/*
  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 | {16, 2, 2}>;

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 + 2*x^3 - 11*x^2 - 2*x + 11;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, 4/11*e^3 + 10/11*e^2 - 39/11*e - 2, -1/11*e^3 + 3/11*e^2 + 18/11*e - 4, -3/11*e^3 - 13/11*e^2 + 10/11*e + 2, 0, 1, 1/11*e^3 - 3/11*e^2 - 29/11*e + 4, -1/11*e^3 + 3/11*e^2 + 29/11*e, -5/11*e^3 - 7/11*e^2 + 46/11*e + 1, -7/11*e^3 - 23/11*e^2 + 38/11*e + 7, -1/11*e^3 + 3/11*e^2 + 7/11*e - 6, -7/11*e^3 - 23/11*e^2 + 49/11*e + 2, -2/11*e^3 - 16/11*e^2 + 3/11*e + 11, 6/11*e^3 + 26/11*e^2 - 31/11*e - 3, 10/11*e^3 + 36/11*e^2 - 48/11*e - 10, 6/11*e^3 + 4/11*e^2 - 64/11*e + 2, 13/11*e^3 + 38/11*e^2 - 69/11*e - 6, -8/11*e^3 - 20/11*e^2 + 78/11*e + 6, -2*e + 2, 15/11*e^3 + 32/11*e^2 - 127/11*e - 2, -16/11*e^3 - 40/11*e^2 + 112/11*e + 4, -14/11*e^3 - 35/11*e^2 + 131/11*e - 3, -2/11*e^3 - 5/11*e^2 - 19/11*e - 9, 15/11*e^3 + 54/11*e^2 - 83/11*e - 19, 5/11*e^3 - 4/11*e^2 - 57/11*e + 1, -1/11*e^3 - 8/11*e^2 + 7/11*e + 4, 5/11*e^3 + 18/11*e^2 - 35/11*e - 4, -14/11*e^3 - 35/11*e^2 + 98/11*e - 1, 3/11*e^3 - 20/11*e^2 - 54/11*e + 15, 21/11*e^3 + 80/11*e^2 - 114/11*e - 19, 13/11*e^3 + 16/11*e^2 - 124/11*e + 3, 19/11*e^3 + 64/11*e^2 - 100/11*e - 15, 6/11*e^3 + 15/11*e^2 + 24/11*e + 5, 30/11*e^3 + 75/11*e^2 - 276/11*e - 7, 3/11*e^3 + 24/11*e^2 + 23/11*e - 11, 1/11*e^3 - 14/11*e^2 - 51/11*e + 5, 15/11*e^3 + 32/11*e^2 - 138/11*e - 3, -3/11*e^3 - 35/11*e^2 - 67/11*e + 16, -5/11*e^3 + 15/11*e^2 + 123/11*e - 8, 9/11*e^3 + 28/11*e^2 - 30/11*e - 5, -14/11*e^3 - 35/11*e^2 + 153/11*e - 5, -2/11*e^3 - 5/11*e^2 + 69/11*e + 7, 18/11*e^3 + 45/11*e^2 - 181/11*e - 3, 6/11*e^3 + 15/11*e^2 - 97/11*e - 15, 10/11*e^3 - 8/11*e^2 - 191/11*e + 10, 2/11*e^3 + 38/11*e^2 + 107/11*e - 16, -17/11*e^3 - 37/11*e^2 + 75/11*e - 2, -39/11*e^3 - 103/11*e^2 + 317/11*e + 14, -6/11*e^3 - 26/11*e^2 - 57/11*e - 1, -2/11*e^3 + 6/11*e^2 + 25/11*e - 23, -10/11*e^3 - 36/11*e^2 + 59/11*e - 9, -30/11*e^3 - 64/11*e^2 + 309/11*e + 1, -8/11*e^3 - 20/11*e^2 + 56/11*e - 8, -8/11*e^3 - 20/11*e^2 + 56/11*e - 8, -7/11*e^3 - 1/11*e^2 + 93/11*e, -9/11*e^3 - 39/11*e^2 + 19/11*e + 16, -16/11*e^3 - 106/11*e^2 + 2/11*e + 34, 16/11*e^3 + 106/11*e^2 - 2/11*e - 42, 6/11*e^3 + 15/11*e^2 - 42/11*e - 7, -31/11*e^3 - 72/11*e^2 + 305/11*e + 18, -5/11*e^3 - 18/11*e^2 - 53/11*e + 10, 18/11*e^3 + 23/11*e^2 - 148/11*e - 1, 34/11*e^3 + 107/11*e^2 - 216/11*e - 29, -35/11*e^3 - 93/11*e^2 + 256/11*e + 22, -25/11*e^3 - 57/11*e^2 + 164/11*e + 12, -2*e^3 - 4*e^2 + 20*e + 22, -10/11*e^3 - 36/11*e^2 + 4/11*e + 26, -3/11*e^3 - 13/11*e^2 + 87/11*e + 6, 27/11*e^3 + 73/11*e^2 - 255/11*e - 14, 6/11*e^3 + 70/11*e^2 + 123/11*e - 27, 6/11*e^3 - 40/11*e^2 - 207/11*e + 23, 37/11*e^3 + 98/11*e^2 - 281/11*e - 19, 10/11*e^3 + 14/11*e^2 - 180/11*e - 8, -18/11*e^3 - 34/11*e^2 + 236/11*e - 4, 23/11*e^3 + 52/11*e^2 - 139/11*e - 7, 23/11*e^3 + 63/11*e^2 - 117/11*e - 16, 6/11*e^3 + 26/11*e^2 - 42/11*e - 24, -6/11*e^3 - 26/11*e^2 + 42/11*e - 8, 3*e^3 + 7*e^2 - 25*e - 16, -5/11*e^3 + 4/11*e^2 + 134/11*e - 13, 15/11*e^3 + 32/11*e^2 - 248/11*e - 9, -31/11*e^3 - 72/11*e^2 + 360/11*e + 9, 13/11*e^3 + 16/11*e^2 - 190/11*e - 7, -2*e^3 - 9*e^2 + 7*e + 21, 19/11*e^3 + 9/11*e^2 - 221/11*e + 20, 29/11*e^3 + 111/11*e^2 - 115/11*e - 20, -2/11*e^3 + 39/11*e^2 + 91/11*e - 29, 17/11*e^3 + 26/11*e^2 - 185/11*e + 11, -15/11*e^3 - 43/11*e^2 + 72/11*e - 19, -21/11*e^3 - 47/11*e^2 + 180/11*e - 21, e^3 + 4*e^2 - e - 1, -8/11*e^3 - 75/11*e^2 - 54/11*e + 42, -28/11*e^3 - 114/11*e^2 + 196/11*e + 36, 20/11*e^3 + 94/11*e^2 - 140/11*e - 28, 12/11*e^3 + 85/11*e^2 + 26/11*e - 18, 19/11*e^3 + 53/11*e^2 - 210/11*e - 9, -1/11*e^3 + 3/11*e^2 - 26/11*e - 27, -19/11*e^3 - 53/11*e^2 + 166/11*e - 13, -15/11*e^3 - 43/11*e^2 + 182/11*e + 13, 2*e^3 + 10*e^2 - 38, 18/11*e^3 - 10/11*e^2 - 280/11*e + 14, 14/11*e^3 + 68/11*e^2 - 32/11*e - 14, 2/11*e^3 - 28/11*e^2 - 80/11*e + 22, 1/11*e^3 + 8/11*e^2 + 26/11*e - 21, 7/11*e^3 + 12/11*e^2 - 82/11*e - 19, -26/11*e^3 - 142/11*e^2 + 61/11*e + 45, -13/11*e^3 - 27/11*e^2 + 102/11*e - 26, -15/11*e^3 - 43/11*e^2 + 94/11*e - 20, 14/11*e^3 + 112/11*e^2 + 23/11*e - 45, 26/11*e^3 + 87/11*e^2 - 149/11*e - 29, 14/11*e^3 + 13/11*e^2 - 131/11*e - 3, -43/11*e^3 - 80/11*e^2 + 400/11*e + 3, -37/11*e^3 - 120/11*e^2 + 160/11*e + 25, 7/11*e^3 + 78/11*e^2 + 94/11*e - 27, -7/11*e^3 - 78/11*e^2 - 94/11*e + 35, 10/11*e^3 + 14/11*e^2 - 213/11*e + 10, -30/11*e^3 - 64/11*e^2 + 353/11*e + 20, 16/11*e^3 - 4/11*e^2 - 288/11*e + 24, 4*e^2 + 16*e - 8, 29/11*e^3 + 56/11*e^2 - 203/11*e, 29/11*e^3 + 78/11*e^2 - 247/11*e - 24, 7/11*e^3 + 12/11*e^2 - 5/11*e - 8, 47/11*e^3 + 134/11*e^2 - 329/11*e - 24, -61/11*e^3 - 180/11*e^2 + 537/11*e + 25, 9/11*e^3 + 50/11*e^2 - 173/11*e - 35, -2*e^3 - 11*e^2 + 3*e + 39, -9/11*e^3 - 61/11*e^2 + 107/11*e + 32, 49/11*e^3 + 161/11*e^2 - 387/11*e - 32, 6/11*e^3 + 81/11*e^2 + 79/11*e - 35, -26/11*e^3 - 109/11*e^2 + 127/11*e + 51, 2/11*e^3 + 49/11*e^2 + 41/11*e - 3, 25/11*e^3 + 134/11*e^2 - 32/11*e - 31, -1/11*e^3 - 74/11*e^2 - 136/11*e + 47, -12/11*e^3 - 63/11*e^2 + 40/11*e + 16, -12/11*e^3 + 3/11*e^2 + 172/11*e - 18, -16/11*e^3 - 73/11*e^2 + 24/11*e + 14, 8/11*e^3 + 53/11*e^2 - 12/11*e - 24, -48/11*e^3 - 131/11*e^2 + 380/11*e - 3, -16/11*e^3 - 29/11*e^2 + 156/11*e - 19, -12/11*e^3 - 41/11*e^2 + 40/11*e - 11, -20/11*e^3 - 39/11*e^2 + 96/11*e - 27, 17/11*e^3 + 70/11*e^2 - 174/11*e - 37, -3*e^3 - 10*e^2 + 26*e + 13, 12/11*e^3 + 30/11*e^2 - 117/11*e - 10, 3*e - 4, -52/11*e^3 - 152/11*e^2 + 408/11*e + 20, -3/11*e^3 - 35/11*e^2 - 199/11*e + 8, -53/11*e^3 - 105/11*e^2 + 591/11*e + 8, -12/11*e^3 - 8/11*e^2 + 40/11*e - 20, -28/11*e^3 - 70/11*e^2 + 196/11*e - 16, 39/11*e^3 + 81/11*e^2 - 361/11*e - 12, 25/11*e^3 + 79/11*e^2 - 87/11*e - 20, -82/11*e^3 - 227/11*e^2 + 585/11*e + 19, -54/11*e^3 - 113/11*e^2 + 367/11*e - 15, -17/11*e^3 - 59/11*e^2 + 141/11*e + 24, -4*e^3 - 9*e^2 + 39*e + 9, -12/11*e^3 - 41/11*e^2 - 37/11*e + 3, 9/11*e^3 + 39/11*e^2 - 85/11*e - 4, 9/11*e^3 + 6/11*e^2 - 63/11*e + 47, 27/11*e^3 + 84/11*e^2 - 189/11*e + 23, -6/11*e^3 - 15/11*e^2 + 42/11*e - 30, 28/11*e^3 + 59/11*e^2 - 251/11*e + 27, 18/11*e^3 + 67/11*e^2 - 93/11*e - 5, 6/11*e^3 - 7/11*e^2 - 75/11*e + 21, 20/11*e^3 + 61/11*e^2 - 85/11*e + 21, 15/11*e^3 + 54/11*e^2 + 38/11*e - 1, 49/11*e^3 + 106/11*e^2 - 486/11*e - 3, -20/11*e^3 - 17/11*e^2 + 228/11*e - 17, -24/11*e^3 - 93/11*e^2 + 80/11*e + 15, 69/11*e^3 + 211/11*e^2 - 439/11*e - 38, 43/11*e^3 + 69/11*e^2 - 345/11*e + 10, -3/11*e^3 + 20/11*e^2 + 153/11*e + 5, -43/11*e^3 - 113/11*e^2 + 378/11*e + 43, -9/11*e^3 - 17/11*e^2 - 14/11*e + 21, 15/11*e^3 + 10/11*e^2 - 237/11*e + 21, -37/11*e^3 - 120/11*e^2 + 402/11*e + 29, 45/11*e^3 + 140/11*e^2 - 458/11*e - 37, 16/11*e^3 + 62/11*e^2 - 112/11*e - 22, -8/11*e^3 - 42/11*e^2 + 56/11*e + 10, 10/11*e^3 + 58/11*e^2 - 114/11*e - 20, -42/11*e^3 - 138/11*e^2 + 338/11*e + 36, -37/11*e^3 - 109/11*e^2 + 314/11*e + 3, 1/11*e^3 + 19/11*e^2 - 62/11*e - 31, 19/11*e^3 - 2/11*e^2 - 287/11*e + 20, 17/11*e^3 + 92/11*e^2 + 35/11*e - 24, -71/11*e^3 - 205/11*e^2 + 398/11*e + 41, -7*e^3 - 15*e^2 + 58*e + 19, -38/11*e^3 - 62/11*e^2 + 343/11*e + 25, -46/11*e^3 - 148/11*e^2 + 245/11*e + 59, -35/11*e^3 - 104/11*e^2 + 322/11*e + 39, e^3 + 4*e^2 - 14*e + 1, -7/11*e^3 - 45/11*e^2 + 5/11*e + 20, 7/11*e^3 + 45/11*e^2 - 5/11*e - 12, -65/11*e^3 - 223/11*e^2 + 444/11*e + 48, -35/11*e^3 - 71/11*e^2 + 366/11*e - 20, -9/11*e^3 - 39/11*e^2 - 58/11*e - 18, -3/11*e^3 + 53/11*e^2 + 32/11*e - 38, 8/11*e^3 + 64/11*e^2 - 56/11*e - 48, -40/11*e^3 - 144/11*e^2 + 280/11*e + 16, -16/11*e^3 - 139/11*e^2 - 42/11*e + 69, 4/11*e^3 + 65/11*e^2 + 38/11*e - 59, -32/11*e^3 - 135/11*e^2 + 158/11*e + 9, 36/11*e^3 + 189/11*e^2 - 98/11*e - 47, -7/11*e^3 - 34/11*e^2 + 104/11*e + 15, 31/11*e^3 + 94/11*e^2 - 272/11*e - 19, -7/11*e^3 + 21/11*e^2 + 93/11*e - 28, 4*e^3 + 16*e^2 - 20*e - 64, 4/11*e^3 - 56/11*e^2 - 116/11*e + 16, -3*e^3 - 11*e^2 + 17*e + 20, -46/11*e^3 - 159/11*e^2 + 322/11*e + 50, 24/11*e^3 + 82/11*e^2 - 168/11*e - 40, -2*e^2 - 8, 2/11*e^3 + 49/11*e^2 - 14/11*e - 14, e^3 + 7*e^2 + e - 20, -e^3 - 7*e^2 - e + 36, 28/11*e^3 + 92/11*e^2 - 141/11*e + 18, 24/11*e^3 + 38/11*e^2 - 223/11*e + 40, -19/11*e^3 - 64/11*e^2 + 210/11*e + 3, -101/11*e^3 - 258/11*e^2 + 718/11*e + 35, -91/11*e^3 - 222/11*e^2 + 626/11*e + 25, 27/11*e^3 + 84/11*e^2 - 266/11*e - 35, -12/11*e^3 - 19/11*e^2 + 172/11*e + 30, 8/11*e^3 + 9/11*e^2 - 144/11*e + 30, 24/11*e^3 + 49/11*e^2 - 168/11*e - 4, 36/11*e^3 + 101/11*e^2 - 252/11*e - 20, -16/11*e^3 - 150/11*e^2 - 75/11*e + 74, 36/11*e^3 + 200/11*e^2 - 65/11*e - 52, e^3 + 8*e^2 + 4*e - 23, -e^3 - 8*e^2 - 4*e + 43, 24/11*e^3 + 60/11*e^2 - 58/11*e - 22, 64/11*e^3 + 160/11*e^2 - 558/11*e - 42, -5/11*e^3 - 29/11*e^2 + 189/11*e + 30, -20/11*e^3 - 94/11*e^2 + 162/11*e + 26, 36/11*e^3 + 134/11*e^2 - 274/11*e - 42, 69/11*e^3 + 189/11*e^2 - 637/11*e - 22, -4/11*e^3 - 21/11*e^2 + 105/11*e + 3, 43/11*e^3 + 157/11*e^2 - 213/11*e - 54, 21/11*e^3 + 3/11*e^2 - 235/11*e + 2, 36/11*e^3 + 101/11*e^2 - 329/11*e - 27, 14/11*e^3 + 46/11*e^2 - 274/11*e - 24, -92/11*e^3 - 230/11*e^2 + 710/11*e + 26, -68/11*e^3 - 170/11*e^2 + 410/11*e + 14, -62/11*e^3 - 166/11*e^2 + 610/11*e + 24, 6/11*e^3 + 92/11*e^2 + 277/11*e - 45, 38/11*e^3 + 18/11*e^2 - 585/11*e + 9, 34/11*e^3 + 107/11*e^2 - 315/11*e - 43, -18/11*e^3 - 67/11*e^2 + 203/11*e + 3, -3/11*e^3 - 13/11*e^2 - 56/11*e + 35, 20/11*e^3 + 116/11*e^2 - 8/11*e - 26, -4/11*e^3 - 76/11*e^2 - 104/11*e + 46, -25/11*e^3 - 57/11*e^2 + 252/11*e + 41, -8/11*e^3 - 31/11*e^2 + 177/11*e + 43, 40/11*e^3 + 100/11*e^2 - 236/11*e - 44, 56/11*e^3 + 140/11*e^2 - 436/11*e - 52, 48/11*e^3 + 131/11*e^2 - 457/11*e + 5, 40/11*e^3 + 166/11*e^2 - 104/11*e - 40, -90/11*e^3 - 203/11*e^2 + 828/11*e + 28, -42/11*e^3 - 127/11*e^2 + 96/11*e + 24, 32/11*e^3 + 14/11*e^2 - 400/11*e + 24, -38/11*e^3 - 150/11*e^2 + 68/11*e + 26, -50/11*e^3 - 70/11*e^2 + 548/11*e - 18, 41/11*e^3 + 119/11*e^2 - 287/11*e - 36, 23/11*e^3 + 41/11*e^2 - 161/11*e - 12, -15/11*e^3 - 76/11*e^2 - 49/11*e + 55, -29/11*e^3 - 34/11*e^2 + 357/11*e + 27, -89/11*e^3 - 239/11*e^2 + 678/11*e + 39, -51/11*e^3 - 111/11*e^2 + 302/11*e + 5, 12, 60/11*e^3 + 150/11*e^2 - 420/11*e - 34, 82/11*e^3 + 205/11*e^2 - 629/11*e - 9, 62/11*e^3 + 155/11*e^2 - 379/11*e + 1, -26/11*e^3 + 1/11*e^2 + 358/11*e + 7, -34/11*e^3 - 151/11*e^2 + 62/11*e + 71, -17/11*e^3 - 81/11*e^2 + 284/11*e + 49, 31/11*e^3 + 72/11*e^2 - 327/11*e - 35, -3/11*e^3 - 2/11*e^2 + 131/11*e - 23, 85/11*e^3 + 251/11*e^2 - 760/11*e - 37, 47/11*e^3 + 123/11*e^2 - 395/11*e - 8, 17/11*e^3 + 37/11*e^2 - 53/11*e + 12, -10/11*e^3 - 113/11*e^2 - 7/11*e + 71, 58/11*e^3 + 233/11*e^2 - 329/11*e - 43, 51/11*e^3 + 177/11*e^2 - 236/11*e - 43, 41/11*e^3 + 53/11*e^2 - 408/11*e + 7, -13/11*e^3 + 72/11*e^2 + 344/11*e - 15, -32/11*e^3 - 190/11*e^2 + 70/11*e + 54, 32/11*e^3 + 190/11*e^2 - 70/11*e - 78, -35/11*e^3 - 192/11*e^2 - 8/11*e + 91, 14/11*e^3 - 9/11*e^2 - 65/11*e + 5, 74/11*e^3 + 229/11*e^2 - 551/11*e - 65, -10/11*e^3 - 14/11*e^2 + 213/11*e + 13, 30/11*e^3 + 64/11*e^2 - 353/11*e + 3, -74/11*e^3 - 152/11*e^2 + 573/11*e - 2, -90/11*e^3 - 258/11*e^2 + 575/11*e + 36, -1/11*e^3 + 69/11*e^2 + 7/11*e - 68, -30/11*e^3 - 42/11*e^2 + 210/11*e - 12, -6*e^3 - 18*e^2 + 42*e + 36, -79/11*e^3 - 269/11*e^2 + 553/11*e + 36, -38/11*e^3 - 84/11*e^2 + 288/11*e - 22, -42/11*e^3 - 116/11*e^2 + 272/11*e - 10, 6*e^2 + 22*e - 38, 16/11*e^3 - 26/11*e^2 - 354/11*e + 14, 85/11*e^3 + 174/11*e^2 - 606/11*e + 17, 23/11*e^3 + 41/11*e^2 - 359/11*e - 36, -31/11*e^3 - 61/11*e^2 + 415/11*e - 24, 123/11*e^3 + 346/11*e^2 - 850/11*e - 37];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {16, 2, 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;