/*
  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 | {9, 3, -w + 3}>;

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^8 - 30*x^6 + 236*x^4 - 584*x^2 + 256;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, -9/352*e^7 + 125/176*e^5 - 397/88*e^3 + 277/44*e, -1, 9/176*e^6 - 57/44*e^4 + 265/44*e^2 - 34/11, e, 9/176*e^6 - 57/44*e^4 + 265/44*e^2 - 23/11, -1/44*e^7 + 29/44*e^5 - 207/44*e^3 + 205/22*e, 9/352*e^7 - 125/176*e^5 + 419/88*e^3 - 475/44*e, -1/88*e^7 + 29/88*e^5 - 109/44*e^3 + 54/11*e, -5/352*e^7 + 89/176*e^5 - 443/88*e^3 + 567/44*e, -13/176*e^7 + 183/88*e^5 - 151/11*e^3 + 230/11*e, -7/352*e^7 + 85/176*e^5 - 189/88*e^3 + 103/44*e, 3/176*e^7 - 19/44*e^5 + 81/44*e^3 + 40/11*e, 7/176*e^7 - 12/11*e^5 + 299/44*e^3 - 123/11*e, -5/88*e^6 + 14/11*e^4 - 91/22*e^2 + 6/11, -1/88*e^6 + 9/44*e^4 + 14/11*e^2 - 100/11, -1/44*e^6 + 29/44*e^4 - 109/22*e^2 + 174/11, 7/176*e^7 - 85/88*e^5 + 167/44*e^3 + 73/22*e, -1/44*e^7 + 9/22*e^5 + 17/11*e^3 - 200/11*e, -3/88*e^6 + 49/44*e^4 - 101/11*e^2 + 140/11, 13/176*e^6 - 75/44*e^4 + 241/44*e^2 + 12/11, -19/176*e^7 + 31/11*e^5 - 164/11*e^3 + 333/22*e, 1/32*e^7 - 13/16*e^5 + 31/8*e^3 + 13/4*e, 1/22*e^6 - 69/44*e^4 + 317/22*e^2 - 260/11, 7/88*e^6 - 24/11*e^4 + 277/22*e^2 - 70/11, -9/88*e^6 + 125/44*e^4 - 193/11*e^2 + 266/11, -e^2 + 18, -9/88*e^6 + 103/44*e^4 - 83/11*e^2 - 20/11, 23/176*e^6 - 153/44*e^4 + 819/44*e^2 - 82/11, -7/44*e^6 + 48/11*e^4 - 288/11*e^2 + 338/11, -1/88*e^7 + 29/88*e^5 - 30/11*e^3 + 207/22*e, 13/352*e^7 - 161/176*e^5 + 329/88*e^3 + 299/44*e, 7/352*e^7 - 107/176*e^5 + 409/88*e^3 - 169/44*e, -27/352*e^7 + 353/176*e^5 - 993/88*e^3 + 831/44*e, 7/352*e^7 - 107/176*e^5 + 409/88*e^3 - 81/44*e, 53/352*e^7 - 697/176*e^5 + 1915/88*e^3 - 1091/44*e, 17/352*e^7 - 241/176*e^5 + 811/88*e^3 - 599/44*e, 1/352*e^7 - 53/176*e^5 + 533/88*e^3 - 989/44*e, -35/352*e^7 + 469/176*e^5 - 1319/88*e^3 + 713/44*e, 21/352*e^7 - 255/176*e^5 + 479/88*e^3 + 43/44*e, 3/88*e^6 - 49/44*e^4 + 123/11*e^2 - 294/11, 1/88*e^6 + 1/22*e^4 - 105/22*e^2 + 188/11, 37/176*e^6 - 249/44*e^4 + 1417/44*e^2 - 350/11, 17/88*e^6 - 219/44*e^4 + 268/11*e^2 - 236/11, -5/176*e^6 + 7/11*e^4 - 135/44*e^2 + 14/11, 19/88*e^6 - 259/44*e^4 + 383/11*e^2 - 410/11, -1/44*e^6 + 29/44*e^4 - 87/22*e^2 - 24/11, -59/176*e^6 + 381/44*e^4 - 2011/44*e^2 + 570/11, -1/88*e^6 - 1/22*e^4 + 105/22*e^2 + 54/11, -9/176*e^7 + 103/88*e^5 - 133/44*e^3 - 295/22*e, -31/352*e^7 + 345/176*e^5 - 331/88*e^3 - 999/44*e, 1/88*e^6 - 9/44*e^4 + 19/11*e^2 - 120/11, -25/352*e^7 + 357/176*e^5 - 1159/88*e^3 + 591/44*e, 45/352*e^7 - 603/176*e^5 + 1699/88*e^3 - 681/44*e, 47/352*e^7 - 599/176*e^5 + 1467/88*e^3 - 547/44*e, 3/22*e^7 - 315/88*e^5 + 423/22*e^3 - 449/22*e, -9/88*e^6 + 57/22*e^4 - 243/22*e^2 - 64/11, -1/22*e^6 + 47/44*e^4 - 31/22*e^2 - 136/11, 1/2*e^4 - 8*e^2 - 6, 1/44*e^6 - 7/44*e^4 - 155/22*e^2 + 266/11, -1/176*e^6 - 23/44*e^4 + 699/44*e^2 - 490/11, -27/352*e^7 + 353/176*e^5 - 1037/88*e^3 + 1095/44*e, -5/88*e^7 + 67/44*e^5 - 413/44*e^3 + 397/22*e, 35/352*e^7 - 447/176*e^5 + 1055/88*e^3 - 339/44*e, -19/352*e^7 + 215/176*e^5 - 227/88*e^3 - 877/44*e, 3/32*e^7 - 41/16*e^5 + 127/8*e^3 - 93/4*e, 23/352*e^7 - 295/176*e^5 + 643/88*e^3 + 221/44*e, -29/88*e^6 + 101/11*e^4 - 1267/22*e^2 + 708/11, -1/16*e^6 + 3/2*e^4 - 27/4*e^2 + 12, 5/176*e^7 - 7/11*e^5 + 47/44*e^3 + 206/11*e, 7/352*e^7 - 107/176*e^5 + 475/88*e^3 - 1203/44*e, 7/176*e^6 - 35/22*e^4 + 717/44*e^2 - 244/11, 17/352*e^7 - 285/176*e^5 + 1339/88*e^3 - 1523/44*e, 15/88*e^7 - 391/88*e^5 + 1019/44*e^3 - 205/11*e, 19/44*e^6 - 124/11*e^4 + 678/11*e^2 - 754/11, -1/8*e^6 + 11/4*e^4 - 7*e^2 - 6, -1/8*e^7 + 31/8*e^5 - 123/4*e^3 + 58*e, -1/8*e^7 + 27/8*e^5 - 79/4*e^3 + 29*e, -1/4*e^4 + 7/2*e^2 + 2, 5/352*e^7 - 155/176*e^5 + 1279/88*e^3 - 1997/44*e, -1/8*e^6 + 11/4*e^4 - 5*e^2 - 20, -15/44*e^6 + 391/44*e^4 - 997/22*e^2 + 212/11, -47/352*e^7 + 643/176*e^5 - 2017/88*e^3 + 1405/44*e, 1/22*e^6 - 47/44*e^4 + 119/22*e^2 - 260/11, -19/88*e^6 + 62/11*e^4 - 645/22*e^2 + 124/11, 39/176*e^6 - 59/11*e^4 + 877/44*e^2 - 52/11, -7/16*e^6 + 45/4*e^4 - 227/4*e^2 + 52, 1/8*e^6 - 7/2*e^4 + 41/2*e^2 + 10, -17/88*e^6 + 241/44*e^4 - 389/11*e^2 + 522/11, -47/176*e^6 + 305/44*e^4 - 1687/44*e^2 + 620/11, -1/88*e^6 - 13/44*e^4 + 113/11*e^2 - 232/11, 67/352*e^7 - 845/176*e^5 + 1941/88*e^3 - 351/44*e, 1/352*e^7 + 101/176*e^5 - 1337/88*e^3 + 2443/44*e, -3/88*e^7 + 65/88*e^5 - 85/44*e^3 - 3/11*e, 17/88*e^7 - 493/88*e^5 + 1765/44*e^3 - 852/11*e, -5/176*e^7 + 23/88*e^5 + 191/22*e^3 - 624/11*e, -19/88*e^6 + 113/22*e^4 - 359/22*e^2 - 162/11, -25/88*e^6 + 313/44*e^4 - 365/11*e^2 + 206/11, 1/176*e^7 + 35/88*e^5 - 128/11*e^3 + 435/11*e, 7/176*e^7 - 107/88*e^5 + 431/44*e^3 - 455/22*e, 21/352*e^7 - 299/176*e^5 + 1073/88*e^3 - 1211/44*e, 9/352*e^7 - 103/176*e^5 + 111/88*e^3 + 559/44*e, -3/176*e^7 + 27/88*e^5 + 31/22*e^3 - 172/11*e, 1/11*e^7 - 199/88*e^5 + 421/44*e^3 + 118/11*e, 5/44*e^7 - 257/88*e^5 + 639/44*e^3 - 45/11*e, 17/88*e^6 - 115/22*e^4 + 679/22*e^2 - 104/11, 3/16*e^7 - 39/8*e^5 + 101/4*e^3 - 49/2*e, 5/16*e^7 - 67/8*e^5 + 193/4*e^3 - 121/2*e, 3/11*e^6 - 337/44*e^4 + 1055/22*e^2 - 680/11, -7/44*e^7 + 48/11*e^5 - 1207/44*e^3 + 863/22*e, -37/352*e^7 + 377/176*e^5 - 31/88*e^3 - 1797/44*e, -29/176*e^6 + 213/44*e^4 - 1553/44*e^2 + 728/11, 1/11*e^6 - 105/44*e^4 + 249/22*e^2 + 118/11, 23/352*e^7 - 317/176*e^5 + 907/88*e^3 - 329/44*e, -41/352*e^7 + 611/176*e^5 - 2229/88*e^3 + 1763/44*e, 15/88*e^6 - 53/11*e^4 + 669/22*e^2 - 348/11, -6/11*e^6 + 641/44*e^4 - 1791/22*e^2 + 920/11, -1/11*e^6 + 83/44*e^4 - 51/22*e^2 + 36/11, 63/176*e^6 - 97/11*e^4 + 1745/44*e^2 - 348/11, -1/8*e^6 + 3*e^4 - 23/2*e^2 + 14, -5/32*e^7 + 65/16*e^5 - 173/8*e^3 + 105/4*e, 19/176*e^7 - 281/88*e^5 + 1085/44*e^3 - 1191/22*e, 17/44*e^6 - 104/11*e^4 + 448/11*e^2 - 120/11, -41/176*e^7 + 567/88*e^5 - 911/22*e^3 + 744/11*e, 15/44*e^7 - 815/88*e^5 + 2489/44*e^3 - 993/11*e, 21/352*e^7 - 343/176*e^5 + 1491/88*e^3 - 1629/44*e, 129/352*e^7 - 1777/176*e^5 + 5617/88*e^3 - 4293/44*e, -75/352*e^7 + 1005/176*e^5 - 2927/88*e^3 + 1817/44*e, -5/32*e^7 + 61/16*e^5 - 123/8*e^3 - 37/4*e, 7/88*e^6 - 107/44*e^4 + 188/11*e^2 - 26/11, 15/88*e^6 - 95/22*e^4 + 449/22*e^2 + 312/11, 7/44*e^6 - 85/22*e^4 + 145/11*e^2 + 212/11, -7/176*e^6 + 35/22*e^4 - 717/44*e^2 + 200/11, -19/176*e^7 + 135/44*e^5 - 449/22*e^3 + 575/22*e, -1/88*e^6 - 1/22*e^4 + 105/22*e^2 + 10/11, -9/88*e^6 + 81/44*e^4 + 16/11*e^2 - 328/11, 63/352*e^7 - 809/176*e^5 + 2053/88*e^3 - 839/44*e, 1/176*e^7 - 53/88*e^5 + 283/22*e^3 - 654/11*e, 3/11*e^6 - 271/44*e^4 + 395/22*e^2 + 222/11, -1/11*e^6 + 18/11*e^4 + 13/11*e^2 - 338/11, -9/176*e^7 + 59/88*e^5 + 181/22*e^3 - 516/11*e, 17/44*e^6 - 197/22*e^4 + 327/11*e^2 + 298/11, -1/8*e^6 + 3*e^4 - 27/2*e^2 + 12, 3/88*e^7 - 43/88*e^5 - 139/22*e^3 + 1271/22*e, -3/32*e^7 + 49/16*e^5 - 219/8*e^3 + 277/4*e, -3/11*e^6 + 76/11*e^4 - 357/11*e^2 + 196/11, -2/11*e^6 + 58/11*e^4 - 414/11*e^2 + 600/11, 13/176*e^6 - 27/11*e^4 + 967/44*e^2 - 208/11, -3/88*e^6 + 27/44*e^4 - 46/11*e^2 + 580/11, 23/88*e^6 - 295/44*e^4 + 338/11*e^2 - 230/11, 1/8*e^7 - 33/8*e^5 + 38*e^3 - 179/2*e, 7/352*e^7 - 107/176*e^5 + 431/88*e^3 - 719/44*e, 9/176*e^7 - 57/44*e^5 + 243/44*e^3 - 56/11*e, 29/176*e^7 - 415/88*e^5 + 1421/44*e^3 - 1225/22*e, -3/8*e^6 + 41/4*e^4 - 61*e^2 + 84, 9/176*e^7 - 169/88*e^5 + 925/44*e^3 - 1311/22*e, 51/352*e^7 - 745/176*e^5 + 2631/88*e^3 - 1841/44*e, -67/176*e^6 + 225/22*e^4 - 2513/44*e^2 + 522/11, 17/352*e^7 - 153/176*e^5 - 201/88*e^3 + 677/44*e, -83/176*e^7 + 1121/88*e^5 - 838/11*e^3 + 1218/11*e, 3/4*e^6 - 79/4*e^4 + 211/2*e^2 - 102, 63/352*e^7 - 699/176*e^5 + 623/88*e^3 + 2197/44*e, -13/176*e^7 + 43/22*e^5 - 140/11*e^3 + 801/22*e, -15/176*e^7 + 157/88*e^5 - 16/11*e^3 - 310/11*e, -63/352*e^7 + 985/176*e^5 - 4099/88*e^3 + 4293/44*e, 57/352*e^7 - 887/176*e^5 + 3673/88*e^3 - 4211/44*e, 7/176*e^7 - 151/88*e^5 + 937/44*e^3 - 1489/22*e, -8/11*e^6 + 829/44*e^4 - 2135/22*e^2 + 838/11, -1/44*e^6 + 31/22*e^4 - 225/11*e^2 + 614/11, 43/88*e^6 - 563/44*e^4 + 729/11*e^2 - 452/11, -31/88*e^6 + 433/44*e^4 - 666/11*e^2 + 948/11, 61/176*e^7 - 103/11*e^5 + 1203/22*e^3 - 1589/22*e, -61/352*e^7 + 813/176*e^5 - 2219/88*e^3 + 555/44*e, -37/352*e^7 + 641/176*e^5 - 3067/88*e^3 + 3615/44*e, -17/176*e^7 + 241/88*e^5 - 200/11*e^3 + 426/11*e, -5/44*e^6 + 45/22*e^4 + 96/11*e^2 - 648/11, 6/11*e^6 - 619/44*e^4 + 1571/22*e^2 - 788/11, 79/176*e^6 - 515/44*e^4 + 2595/44*e^2 - 296/11, -5/22*e^6 + 123/22*e^4 - 237/11*e^2 - 174/11, 37/352*e^7 - 487/176*e^5 + 1527/88*e^3 - 2097/44*e, -21/176*e^7 + 155/44*e^5 - 282/11*e^3 + 749/22*e, 3/8*e^6 - 21/2*e^4 + 133/2*e^2 - 94, -15/176*e^6 + 21/11*e^4 - 405/44*e^2 + 504/11, 49/352*e^7 - 551/176*e^5 + 641/88*e^3 + 1193/44*e, 111/352*e^7 - 1549/176*e^5 + 4955/88*e^3 - 3673/44*e, -21/176*e^7 + 321/88*e^5 - 685/22*e^3 + 1018/11*e, 69/352*e^7 - 929/176*e^5 + 2787/88*e^3 - 2131/44*e, 5/176*e^7 - 25/22*e^5 + 663/44*e^3 - 806/11*e, -19/176*e^7 + 193/88*e^5 - 29/44*e^3 - 679/22*e, 29/352*e^7 - 349/176*e^5 + 651/88*e^3 + 205/44*e, -109/352*e^7 + 1487/176*e^5 - 4571/88*e^3 + 3741/44*e, -35/88*e^6 + 229/22*e^4 - 1209/22*e^2 + 724/11, 3/8*e^6 - 37/4*e^4 + 36*e^2, -73/352*e^7 + 1009/176*e^5 - 3269/88*e^3 + 2677/44*e, -7/88*e^6 + 85/44*e^4 - 78/11*e^2 + 378/11, 3/11*e^6 - 76/11*e^4 + 324/11*e^2 - 86/11, -131/352*e^7 + 1795/176*e^5 - 5451/88*e^3 + 3543/44*e, -3/8*e^5 + 29/4*e^3 - 7*e, -5/352*e^7 + 45/176*e^5 - 91/88*e^3 + 567/44*e, 19/88*e^6 - 51/11*e^4 + 227/22*e^2 + 118/11, 7/32*e^7 - 89/16*e^5 + 213/8*e^3 - 75/4*e, 23/352*e^7 - 427/176*e^5 + 2425/88*e^3 - 4069/44*e, 43/88*e^6 - 563/44*e^4 + 762/11*e^2 - 826/11, 1/352*e^7 + 79/176*e^5 - 1073/88*e^3 + 1717/44*e, -3/352*e^7 + 71/176*e^5 - 477/88*e^3 + 305/44*e, -41/176*e^6 + 289/44*e^4 - 1877/44*e^2 + 744/11, -41/88*e^6 + 545/44*e^4 - 812/11*e^2 + 1114/11, -15/22*e^6 + 369/22*e^4 - 788/11*e^2 + 160/11, 21/44*e^6 - 133/11*e^4 + 655/11*e^2 - 266/11, -19/352*e^7 + 391/176*e^5 - 2537/88*e^3 + 4777/44*e, -51/176*e^6 + 78/11*e^4 - 1289/44*e^2 + 420/11, -43/88*e^6 + 541/44*e^4 - 630/11*e^2 + 518/11, 7/176*e^7 - 37/44*e^5 - 16/11*e^3 + 1041/22*e, 61/352*e^7 - 791/176*e^5 + 2021/88*e^3 - 775/44*e, 1/44*e^7 - 135/88*e^5 + 549/22*e^3 - 1855/22*e, -65/352*e^7 + 1003/176*e^5 - 3977/88*e^3 + 4027/44*e, 6/11*e^7 - 641/44*e^5 + 923/11*e^3 - 1195/11*e, -125/352*e^7 + 1653/176*e^5 - 4519/88*e^3 + 2339/44*e, 17/44*e^6 - 219/22*e^4 + 514/11*e^2 - 384/11, -45/88*e^6 + 625/44*e^4 - 987/11*e^2 + 1088/11, 7/32*e^7 - 91/16*e^5 + 225/8*e^3 - 53/4*e, -5/22*e^7 + 67/11*e^5 - 1509/44*e^3 + 763/22*e, 21/352*e^7 - 431/176*e^5 + 2591/88*e^3 - 4049/44*e, 47/176*e^6 - 79/11*e^4 + 1885/44*e^2 - 664/11, -7/44*e^6 + 137/44*e^4 + 51/22*e^2 - 674/11, -3/176*e^7 + 41/44*e^5 - 653/44*e^3 + 653/11*e, 21/176*e^7 - 36/11*e^5 + 941/44*e^3 - 424/11*e, 63/176*e^6 - 355/44*e^4 + 843/44*e^2 + 576/11, -19/44*e^6 + 463/44*e^4 - 905/22*e^2 - 214/11, 137/176*e^6 - 897/44*e^4 + 4821/44*e^2 - 1026/11, -3/16*e^6 + 11/2*e^4 - 157/4*e^2 + 54, 41/352*e^7 - 413/176*e^5 - 191/88*e^3 + 2923/44*e, 4/11*e^6 - 365/44*e^4 + 545/22*e^2 + 340/11, -21/88*e^6 + 321/44*e^4 - 630/11*e^2 + 1134/11, -47/352*e^7 + 511/176*e^5 - 411/88*e^3 - 1257/44*e, -1/88*e^7 - 1/22*e^5 + 47/11*e^3 - 1/11*e, 13/44*e^6 - 311/44*e^4 + 493/22*e^2 + 620/11, -71/176*e^6 + 117/11*e^4 - 2357/44*e^2 + 476/11, -105/352*e^7 + 1473/176*e^5 - 4969/88*e^3 + 4801/44*e, 2/11*e^6 - 47/11*e^4 + 172/11*e^2 + 148/11, -1/8*e^7 + 35/8*e^5 - 173/4*e^3 + 106*e, -21/352*e^7 + 409/176*e^5 - 2283/88*e^3 + 3323/44*e, -23/88*e^6 + 60/11*e^4 - 137/22*e^2 - 606/11, 23/88*e^7 - 645/88*e^5 + 1017/22*e^3 - 1153/22*e, 37/88*e^7 - 509/44*e^5 + 1615/22*e^3 - 1305/11*e, 41/352*e^7 - 545/176*e^5 + 1437/88*e^3 + 107/44*e, 17/88*e^7 - 115/22*e^5 + 356/11*e^3 - 764/11*e, 25/176*e^7 - 81/22*e^5 + 409/22*e^3 - 129/22*e, 107/352*e^7 - 1403/176*e^5 + 3725/88*e^3 - 2005/44*e, 61/352*e^7 - 879/176*e^5 + 3275/88*e^3 - 4625/44*e, -2/11*e^7 + 365/88*e^5 - 457/44*e^3 - 346/11*e, -39/88*e^6 + 247/22*e^4 - 1207/22*e^2 + 434/11, -63/88*e^6 + 831/44*e^4 - 1120/11*e^2 + 938/11, 17/44*e^6 - 219/22*e^4 + 569/11*e^2 - 692/11, -13/88*e^6 + 97/22*e^4 - 725/22*e^2 + 724/11, -43/352*e^7 + 519/176*e^5 - 897/88*e^3 - 1275/44*e, -5/22*e^6 + 145/22*e^4 - 512/11*e^2 + 486/11, 3/16*e^6 - 5*e^4 + 121/4*e^2 - 2, 1/44*e^7 - 7/44*e^5 - 277/44*e^3 + 543/22*e, -9/88*e^6 + 81/44*e^4 + 27/11*e^2 + 134/11, -35/176*e^6 + 185/44*e^4 - 263/44*e^2 - 166/11, 125/352*e^7 - 1807/176*e^5 + 6345/88*e^3 - 5639/44*e, 115/352*e^7 - 1541/176*e^5 + 4491/88*e^3 - 3009/44*e, 3/32*e^7 - 49/16*e^5 + 213/8*e^3 - 239/4*e, 5/352*e^7 + 21/176*e^5 - 833/88*e^3 + 2007/44*e, -113/176*e^6 + 178/11*e^4 - 3447/44*e^2 + 400/11, -7/44*e^6 + 137/44*e^4 + 51/22*e^2 - 278/11, -3/44*e^6 + 87/44*e^4 - 217/22*e^2 - 138/11, 25/44*e^6 - 615/44*e^4 + 1295/22*e^2 - 258/11, -75/88*e^6 + 939/44*e^4 - 1117/11*e^2 + 684/11, 23/44*e^6 - 557/44*e^4 + 1143/22*e^2 - 64/11, 21/44*e^6 - 255/22*e^4 + 490/11*e^2 + 394/11, 39/88*e^6 - 129/11*e^4 + 1383/22*e^2 - 786/11, -9/22*e^6 + 423/44*e^4 - 675/22*e^2 - 454/11, 151/352*e^7 - 1997/176*e^5 + 5573/88*e^3 - 3391/44*e, -11/32*e^7 + 159/16*e^5 - 561/8*e^3 + 509/4*e, -3/8*e^6 + 37/4*e^4 - 34*e^2 - 34, -5/8*e^6 + 33/2*e^4 - 173/2*e^2 + 88, -17/88*e^6 + 175/44*e^4 - 92/11*e^2 - 28/11, -1/32*e^7 + 23/16*e^5 - 163/8*e^3 + 305/4*e, 5/16*e^7 - 67/8*e^5 + 99/2*e^3 - 78*e, -13/44*e^6 + 355/44*e^4 - 1043/22*e^2 + 172/11, -171/176*e^6 + 1105/44*e^4 - 5651/44*e^2 + 1306/11, 47/352*e^7 - 599/176*e^5 + 1599/88*e^3 - 1691/44*e, 17/352*e^7 - 307/176*e^5 + 1603/88*e^3 - 2117/44*e, -1/32*e^7 + 9/16*e^5 + 29/8*e^3 - 157/4*e, 105/352*e^7 - 1363/176*e^5 + 3517/88*e^3 - 1787/44*e, -29/88*e^7 + 797/88*e^5 - 617/11*e^3 + 1735/22*e, -61/352*e^7 + 769/176*e^5 - 1801/88*e^3 + 181/44*e, -17/352*e^7 + 197/176*e^5 - 305/88*e^3 + 621/44*e, 3/22*e^7 - 381/88*e^5 + 819/22*e^3 - 2143/22*e, -45/176*e^6 + 63/11*e^4 - 511/44*e^2 - 402/11, -7/22*e^6 + 329/44*e^4 - 415/22*e^2 - 842/11, 25/176*e^7 - 269/88*e^5 + 213/44*e^3 + 795/22*e, 9/176*e^6 - 17/11*e^4 + 375/44*e^2 + 98/11, -49/176*e^6 + 325/44*e^4 - 1917/44*e^2 + 124/11, -29/88*e^7 + 731/88*e^5 - 419/11*e^3 + 327/22*e, -1/44*e^6 + 73/44*e^4 - 571/22*e^2 + 218/11, 9/88*e^6 - 147/44*e^4 + 281/11*e^2 - 640/11, -17/176*e^6 + 115/44*e^4 - 1009/44*e^2 + 976/11, 79/88*e^6 - 252/11*e^4 + 2529/22*e^2 - 1208/11, -1/32*e^7 + 13/16*e^5 - 41/8*e^3 + 57/4*e, 21/176*e^6 - 155/44*e^4 + 1293/44*e^2 - 974/11, 5/44*e^6 - 79/44*e^4 - 247/22*e^2 + 912/11, 43/176*e^7 - 673/88*e^5 + 2811/44*e^3 - 3125/22*e];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {9, 3, -w + 3}>] := 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;