719, 1/3*w^3 + w^2 - 13/3*w - 6],
[719, 719, 2/3*w^3 - w^2 - 17/3*w + 4],
[739, 739, 2/3*w^3 + 2/3*w^2 - 19/3*w - 11],
[739, 739, -2/3*w^3 + 8/3*w^2 + 3*w - 16],
[751, 751, -1/3*w^3 + w^2 + 4/3*w - 9],
[751, 751, 1/3*w^3 - 7/3*w - 7],
[761, 761, 1/3*w^3 - 16/3*w - 4],
[761, 761, w^3 - 1/3*w^2 - 29/3*w - 12],
[761, 761, -w^3 + 8/3*w^2 + 22/3*w - 21],
[761, 761, -1/3*w^3 + w^2 + 13/3*w - 9],
[769, 769, w^3 - 7/3*w^2 - 23/3*w + 8],
[769, 769, 1/3*w^3 + 4/3*w^2 - 14/3*w - 14],
[769, 769, -1/3*w^3 + 7/3*w^2 + w - 17],
[769, 769, 1/3*w^3 - 3*w^2 + 11/3*w + 13],
[811, 811, 1/3*w^3 + w^2 - 10/3*w - 12],
[811, 811, -1/3*w^3 + 2*w^2 + 1/3*w - 14],
[821, 821, -1/3*w^3 + 13/3*w - 3],
[821, 821, -1/3*w^3 + w^2 + 10/3*w - 1],
[839, 839, -w^3 - 5/3*w^2 + 32/3*w + 18],
[839, 839, 1/3*w^3 + 3*w^2 - 7/3*w - 22],
[839, 839, 1/3*w^3 - 4*w^2 + 14/3*w + 21],
[839, 839, -w^3 + 14/3*w^2 + 13/3*w - 26],
[841, 29, -5/3*w^2 + 5/3*w + 11],
[859, 859, -w^3 + 5/3*w^2 + 25/3*w - 1],
[859, 859, -w^3 + 4/3*w^2 + 26/3*w - 8],
[911, 911, -1/3*w^3 + 1/3*w^2 + 5*w + 1],
[911, 911, 1/3*w^3 - 2/3*w^2 - 14/3*w + 6],
[919, 919, 2/3*w^3 - 2/3*w^2 - 5*w - 7],
[919, 919, -2/3*w^3 + 7/3*w^2 + 10/3*w - 7],
[919, 919, 2/3*w^3 + 1/3*w^2 - 6*w - 2],
[919, 919, -2/3*w^3 + 4/3*w^2 + 13/3*w - 12],
[941, 941, 5/3*w^2 + 1/3*w - 9],
[941, 941, 5/3*w^2 - 11/3*w - 7],
[961, 31, 5/3*w^2 - 5/3*w - 13],
[971, 971, -1/3*w^3 + w^2 + 13/3*w - 8],
[971, 971, 2/3*w^3 - 2/3*w^2 - 5*w + 2],
[971, 971, -2/3*w^3 + 4/3*w^2 + 13/3*w - 3],
[971, 971, 1/3*w^3 - 16/3*w - 3],
[991, 991, 10/3*w^2 - 22/3*w - 19],
[991, 991, 10/3*w^2 + 2/3*w - 23],
[1009, 1009, -2*w^2 + 3*w + 16],
[1009, 1009, 2*w^2 - w - 17],
[1019, 1019, 1/3*w^3 + w^2 - 19/3*w - 14],
[1019, 1019, -1/3*w^3 + 2*w^2 + 10/3*w - 19],
[1021, 1021, 2/3*w^3 - 3*w^2 - 14/3*w + 19],
[1021, 1021, 11/3*w^2 + 4/3*w - 27],
[1021, 1021, 11/3*w^2 - 26/3*w - 22],
[1021, 1021, -2/3*w^3 - w^2 + 26/3*w + 12],
[1031, 1031, -2/3*w^3 + 5/3*w^2 + 4*w + 1],
[1031, 1031, -2/3*w^3 + 1/3*w^2 + 16/3*w - 6],
[1049, 1049, -1/3*w^3 - 2/3*w^2 + 4*w - 1],
[1049, 1049, 1/3*w^3 + 5/3*w^2 - 6*w - 14],
[1051, 1051, 1/3*w^3 - 16/3*w - 2],
[1051, 1051, -1/3*w^3 + w^2 + 13/3*w - 7],
[1061, 1061, 11/3*w^2 + 1/3*w - 27],
[1061, 1061, 11/3*w^2 - 23/3*w - 23],
[1069, 1069, w^2 + 2*w - 11],
[1069, 1069, w^2 - 4*w - 8],
[1109, 1109, w^3 - 5/3*w^2 - 19/3*w - 1],
[1109, 1109, -w^3 - w^2 + 12*w + 16],
[1129, 1129, -w^3 + 5/3*w^2 + 25/3*w - 11],
[1129, 1129, -w^3 + 4/3*w^2 + 26/3*w + 2],
[1151, 1151, 1/3*w^3 + w^2 - 16/3*w - 7],
[1151, 1151, -1/3*w^3 + 2*w^2 + 7/3*w - 11],
[1171, 1171, w^3 - 8/3*w^2 - 16/3*w + 1],
[1171, 1171, -w^3 + 1/3*w^2 + 23/3*w - 6],
[1201, 1201, -1/3*w^3 + 2/3*w^2 + 14/3*w - 4],
[1201, 1201, -1/3*w^3 - 4/3*w^2 + 8/3*w + 13],
[1201, 1201, 1/3*w^3 - 7/3*w^2 + w + 14],
[1201, 1201, 1/3*w^3 - 1/3*w^2 - 5*w + 1],
[1231, 1231, -1/3*w^3 + 5/3*w^2 + 8/3*w - 8],
[1231, 1231, 1/3*w^3 + 2/3*w^2 - 5*w - 4],
[1249, 1249, -4/3*w^3 + 7/3*w^2 + 9*w + 4],
[1249, 1249, 4/3*w^2 - 13/3*w - 9],
[1249, 1249, 4/3*w^2 + 5/3*w - 12],
[1249, 1249, -4/3*w^3 + 5/3*w^2 + 29/3*w - 14],
[1259, 1259, 2/3*w^3 - w^2 - 20/3*w + 11],
[1259, 1259, -2/3*w^3 + w^2 + 20/3*w + 4],
[1279, 1279, -2/3*w^3 - 1/3*w^2 + 5*w + 8],
[1279, 1279, 1/3*w^3 - w^2 - 7/3*w + 12],
[1279, 1279, -2*w^2 + 3*w + 13],
[1279, 1279, -2/3*w^3 + 7/3*w^2 + 7/3*w - 12],
[1289, 1289, w^3 + 1/3*w^2 - 28/3*w - 8],
[1289, 1289, -2/3*w^3 + 3*w^2 + 8/3*w - 19],
[1289, 1289, 2/3*w^3 + w^2 - 20/3*w - 14],
[1289, 1289, 5/3*w^3 - 11/3*w^2 - 11*w - 1],
[1291, 1291, -w^3 + 2/3*w^2 + 25/3*w - 7],
[1291, 1291, w^3 - 7/3*w^2 - 20/3*w + 1],
[1301, 1301, w^3 - 10/3*w^2 - 14/3*w + 16],
[1301, 1301, -w^3 + 14/3*w^2 + 25/3*w - 34],
[1319, 1319, -5/3*w^3 + 6*w^2 + 17/3*w - 14],
[1319, 1319, 1/3*w^3 - 2/3*w^2 - 8/3*w - 4],
[1319, 1319, -1/3*w^3 + 1/3*w^2 + 3*w - 7],
[1319, 1319, -5/3*w^3 - w^2 + 38/3*w + 4],
[1361, 1361, w^3 - 4*w^2 - 6*w + 23],
[1361, 1361, w^3 + w^2 - 11*w - 14],
[1369, 37, -w^3 + 7*w - 4],
[1369, 37, -1/3*w^3 - 2*w^2 + 10/3*w + 16],
[1381, 1381, -2/3*w^3 + 20/3*w + 1],
[1381, 1381, 2/3*w^3 - 2*w^2 - 14/3*w + 7],
[1399, 1399, -w^3 + 4/3*w^2 + 20/3*w + 4],
[1399, 1399, -w^3 + 5/3*w^2 + 19/3*w - 11],
[1429, 1429, -2/3*w^3 + 2/3*w^2 + 4*w - 7],
[1429, 1429, w^3 + 1/3*w^2 - 19/3*w + 1],
[1451, 1451, w^3 - 4/3*w^2 - 23/3*w + 9],
[1451, 1451, -1/3*w^3 + 2*w^2 - 2/3*w - 12],
[1451, 1451, -1/3*w^3 - w^2 + 7/3*w + 11],
[1451, 1451, 1/3*w^3 - 8/3*w^2 + 10/3*w + 12],
[1459, 1459, -2/3*w^3 + 8/3*w^2 + 5*w - 16],
[1459, 1459, -1/3*w^3 + 8/3*w^2 + 2/3*w - 16],
[1459, 1459, -1/3*w^3 - 5/3*w^2 + 5*w + 13],
[1459, 1459, 2/3*w^3 + 2/3*w^2 - 25/3*w - 9],
[1471, 1471, -2/3*w^3 + 3*w^2 + 5/3*w - 16],
[1471, 1471, -13/3*w^2 - 5/3*w + 33],
[1471, 1471, -13/3*w^2 + 31/3*w + 27],
[1471, 1471, 2/3*w^3 + w^2 - 17/3*w - 12],
[1489, 1489, 1/3*w^3 - 7/3*w^2 - 4*w + 17],
[1489, 1489, 1/3*w^3 + 4/3*w^2 - 23/3*w - 11],
[1531, 1531, -w^3 + 3*w^2 + 7*w - 17],
[1531, 1531, w^3 - 10*w - 8],
[1549, 1549, w^3 + 2/3*w^2 - 35/3*w - 13],
[1549, 1549, -2/3*w^3 + 5*w^2 + 17/3*w - 36],
[1549, 1549, -2/3*w^3 - 3*w^2 + 41/3*w + 26],
[1549, 1549, -w^3 + 11/3*w^2 + 22/3*w - 23],
[1571, 1571, -w^3 + 3*w^2 + 7*w - 23],
[1571, 1571, -1/3*w^3 - 2*w^2 + 4/3*w + 17],
[1571, 1571, -1/3*w^3 + 1/3*w^2 + 6*w - 13],
[1571, 1571, -2/3*w^3 + 4*w^2 + 8/3*w - 23],
[1601, 1601, -w^3 + 4/3*w^2 + 26/3*w - 11],
[1601, 1601, -w^3 + 8/3*w^2 + 25/3*w - 24],
[1601, 1601, -1/3*w^3 - 7/3*w^2 + 23/3*w + 18],
[1601, 1601, -w^3 + 5/3*w^2 + 25/3*w + 2],
[1609, 1609, -5/3*w^3 + 2/3*w^2 + 12*w - 8],
[1609, 1609, 2/3*w^3 + 4/3*w^2 - 7*w - 17],
[1619, 1619, w^3 - 4/3*w^2 - 23/3*w - 3],
[1619, 1619, -w^3 + 5/3*w^2 + 22/3*w - 11],
[1621, 1621, 7/3*w^2 - 13/3*w - 11],
[1621, 1621, -2/3*w^3 + 8/3*w^2 + 6*w - 17],
[1669, 1669, 5/3*w^2 + 1/3*w - 16],
[1669, 1669, 5/3*w^2 - 11/3*w - 14],
[1681, 41, 2*w^2 - 2*w - 17],
[1681, 41, 2*w^2 - 2*w - 13],
[1699, 1699, 1/3*w^3 - 2*w^2 + 5/3*w + 11],
[1699, 1699, -1/3*w^3 - w^2 + 4/3*w + 11],
[1721, 1721, 1/3*w^2 + 8/3*w - 11],
[1721, 1721, 1/3*w^2 - 10/3*w - 8],
[1789, 1789, -2/3*w^2 - 7/3*w - 4],
[1789, 1789, -w^3 + 2*w^2 + 8*w - 13],
[1789, 1789, 4/3*w^2 - 13/3*w - 8],
[1789, 1789, 2/3*w^2 - 11/3*w + 7],
[1811, 1811, 2/3*w^2 - 11/3*w - 9],
[1811, 1811, 2/3*w^2 + 7/3*w - 12],
[1831, 1831, 4/3*w^2 + 5/3*w - 9],
[1831, 1831, -w^3 + 5/3*w^2 + 25/3*w - 2],
[1861, 1861, 14/3*w^2 + 1/3*w - 33],
[1861, 1861, 14/3*w^2 - 29/3*w - 28],
[1871, 1871, -w^3 + 2/3*w^2 + 28/3*w - 3],
[1871, 1871, 1/3*w^3 + w^2 - 22/3*w - 7],
[1871, 1871, -1/3*w^3 + 2*w^2 + 13/3*w - 13],
[1871, 1871, -w^3 + 7/3*w^2 + 23/3*w - 6],
[1879, 1879, -w^3 + 8/3*w^2 + 22/3*w - 13],
[1879, 1879, -w^3 + 1/3*w^2 + 29/3*w + 4],
[1889, 1889, -5/3*w^3 + 2*w^2 + 38/3*w - 16],
[1889, 1889, 5/3*w^3 - 3*w^2 - 35/3*w - 3],
[1901, 1901, w^2 - 3*w - 11],
[1901, 1901, w^2 + w - 13],
[1931, 1931, 2/3*w^3 + 2*w^2 - 38/3*w - 21],
[1931, 1931, w^3 - 2/3*w^2 - 25/3*w - 9],
[1931, 1931, -w^3 + 7/3*w^2 + 20/3*w - 17],
[1931, 1931, -2/3*w^3 + 4*w^2 + 20/3*w - 31],
[1949, 1949, 2/3*w^3 + 5/3*w^2 - 28/3*w - 21],
[1949, 1949, -1/3*w^3 + 3*w^2 + 1/3*w - 17],
[1951, 1951, -1/3*w^3 + 8/3*w^2 + 2/3*w - 19],
[1951, 1951, 1/3*w^3 + 5/3*w^2 - 5*w - 16],
[1999, 1999, -5/3*w^3 + 3*w^2 + 38/3*w - 23],
[1999, 1999, -1/3*w^3 - 5/3*w^2 + 4*w + 19],
[1999, 1999, 1/3*w^3 - 8/3*w^2 + 1/3*w + 21],
[1999, 1999, 5/3*w^3 - 2*w^2 - 41/3*w - 9]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^4 + 9*x^3 + 18*x^2 - 19*x - 40;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [1, 1, e, e, 4*e^3 + 18*e^2 - 8*e - 32, -e^3 - 4*e^2 + 4*e + 13, 2*e^3 + 10*e^2 + e - 20, 2*e^3 + 10*e^2 + e - 20, 3*e^3 + 14*e^2 - 5*e - 30, 3*e^3 + 14*e^2 - 5*e - 30, -5*e^3 - 22*e^2 + 11*e + 38, -5*e^3 - 22*e^2 + 11*e + 38, 7*e^3 + 31*e^2 - 19*e - 70, 7*e^3 + 31*e^2 - 19*e - 70, 11*e^3 + 49*e^2 - 24*e - 90, 11*e^3 + 49*e^2 - 24*e - 90, 8*e^3 + 37*e^2 - 16*e - 82, -2*e^2 - 7*e + 2, -2*e^2 - 7*e + 2, 8*e^3 + 37*e^2 - 16*e - 82, 2*e^3 + 6*e^2 - 14*e - 12, -e^3 - 5*e^2 + 2*e + 20, -e^3 - 5*e^2 + 2*e + 20, 4*e^3 + 16*e^2 - 13*e - 22, 4*e^3 + 16*e^2 - 13*e - 22, -6*e^3 - 26*e^2 + 18*e + 60, -6*e^3 - 26*e^2 + 18*e + 60, 2*e^3 + 9*e^2 - 3*e + 2, 3*e^3 + 10*e^2 - 16*e - 12, 3*e^3 + 10*e^2 - 16*e - 12, -6*e^3 - 27*e^2 + 16*e + 68, -6*e^3 - 27*e^2 + 16*e + 68, 2*e^3 + 13*e^2 + 9*e - 30, 2*e^3 + 13*e^2 + 9*e - 30, 2*e^3 + 7*e^2 - 9*e - 8, 2*e^3 + 7*e^2 - 9*e - 8, 5*e^3 + 20*e^2 - 17*e - 38, e^3 + 8*e^2 + 7*e - 28, e^3 + 8*e^2 + 7*e - 28, 5*e^3 + 20*e^2 - 17*e - 38, -15*e^3 - 66*e^2 + 35*e + 110, -4*e^3 - 17*e^2 + 14*e + 40, -4*e^3 - 17*e^2 + 14*e + 40, -15*e^3 - 66*e^2 + 35*e + 110, -15*e^3 - 69*e^2 + 25*e + 140, -15*e^3 - 69*e^2 + 25*e + 140, 5*e^3 + 22*e^2 - 17*e - 48, 5*e^3 + 22*e^2 - 17*e - 48, -9*e^3 - 43*e^2 + 4*e + 78, -4*e^3 - 21*e^2 - 8*e + 38, -4*e^3 - 21*e^2 - 8*e + 38, -9*e^3 - 43*e^2 + 4*e + 78, -7*e^3 - 32*e^2 + 10*e + 50, -7*e^3 - 32*e^2 + 10*e + 50, 18*e^3 + 79*e^2 - 45*e - 148, 18*e^3 + 79*e^2 - 45*e - 148, -9*e^3 - 43*e^2 + 15*e + 90, -9*e^3 - 43*e^2 + 15*e + 90, 3*e^3 + 14*e^2 - 10*e - 22, -15*e^3 - 70*e^2 + 23*e + 160, -15*e^3 - 70*e^2 + 23*e + 160, -8*e^3 - 40*e^2 + 6*e + 88, -8*e^3 - 40*e^2 + 6*e + 88, -3*e^3 - 15*e^2 + 4*e + 30, -3*e^3 - 15*e^2 + 4*e + 30, 9*e^3 + 38*e^2 - 35*e - 82, 9*e^3 + 38*e^2 - 35*e - 82, -12*e^3 - 55*e^2 + 22*e + 112, -12*e^3 - 55*e^2 + 22*e + 112, -6*e^3 - 26*e^2 + 11*e + 30, -6*e^3 - 26*e^2 + 11*e + 30, -17*e^3 - 70*e^2 + 63*e + 138, -15*e^3 - 68*e^2 + 31*e + 128, -15*e^3 - 68*e^2 + 31*e + 128, -17*e^3 - 70*e^2 + 63*e + 138, 9*e^3 + 43*e^2 - 17*e - 110, -26*e^3 - 115*e^2 + 67*e + 240, -26*e^3 - 115*e^2 + 67*e + 240, 9*e^3 + 43*e^2 - 17*e - 110, 11*e^3 + 48*e^2 - 34*e - 118, 5*e^3 + 25*e^2 - 12*e - 68, 5*e^3 + 25*e^2 - 12*e - 68, 11*e^3 + 48*e^2 - 34*e - 118, -14*e^3 - 61*e^2 + 44*e + 140, 8*e^3 + 38*e^2 - 10*e - 90, 8*e^3 + 38*e^2 - 10*e - 90, -14*e^3 - 61*e^2 + 44*e + 140, 6*e^3 + 32*e^2 + 7*e - 70, 11*e^3 + 44*e^2 - 41*e - 70, 11*e^3 + 44*e^2 - 41*e - 70, 6*e^3 + 32*e^2 + 7*e - 70, 8*e^3 + 39*e^2 - 16*e - 108, 11*e^3 + 48*e^2 - 27*e - 92, 11*e^3 + 48*e^2 - 27*e - 92, 8*e^3 + 39*e^2 - 16*e - 108, -14*e^3 - 58*e^2 + 45*e + 92, 15*e^3 + 69*e^2 - 30*e - 158, 15*e^3 + 69*e^2 - 30*e - 158, -14*e^3 - 58*e^2 + 45*e + 92, 2*e^3 + 16*e^2 + 19*e - 30, 2*e^3 + 16*e^2 + 19*e - 30, -3*e^2 - 12*e + 2, -3*e^2 - 12*e + 2, 4*e^3 + 18*e^2 - 10*e - 40, 4*e^3 + 18*e^2 - 10*e - 40, 2*e^3 + 7*e^2 - 10*e - 18, 9*e^3 + 37*e^2 - 31*e - 62, 9*e^3 + 37*e^2 - 31*e - 62, 2*e^3 + 7*e^2 - 10*e - 18, -21*e^3 - 95*e^2 + 45*e + 200, -21*e^3 - 95*e^2 + 45*e + 200, 18*e^3 + 81*e^2 - 39*e - 152, 18*e^3 + 81*e^2 - 39*e - 152, -15*e^3 - 68*e^2 + 32*e + 140, -15*e^3 - 68*e^2 + 32*e + 140, -3*e^3 - 16*e^2 - 4*e + 22, -3*e^3 - 16*e^2 - 4*e + 22, 10*e^3 + 43*e^2 - 23*e - 68, 10*e^3 + 43*e^2 - 23*e - 68, -4*e^3 - 22*e^2 - 13*e + 58, 23*e^3 + 106*e^2 - 47*e - 242, 23*e^3 + 106*e^2 - 47*e - 242, -4*e^3 - 22*e^2 - 13*e + 58, 32*e^3 + 141*e^2 - 79*e - 270, 32*e^3 + 141*e^2 - 79*e - 270, 9*e^3 + 36*e^2 - 29*e - 40, -15*e^3 - 65*e^2 + 43*e + 120, -15*e^3 - 65*e^2 + 43*e + 120, 9*e^3 + 36*e^2 - 29*e - 40, 12*e^3 + 47*e^2 - 57*e - 100, 12*e^3 + 47*e^2 - 57*e - 100, -26*e^3 - 114*e^2 + 61*e + 208, -26*e^3 - 114*e^2 + 61*e + 208, e^3 + 7*e^2 + 15*e + 12, 15*e^3 + 69*e^2 - 25*e - 142, 15*e^3 + 69*e^2 - 25*e - 142, e^3 + 7*e^2 + 15*e + 12, 4*e^3 + 12*e^2 - 24*e - 10, -8*e^3 - 41*e^2 + 11*e + 110, -8*e^3 - 41*e^2 + 11*e + 110, 4*e^3 + 12*e^2 - 24*e - 10, -27*e^3 - 119*e^2 + 70*e + 248, -27*e^3 - 119*e^2 + 70*e + 248, 20*e^3 + 88*e^2 - 49*e - 142, 20*e^3 + 88*e^2 - 49*e - 142, 20*e^3 + 83*e^2 - 68*e - 160, -5*e^3 - 19*e^2 + 21*e + 50, -5*e^3 - 19*e^2 + 21*e + 50, 20*e^3 + 83*e^2 - 68*e - 160, e^3 + 3*e^2 - 6*e - 2, 26*e^3 + 115*e^2 - 74*e - 260, 26*e^3 + 115*e^2 - 74*e - 260, 22*e^3 + 97*e^2 - 55*e - 208, 22*e^3 + 97*e^2 - 55*e - 208, -5*e^3 - 28*e^2 - 5*e + 80, -36*e^3 - 166*e^2 + 69*e + 360, -36*e^3 - 166*e^2 + 69*e + 360, -5*e^3 - 28*e^2 - 5*e + 80, 9*e^3 + 40*e^2 - 15*e - 72, 9*e^3 + 40*e^2 - 15*e - 72, -7*e^3 - 33*e^2 + 5*e + 102, 18*e^3 + 72*e^2 - 71*e - 132, 15*e^3 + 64*e^2 - 40*e - 118, 15*e^3 + 64*e^2 - 40*e - 118, 18*e^3 + 72*e^2 - 71*e - 132, -20*e^3 - 94*e^2 + 37*e + 212, -20*e^3 - 94*e^2 + 37*e + 212, -13*e^3 - 56*e^2 + 36*e + 110, -13*e^3 - 56*e^2 + 36*e + 110, -e^3 - 10*e^2 - 28*e, -e^3 - 10*e^2 - 28*e, 25*e^3 + 120*e^2 - 26*e - 232, 5*e^3 + 29*e^2 + 17*e - 52, 5*e^3 + 29*e^2 + 17*e - 52, 25*e^3 + 120*e^2 - 26*e - 232, 17*e^3 + 77*e^2 - 33*e - 128, 17*e^3 + 77*e^2 - 33*e - 128, 44*e^3 + 195*e^2 - 110*e - 410, 44*e^3 + 195*e^2 - 110*e - 410, -e^3 - 6*e^2 + 4*e + 38, -e^3 - 6*e^2 + 4*e + 38, 12*e^3 + 53*e^2 - 37*e - 138, 12*e^3 + 53*e^2 - 37*e - 138, 23*e^3 + 107*e^2 - 32*e - 190, 23*e^3 + 107*e^2 - 32*e - 190, 28*e^3 + 121*e^2 - 81*e - 230, 28*e^3 + 121*e^2 - 81*e - 230, -32*e^3 - 147*e^2 + 64*e + 330, -32*e^3 - 147*e^2 + 64*e + 330, -6*e^3 - 28*e^2 - 3*e + 38, -6*e^3 - 28*e^2 - 3*e + 38, -4*e^3 - 8*e^2 + 46*e + 12, -4*e^3 - 8*e^2 + 46*e + 12, -14*e^3 - 65*e^2 + 22*e + 142, -5*e^3 - 21*e^2 + 19*e + 38, -5*e^3 - 21*e^2 + 19*e + 38, -14*e^3 - 65*e^2 + 22*e + 142, -18*e^3 - 77*e^2 + 61*e + 142, -18*e^3 - 77*e^2 + 61*e + 142, -4*e^3 - 15*e^2 + 14*e + 10, -15*e^3 - 71*e^2 + 10*e + 150, -15*e^3 - 71*e^2 + 10*e + 150, -4*e^3 - 15*e^2 + 14*e + 10, -8*e^3 - 46*e^2 - 21*e + 100, -8*e^3 - 46*e^2 - 21*e + 100, 22*e^3 + 104*e^2 - 31*e - 230, -16*e^3 - 78*e^2 + 17*e + 160, -16*e^3 - 78*e^2 + 17*e + 160, 22*e^3 + 104*e^2 - 31*e - 230, 4*e^3 + 21*e^2 + 6*e - 30, -13*e^3 - 60*e^2 + 25*e + 130, -13*e^3 - 60*e^2 + 25*e + 130, 4*e^3 + 21*e^2 + 6*e - 30, -32*e^3 - 146*e^2 + 66*e + 328, -32*e^3 - 146*e^2 + 66*e + 328, 5*e - 2, 5*e - 2, -23*e^3 - 105*e^2 + 39*e + 220, 9*e^3 + 44*e^2 - 9*e - 110, 9*e^3 + 44*e^2 - 9*e - 110, -23*e^3 - 105*e^2 + 39*e + 220, -11*e^3 - 48*e^2 + 28*e + 52, -11*e^3 - 48*e^2 + 28*e + 52, 31*e^3 + 145*e^2 - 50*e - 290, 31*e^3 + 145*e^2 - 50*e - 290, -7*e^3 - 25*e^2 + 28*e + 42, -7*e^3 - 25*e^2 + 28*e + 42, 5*e^3 + 20*e^2 - 16*e - 50, 5*e^3 + 20*e^2 - 16*e - 50, -16*e^3 - 76*e^2 + 12*e + 160, -16*e^3 - 76*e^2 + 12*e + 160, 34*e^3 + 146*e^2 - 103*e - 292, -19*e^3 - 83*e^2 + 60*e + 212, -19*e^3 - 83*e^2 + 60*e + 212, 34*e^3 + 146*e^2 - 103*e - 292, 15*e^3 + 69*e^2 - 24*e - 180, -15*e^3 - 62*e^2 + 45*e + 100, -15*e^3 - 62*e^2 + 45*e + 100, 15*e^3 + 69*e^2 - 24*e - 180, -30*e^3 - 128*e^2 + 97*e + 248, 8*e^3 + 31*e^2 - 46*e - 82, 8*e^3 + 31*e^2 - 46*e - 82, -30*e^3 - 128*e^2 + 97*e + 248, -7*e^3 - 32*e^2 + 29*e + 70, -7*e^3 - 32*e^2 + 29*e + 70, -5*e^3 - 28*e^2 - 5*e + 32, -5*e^3 - 28*e^2 - 5*e + 32, -4*e^3 - 21*e^2 - 8*e + 30, -26*e^3 - 113*e^2 + 76*e + 250, -26*e^3 - 113*e^2 + 76*e + 250, -4*e^3 - 21*e^2 - 8*e + 30, 21*e^3 + 101*e^2 - 19*e - 188, 5*e^3 + 30*e^2 + 11*e - 92, 5*e^3 + 30*e^2 + 11*e - 92, 21*e^3 + 101*e^2 - 19*e - 188, -26*e^3 - 121*e^2 + 42*e + 232, 6*e^3 + 31*e^2 + 4*e - 58, 6*e^3 + 31*e^2 + 4*e - 58, -26*e^3 - 121*e^2 + 42*e + 232, 16*e^3 + 70*e^2 - 37*e - 120, 16*e^3 + 70*e^2 - 37*e - 120, 22*e^3 + 91*e^2 - 81*e - 180, 22*e^3 + 91*e^2 - 81*e - 180, 15*e^3 + 65*e^2 - 30*e - 108, 15*e^3 + 65*e^2 - 30*e - 108, 42*e^3 + 196*e^2 - 60*e - 370, 42*e^3 + 196*e^2 - 60*e - 370, -43*e^3 - 201*e^2 + 58*e + 402, 38*e^3 + 168*e^2 - 83*e - 258, -22*e^3 - 89*e^2 + 88*e + 150, -22*e^3 - 89*e^2 + 88*e + 150, 48*e^3 + 209*e^2 - 131*e - 418, 48*e^3 + 209*e^2 - 131*e - 418, 30*e^3 + 145*e^2 - 38*e - 310, -6*e^3 - 32*e^2 + e + 90, -6*e^3 - 32*e^2 + e + 90, 30*e^3 + 145*e^2 - 38*e - 310, -19*e^3 - 77*e^2 + 82*e + 148, -19*e^3 - 77*e^2 + 82*e + 148, 18*e^3 + 82*e^2 - 32*e - 178, 18*e^3 + 82*e^2 - 32*e - 178, -18*e^3 - 84*e^2 + 34*e + 202, -18*e^3 - 84*e^2 + 34*e + 202, 21*e^3 + 86*e^2 - 78*e - 152, -34*e^3 - 157*e^2 + 77*e + 378, -34*e^3 - 157*e^2 + 77*e + 378, 21*e^3 + 86*e^2 - 78*e - 152, 13*e^3 + 69*e^2 + 3*e - 120, 13*e^3 + 69*e^2 + 3*e - 120, -3*e^3 - 16*e^2 - 21*e - 10, -3*e^3 - 16*e^2 - 21*e - 10, 42*e^3 + 183*e^2 - 110*e - 362, 42*e^3 + 183*e^2 - 110*e - 362, 41*e^3 + 190*e^2 - 77*e - 428, 10*e^3 + 43*e^2 - 20*e - 68, 10*e^3 + 43*e^2 - 20*e - 68, 41*e^3 + 190*e^2 - 77*e - 428, -46*e^3 - 201*e^2 + 123*e + 380, -46*e^3 - 201*e^2 + 123*e + 380, -e^2 - 9*e + 8, -e^2 - 9*e + 8, 36*e^3 + 159*e^2 - 78*e - 280, 15*e^3 + 63*e^2 - 57*e - 140, 15*e^3 + 63*e^2 - 57*e - 140, 36*e^3 + 159*e^2 - 78*e - 280];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {5, 5, 2/3*w^2 + 1/3*w - 5}>] := -1;
ALEigenvalues[ideal<ZF | {5, 5, 2/3*w^2 - 5/3*w - 4}>] := -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;