/*
  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![41, 9, -14, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {16, 4, -1/6*w^3 + 1/3*w + 5/6}>;

primesArray := [
[4, 2, 1/6*w^3 - 7/3*w - 17/6],
[4, 2, w - 3],
[5, 5, -1/6*w^3 + w^2 + 1/3*w - 25/6],
[9, 3, 1/6*w^3 - 7/3*w + 13/6],
[9, 3, -w - 2],
[11, 11, -1/6*w^3 + 7/3*w + 11/6],
[11, 11, -w + 2],
[41, 41, -w],
[41, 41, 1/6*w^3 - 7/3*w + 1/6],
[59, 59, -1/6*w^3 + 1/3*w + 23/6],
[59, 59, -1/3*w^3 + 11/3*w + 11/3],
[61, 61, -5/3*w^3 - 3*w^2 + 46/3*w + 79/3],
[61, 61, 5/6*w^3 + w^2 - 23/3*w - 55/6],
[61, 61, w^3 + 2*w^2 - 9*w - 17],
[61, 61, -1/6*w^3 + 10/3*w + 35/6],
[71, 71, -1/2*w^3 + 2*w^2 + 4*w - 33/2],
[71, 71, 1/6*w^3 - w^2 - 4/3*w + 67/6],
[79, 79, 1/2*w^3 - 3*w + 1/2],
[79, 79, -2/3*w^3 + 19/3*w - 2/3],
[89, 89, 1/2*w^3 + w^2 - 5*w - 15/2],
[89, 89, 1/3*w^3 - 11/3*w + 7/3],
[101, 101, -5/6*w^3 - w^2 + 23/3*w + 61/6],
[101, 101, 1/2*w^3 - w^2 - 3*w + 9/2],
[109, 109, 1/2*w^3 - w^2 - 4*w + 11/2],
[109, 109, -2/3*w^3 - w^2 + 16/3*w + 28/3],
[121, 11, -1/2*w^3 + 4*w + 1/2],
[131, 131, 2*w - 7],
[131, 131, -7/6*w^3 - 2*w^2 + 28/3*w + 89/6],
[139, 139, -1/6*w^3 + w^2 + 4/3*w - 31/6],
[139, 139, 1/3*w^3 + w^2 - 8/3*w - 29/3],
[149, 149, 1/6*w^3 - 2*w^2 + 5/3*w + 37/6],
[151, 151, -7/6*w^3 + 3*w^2 + 28/3*w - 133/6],
[151, 151, 5/3*w^3 + 3*w^2 - 40/3*w - 67/3],
[169, 13, -1/6*w^3 - 2/3*w + 29/6],
[169, 13, 7/6*w^3 - 4*w^2 - 19/3*w + 139/6],
[181, 181, -2/3*w^3 + 13/3*w + 10/3],
[181, 181, 5/6*w^3 - 23/3*w - 19/6],
[191, 191, 5/6*w^3 + w^2 - 32/3*w - 97/6],
[191, 191, w^3 - 9*w - 5],
[191, 191, -w^2 + 4*w - 2],
[191, 191, 5/6*w^3 + w^2 - 17/3*w - 43/6],
[199, 199, 1/2*w^3 - 2*w - 7/2],
[199, 199, 7/6*w^3 - 5*w^2 - 7/3*w + 109/6],
[211, 211, -1/6*w^3 + w^2 + 7/3*w - 79/6],
[211, 211, 1/2*w^3 + 2*w^2 - 6*w - 37/2],
[229, 229, -1/2*w^3 + 5*w - 5/2],
[229, 229, 1/2*w^3 - 2*w^2 - 2*w + 23/2],
[229, 229, 1/6*w^3 + w^2 - 10/3*w - 53/6],
[229, 229, -7/6*w^3 - 2*w^2 + 34/3*w + 107/6],
[239, 239, 2*w^3 + 3*w^2 - 22*w - 36],
[239, 239, -7/3*w^3 - 3*w^2 + 74/3*w + 113/3],
[241, 241, -1/3*w^3 - w^2 + 2/3*w + 11/3],
[241, 241, -1/2*w^3 + w^2 + 6*w - 23/2],
[271, 271, -5/6*w^3 - w^2 + 20/3*w + 49/6],
[271, 271, 1/6*w^3 - 1/3*w - 35/6],
[281, 281, -1/6*w^3 - w^2 + 7/3*w + 11/6],
[281, 281, -w^3 + 9*w - 1],
[281, 281, -1/2*w^3 + 8*w + 19/2],
[281, 281, w^3 - 4*w^2 - 4*w + 20],
[311, 311, 2*w^2 - 13],
[311, 311, -1/3*w^3 - 2*w^2 + 8/3*w + 50/3],
[331, 331, 1/2*w^3 + w^2 - 4*w - 5/2],
[331, 331, -1/2*w^3 - w^2 + 5*w + 27/2],
[331, 331, 1/6*w^3 - 2*w^2 + 2/3*w + 73/6],
[331, 331, 5/6*w^3 + 2*w^2 - 26/3*w - 103/6],
[349, 349, -1/2*w^3 - w^2 + 4*w + 21/2],
[349, 349, 1/6*w^3 + w^2 - 1/3*w - 59/6],
[361, 19, 2/3*w^3 - 16/3*w - 7/3],
[361, 19, 1/6*w^3 - 4/3*w - 29/6],
[379, 379, 1/6*w^3 + w^2 - 10/3*w - 17/6],
[379, 379, -5/6*w^3 - 2*w^2 + 23/3*w + 115/6],
[379, 379, -1/3*w^3 + 2*w^2 + 5/3*w - 31/3],
[379, 379, 1/3*w^3 + w^2 - 14/3*w - 35/3],
[389, 389, -1/3*w^3 + 2/3*w + 14/3],
[389, 389, 1/2*w^3 + 2*w^2 - 4*w - 35/2],
[401, 401, 1/3*w^3 + 2*w^2 - 14/3*w - 38/3],
[401, 401, 5/6*w^3 - 4*w^2 - 5/3*w + 101/6],
[401, 401, -1/3*w^3 - 2*w^2 + 14/3*w + 50/3],
[401, 401, -5/6*w^3 + 2*w^2 + 5/3*w - 29/6],
[409, 409, -1/2*w^3 + 6*w - 7/2],
[409, 409, 5/6*w^3 + 2*w^2 - 26/3*w - 91/6],
[439, 439, 1/3*w^3 + 2*w^2 - 14/3*w - 32/3],
[439, 439, 1/3*w^3 + 2*w^2 - 14/3*w - 56/3],
[449, 449, -11/6*w^3 - 3*w^2 + 47/3*w + 157/6],
[449, 449, -7/6*w^3 + 3*w^2 + 25/3*w - 109/6],
[461, 461, 1/6*w^3 + 2*w^2 - 4/3*w - 77/6],
[461, 461, 2/3*w^3 + 2*w^2 - 22/3*w - 55/3],
[479, 479, 2/3*w^3 - 2*w^2 - 13/3*w + 32/3],
[479, 479, -7/6*w^3 - 2*w^2 + 31/3*w + 113/6],
[491, 491, 1/3*w^3 + 2*w^2 - 11/3*w - 35/3],
[491, 491, 1/6*w^3 + 2*w^2 - 7/3*w - 107/6],
[499, 499, 4/3*w^3 + 2*w^2 - 38/3*w - 53/3],
[499, 499, 5/6*w^3 - 4*w^2 - 2/3*w + 101/6],
[509, 509, -2/3*w^3 + 31/3*w + 40/3],
[509, 509, 1/6*w^3 - 19/3*w + 85/6],
[521, 521, -2/3*w^3 - 2*w^2 + 19/3*w + 52/3],
[521, 521, -w^2 - 2*w + 10],
[529, 23, -1/3*w^3 + 2*w^2 + 5/3*w - 49/3],
[529, 23, 5/6*w^3 + 2*w^2 - 23/3*w - 79/6],
[541, 541, -1/6*w^3 + 10/3*w - 13/6],
[541, 541, 1/6*w^3 - 10/3*w - 11/6],
[569, 569, 1/2*w^3 + w^2 - 5*w - 9/2],
[569, 569, -1/6*w^3 + w^2 + 1/3*w - 61/6],
[599, 599, 1/6*w^3 + w^2 - 1/3*w - 65/6],
[599, 599, -1/6*w^3 + w^2 + 7/3*w - 25/6],
[619, 619, 7/2*w^3 + 6*w^2 - 33*w - 113/2],
[619, 619, -7/6*w^3 - 2*w^2 + 31/3*w + 89/6],
[631, 631, -1/2*w^3 + 3*w^2 + w - 29/2],
[631, 631, -1/6*w^3 + w^2 - 8/3*w + 29/6],
[631, 631, -2/3*w^3 + 25/3*w + 16/3],
[631, 631, -11/6*w^3 - 3*w^2 + 47/3*w + 145/6],
[659, 659, 5/6*w^3 - 3*w^2 - 20/3*w + 155/6],
[659, 659, 7/6*w^3 - 2*w^2 - 16/3*w + 49/6],
[659, 659, -7/6*w^3 + 34/3*w + 29/6],
[659, 659, -5/6*w^3 + w^2 + 14/3*w - 35/6],
[661, 661, 1/3*w^3 - 20/3*w + 28/3],
[661, 661, 1/3*w^3 - 20/3*w - 26/3],
[691, 691, 1/2*w^3 + 2*w^2 - 5*w - 35/2],
[691, 691, 2*w^2 - 15],
[691, 691, 1/2*w^3 + 2*w^2 - 6*w - 31/2],
[691, 691, 2*w^2 - w - 12],
[701, 701, 1/3*w^3 + 2*w^2 - 11/3*w - 53/3],
[701, 701, 1/6*w^3 + 2*w^2 - 7/3*w - 71/6],
[709, 709, -11/6*w^3 - 3*w^2 + 50/3*w + 151/6],
[709, 709, 7/6*w^3 + w^2 - 25/3*w - 71/6],
[709, 709, 7/6*w^3 - w^2 - 31/3*w + 19/6],
[709, 709, 13/6*w^3 + 4*w^2 - 58/3*w - 209/6],
[739, 739, 7/6*w^3 - 4*w^2 - 28/3*w + 199/6],
[739, 739, 5/3*w^3 - 5*w^2 - 40/3*w + 119/3],
[739, 739, 5/6*w^3 - 4*w^2 - 8/3*w + 107/6],
[739, 739, -5/6*w^3 + w^2 + 26/3*w - 35/6],
[809, 809, 11/6*w^3 + 3*w^2 - 56/3*w - 175/6],
[809, 809, -5/6*w^3 + 2*w^2 + 17/3*w - 89/6],
[809, 809, -4/3*w^3 + 4*w^2 + 14/3*w - 49/3],
[809, 809, -5/6*w^3 + w^2 + 11/3*w - 23/6],
[811, 811, -1/3*w^3 + 17/3*w + 17/3],
[811, 811, -1/6*w^3 + 13/3*w - 37/6],
[821, 821, w^3 - w^2 - 8*w + 3],
[821, 821, -2*w^3 - 3*w^2 + 18*w + 28],
[821, 821, 7/6*w^3 + w^2 - 28/3*w - 71/6],
[821, 821, 1/2*w^3 - w^2 - 7*w + 35/2],
[829, 829, 5/6*w^3 + w^2 - 20/3*w - 73/6],
[829, 829, 1/6*w^3 + w^2 - 1/3*w - 71/6],
[839, 839, 1/2*w^3 - 10*w + 33/2],
[839, 839, 1/2*w^3 - 10*w - 31/2],
[841, 29, 5/6*w^3 - 20/3*w - 19/6],
[841, 29, 1/6*w^3 - 4/3*w - 35/6],
[859, 859, -1/6*w^3 + 2*w^2 + 7/3*w - 121/6],
[859, 859, -2*w^3 - 4*w^2 + 16*w + 27],
[859, 859, 1/2*w^3 - w^2 - 4*w - 1/2],
[859, 859, 4/3*w^3 - 4*w^2 - 32/3*w + 97/3],
[881, 881, -5/6*w^3 + w^2 + 14/3*w - 11/6],
[881, 881, 4/3*w^3 + w^2 - 38/3*w - 38/3],
[911, 911, 2*w^2 - 3*w - 4],
[911, 911, -2/3*w^3 + 19/3*w - 20/3],
[911, 911, 2/3*w^3 + w^2 - 22/3*w - 46/3],
[911, 911, -2/3*w^3 + 4*w^2 - 2/3*w - 35/3],
[919, 919, -1/6*w^3 + 3*w^2 - 11/3*w - 43/6],
[919, 919, -13/6*w^3 - 4*w^2 + 55/3*w + 179/6],
[919, 919, 5/6*w^3 - 2*w^2 - 8/3*w + 47/6],
[919, 919, 4/3*w^3 - 4*w^2 - 29/3*w + 88/3],
[929, 929, -7/6*w^3 + 3*w^2 + 10/3*w - 61/6],
[929, 929, -1/2*w^3 + 2*w^2 + 4*w - 25/2],
[929, 929, -2/3*w^3 + 22/3*w + 19/3],
[929, 929, 2*w^2 - 4*w - 3],
[961, 31, 5/6*w^3 - 20/3*w - 13/6],
[961, 31, -5/6*w^3 + 20/3*w + 7/6],
[971, 971, -7/6*w^3 + 6*w^2 + 4/3*w - 139/6],
[971, 971, -2/3*w^3 + 2*w^2 + 1/3*w - 14/3],
[991, 991, 7/3*w^3 + 4*w^2 - 59/3*w - 95/3],
[991, 991, -1/6*w^3 + w^2 - 11/3*w + 47/6],
[1009, 1009, -7/6*w^3 - w^2 + 31/3*w + 35/6],
[1009, 1009, -5/6*w^3 + w^2 + 17/3*w - 53/6],
[1021, 1021, -11/6*w^3 - 3*w^2 + 44/3*w + 139/6],
[1021, 1021, 4/3*w^3 - 3*w^2 - 32/3*w + 64/3],
[1021, 1021, -w^3 + 11*w + 3],
[1021, 1021, 1/2*w^3 - w - 7/2],
[1031, 1031, -w^3 + 6*w + 2],
[1031, 1031, 4/3*w^3 - 38/3*w - 5/3],
[1049, 1049, -w^3 + 4*w^2 + 3*w - 19],
[1049, 1049, -5/2*w^3 - 4*w^2 + 25*w + 79/2],
[1049, 1049, -2*w^3 - 4*w^2 + 18*w + 31],
[1049, 1049, -w^3 + 4*w^2 + 6*w - 28],
[1051, 1051, -3/2*w^3 - 2*w^2 + 13*w + 33/2],
[1051, 1051, 1/6*w^3 - w^2 + 5/3*w - 17/6],
[1061, 1061, 7/6*w^3 - 5*w^2 - 13/3*w + 157/6],
[1061, 1061, -1/2*w^3 + 9*w + 23/2],
[1091, 1091, -4/3*w^3 + 2*w^2 + 17/3*w - 16/3],
[1091, 1091, -1/6*w^3 + 2*w^2 - 8/3*w - 13/6],
[1109, 1109, -1/6*w^3 + 2*w^2 - 2/3*w - 91/6],
[1109, 1109, -5/6*w^3 - 2*w^2 + 26/3*w + 85/6],
[1151, 1151, -1/2*w^3 + 5*w - 13/2],
[1151, 1151, -7/6*w^3 - w^2 + 31/3*w + 59/6],
[1171, 1171, 5/6*w^3 - w^2 - 17/3*w + 23/6],
[1171, 1171, 7/6*w^3 + w^2 - 31/3*w - 65/6],
[1181, 1181, w^2 - 6*w + 10],
[1181, 1181, -7/6*w^3 - w^2 + 46/3*w + 143/6],
[1229, 1229, -5/3*w^3 - 3*w^2 + 46/3*w + 73/3],
[1229, 1229, 1/6*w^3 - w^2 - 13/3*w + 109/6],
[1229, 1229, 5/6*w^3 - 3*w^2 - 14/3*w + 119/6],
[1229, 1229, -1/6*w^3 + w^2 + 13/3*w + 17/6],
[1231, 1231, -2/3*w^3 + 25/3*w - 8/3],
[1231, 1231, -1/6*w^3 - 5/3*w - 13/6],
[1231, 1231, 1/2*w^3 - w - 3/2],
[1231, 1231, w^3 - 11*w - 1],
[1249, 1249, 1/3*w^3 + 3*w^2 - 14/3*w - 89/3],
[1249, 1249, 7/6*w^3 - 2*w^2 - 25/3*w + 43/6],
[1279, 1279, -17/6*w^3 - 5*w^2 + 80/3*w + 265/6],
[1279, 1279, 4/3*w^3 - 5*w^2 - 20/3*w + 88/3],
[1289, 1289, 5/6*w^3 + 2*w^2 - 23/3*w - 127/6],
[1289, 1289, 1/2*w^3 + w^2 - 2*w - 21/2],
[1291, 1291, -1/3*w^3 + 26/3*w - 46/3],
[1291, 1291, -2/3*w^3 + 34/3*w + 43/3],
[1301, 1301, 19/6*w^3 + 5*w^2 - 88/3*w - 275/6],
[1301, 1301, -5/3*w^3 + 5*w^2 + 28/3*w - 83/3],
[1321, 1321, -2*w^3 - 4*w^2 + 19*w + 34],
[1321, 1321, -7/6*w^3 + 2*w^2 + 28/3*w - 97/6],
[1321, 1321, 5/3*w^3 + 2*w^2 - 46/3*w - 64/3],
[1321, 1321, 5/6*w^3 - 4*w^2 - 11/3*w + 149/6],
[1381, 1381, -7/6*w^3 - 3*w^2 + 34/3*w + 167/6],
[1381, 1381, 1/3*w^3 - 3*w^2 - 2/3*w + 49/3],
[1399, 1399, 7/6*w^3 - 4*w^2 - 22/3*w + 163/6],
[1399, 1399, 13/6*w^3 + 4*w^2 - 58/3*w - 191/6],
[1409, 1409, -3/2*w^3 + 4*w^2 + 10*w - 45/2],
[1409, 1409, 2*w^2 - 9],
[1409, 1409, -5/2*w^3 - 4*w^2 + 22*w + 73/2],
[1409, 1409, 1/3*w^3 + 2*w^2 - 8/3*w - 62/3],
[1439, 1439, 7/6*w^3 - 37/3*w + 7/6],
[1439, 1439, -1/2*w^3 - 3*w^2 + 7*w + 33/2],
[1459, 1459, 2*w^2 - 2*w - 17],
[1459, 1459, -5/6*w^3 + 26/3*w + 55/6],
[1459, 1459, 2/3*w^3 + 2*w^2 - 22/3*w - 37/3],
[1459, 1459, -1/2*w^3 + 2*w + 19/2],
[1511, 1511, 3/2*w^3 - 6*w^2 - 8*w + 75/2],
[1511, 1511, -19/6*w^3 - 6*w^2 + 88/3*w + 305/6],
[1531, 1531, 5/2*w^3 + 3*w^2 - 29*w - 87/2],
[1531, 1531, -13/6*w^3 - 3*w^2 + 61/3*w + 179/6],
[1549, 1549, 1/2*w^3 + 2*w^2 - 6*w - 21/2],
[1549, 1549, w^3 - 10*w + 2],
[1579, 1579, -1/2*w^3 + w^2 + 3*w - 21/2],
[1579, 1579, 5/6*w^3 + w^2 - 23/3*w - 25/6],
[1601, 1601, 1/3*w^3 + w^2 - 20/3*w - 5/3],
[1601, 1601, -7/6*w^3 - 3*w^2 + 31/3*w + 131/6],
[1609, 1609, 5/6*w^3 - w^2 - 23/3*w + 23/6],
[1609, 1609, -11/6*w^3 - 3*w^2 + 50/3*w + 175/6],
[1609, 1609, -5/6*w^3 - w^2 + 17/3*w + 67/6],
[1609, 1609, -w^3 + 3*w^2 + 6*w - 15],
[1619, 1619, 1/3*w^3 + 2*w^2 - 5/3*w - 53/3],
[1619, 1619, 2/3*w^3 - w^2 - 16/3*w + 2/3],
[1619, 1619, 3/2*w^3 + 2*w^2 - 12*w - 39/2],
[1619, 1619, 1/3*w^3 - 2*w^2 - 14/3*w + 70/3],
[1621, 1621, 1/6*w^3 + 3*w^2 - 4/3*w - 113/6],
[1621, 1621, -1/3*w^3 - 3*w^2 + 8/3*w + 77/3],
[1681, 41, w^3 - 8*w - 4],
[1699, 1699, 5/6*w^3 + w^2 - 29/3*w - 37/6],
[1699, 1699, -1/6*w^3 + w^2 - 5/3*w - 49/6],
[1709, 1709, -1/6*w^3 - 2*w^2 + 13/3*w + 35/6],
[1709, 1709, -3/2*w^3 - w^2 + 14*w + 15/2],
[1721, 1721, -5/6*w^3 - w^2 + 35/3*w + 103/6],
[1721, 1721, 5/3*w^3 + 2*w^2 - 43/3*w - 61/3],
[1721, 1721, -7/6*w^3 + 2*w^2 + 25/3*w - 55/6],
[1721, 1721, -1/6*w^3 - w^2 + 19/3*w - 19/6],
[1741, 1741, 1/6*w^3 - 3*w^2 + 5/3*w + 115/6],
[1741, 1741, -1/2*w^3 + 4*w^2 - w - 25/2],
[1741, 1741, 7/6*w^3 + 3*w^2 - 37/3*w - 149/6],
[1741, 1741, -11/6*w^3 + 6*w^2 + 17/3*w - 131/6],
[1789, 1789, 1/6*w^3 - w^2 - 13/3*w + 103/6],
[1789, 1789, w^3 + 2*w^2 - 8*w - 18],
[1789, 1789, -2*w^3 - 3*w^2 + 18*w + 30],
[1789, 1789, -2/3*w^3 + 2*w^2 + 16/3*w - 35/3],
[1801, 1801, -7/6*w^3 - w^2 + 28/3*w + 47/6],
[1801, 1801, -w^3 + w^2 + 8*w - 7],
[1811, 1811, 4/3*w^3 - 29/3*w - 26/3],
[1811, 1811, -7/6*w^3 + w^2 + 31/3*w - 43/6],
[1831, 1831, -1/3*w^3 + 2/3*w + 32/3],
[1831, 1831, -3/2*w^3 - 2*w^2 + 12*w + 33/2],
[1849, 43, -7/6*w^3 - 2*w^2 + 31/3*w + 125/6],
[1849, 43, -1/2*w^3 + 3*w^2 + 5*w - 43/2],
[1871, 1871, -3/2*w^3 - 2*w^2 + 12*w + 29/2],
[1871, 1871, 7/6*w^3 - 2*w^2 - 28/3*w + 91/6],
[1879, 1879, 3/2*w^3 + 3*w^2 - 13*w - 53/2],
[1879, 1879, -5/6*w^3 + 3*w^2 + 17/3*w - 107/6],
[1889, 1889, -1/2*w^3 - w^2 + 6*w + 5/2],
[1889, 1889, w^2 - 2*w - 12],
[1901, 1901, w^2 + 2*w + 2],
[1901, 1901, 1/2*w^3 - 3*w^2 + w + 13/2],
[1901, 1901, -1/6*w^3 + 4*w^2 - 11/3*w - 73/6],
[1901, 1901, 1/6*w^3 - w^2 - 10/3*w + 103/6],
[1931, 1931, 7/6*w^3 - w^2 - 28/3*w + 13/6],
[1931, 1931, 4/3*w^3 + w^2 - 32/3*w - 38/3],
[1931, 1931, -7/6*w^3 - 3*w^2 + 31/3*w + 137/6],
[1931, 1931, 1/2*w^3 - 3*w^2 - 3*w + 43/2],
[1949, 1949, 4/3*w^3 - 3*w^2 - 32/3*w + 58/3],
[1949, 1949, 11/6*w^3 + 3*w^2 - 44/3*w - 151/6],
[1951, 1951, 11/6*w^3 - 5*w^2 - 20/3*w + 113/6],
[1951, 1951, -4/3*w^3 + 26/3*w - 1/3],
[1999, 1999, -1/6*w^3 + 3*w^2 - 2/3*w - 151/6],
[1999, 1999, 1/2*w^3 + 2*w^2 - 4*w - 49/2]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^12 - 33*x^10 + 408*x^8 - 2323*x^6 + 6021*x^4 - 6150*x^2 + 1156;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [0, e, -61/918*e^11 + 1537/918*e^9 - 385/27*e^7 + 43817/918*e^5 - 52679/918*e^3 + 4366/459*e, -4/459*e^11 + 179/918*e^9 - 35/27*e^7 + 962/459*e^5 + 1421/918*e^3 - 67/459*e, -26/153*e^11 + 1291/306*e^9 - 316/9*e^7 + 17422/153*e^5 - 42197/306*e^3 + 5557/153*e, -7/54*e^10 + 169/54*e^8 - 668/27*e^6 + 3911/54*e^4 - 4043/54*e^2 + 469/27, 11/54*e^10 - 263/54*e^8 + 1024/27*e^6 - 5845/54*e^4 + 5839/54*e^2 - 539/27, 5/9*e^10 - 247/18*e^8 + 1019/9*e^6 - 3250/9*e^4 + 7583/18*e^2 - 766/9, 2/3*e^10 - 33/2*e^8 + 409/3*e^6 - 1300/3*e^4 + 983/2*e^2 - 260/3, -13/459*e^11 + 310/459*e^9 - 143/27*e^7 + 7487/459*e^5 - 11786/459*e^3 + 12137/459*e, 19/459*e^11 - 559/459*e^9 + 344/27*e^7 - 25454/459*e^5 + 39293/459*e^3 - 9971/459*e, -53/54*e^10 + 1295/54*e^8 - 5239/27*e^6 + 32173/54*e^4 - 35425/54*e^2 + 3416/27, 31/54*e^10 - 769/54*e^8 + 3191/27*e^6 - 20483/54*e^4 + 23747/54*e^2 - 2122/27, 20/27*e^10 - 985/54*e^8 + 4037/27*e^6 - 12694/27*e^4 + 28877/54*e^2 - 2653/27, -22/27*e^10 + 1079/54*e^8 - 4393/27*e^6 + 13661/27*e^4 - 30673/54*e^2 + 2831/27, -35/54*e^10 + 863/54*e^8 - 3547/27*e^6 + 22471/54*e^4 - 26083/54*e^2 + 2543/27, -13/18*e^10 + 319/18*e^8 - 1301/9*e^6 + 8135/18*e^4 - 9329/18*e^2 + 1033/9, 235/918*e^11 - 5851/918*e^9 + 1441/27*e^7 - 161399/918*e^5 + 204263/918*e^3 - 31405/459*e, -215/918*e^11 + 5327/918*e^9 - 1295/27*e^7 + 139147/918*e^5 - 149599/918*e^3 + 2273/459*e, 38/459*e^11 - 1777/918*e^9 + 391/27*e^7 - 17860/459*e^5 + 40127/918*e^3 - 11680/459*e, -4/51*e^11 + 71/34*e^9 - 58/3*e^7 + 3767/51*e^5 - 3697/34*e^3 + 1990/51*e, -11/9*e^10 + 535/18*e^8 - 2147/9*e^6 + 6502/9*e^4 - 14135/18*e^2 + 1348/9, 2/9*e^10 - 97/18*e^8 + 386/9*e^6 - 1138/9*e^4 + 2261/18*e^2 - 112/9, 3/34*e^11 - 41/17*e^9 + 23*e^7 - 3025/34*e^5 + 1951/17*e^3 - 62/17*e, 127/306*e^11 - 1526/153*e^9 + 703/9*e^7 - 68105/306*e^5 + 31054/153*e^3 + 662/153*e, -1/54*e^10 + 14/27*e^8 - 134/27*e^6 + 1037/54*e^4 - 796/27*e^2 + 310/27, -37/54*e^10 + 919/54*e^8 - 3815/27*e^6 + 24437/54*e^4 - 28133/54*e^2 + 2866/27, -17/54*e^10 + 431/54*e^8 - 1855/27*e^6 + 12661/54*e^4 - 15661/54*e^2 + 1292/27, 5/34*e^11 - 359/102*e^9 + 82/3*e^7 - 2605/34*e^5 + 7219/102*e^3 - 820/51*e, 367/918*e^11 - 9187/918*e^9 + 2284/27*e^7 - 260159/918*e^5 + 336035/918*e^3 - 50572/459*e, -82/459*e^11 + 2026/459*e^9 - 983/27*e^7 + 53228/459*e^5 - 61667/459*e^3 + 10178/459*e, 73/54*e^10 - 1801/54*e^8 + 7406/27*e^6 - 46865/54*e^4 + 53873/54*e^2 - 5350/27, 17/54*e^10 - 413/54*e^8 + 1648/27*e^6 - 9745/54*e^4 + 9577/54*e^2 - 338/27, 7/306*e^11 - 56/153*e^9 + 1/9*e^7 + 5431/306*e^5 - 9263/153*e^3 + 8735/153*e, -37/102*e^11 + 466/51*e^9 - 78*e^7 + 27029/102*e^5 - 17489/51*e^3 + 1579/17*e, -13/54*e^10 + 337/54*e^8 - 1499/27*e^6 + 10781/54*e^4 - 14513/54*e^2 + 2059/27, 37/54*e^10 - 937/54*e^8 + 4022/27*e^6 - 27353/54*e^4 + 34379/54*e^2 - 3901/27, 35/18*e^10 - 863/18*e^8 + 3547/9*e^6 - 22435/18*e^4 + 25777/18*e^2 - 2588/9, -7/54*e^10 + 169/54*e^8 - 668/27*e^6 + 3911/54*e^4 - 3989/54*e^2 + 658/27, -11/18*e^10 + 263/18*e^8 - 1024/9*e^6 + 5863/18*e^4 - 6019/18*e^2 + 620/9, -1/6*e^10 + 9/2*e^8 - 127/3*e^6 + 977/6*e^4 - 447/2*e^2 + 140/3, -7/459*e^11 + 61/459*e^9 + 58/27*e^7 - 10480/459*e^5 + 15721/459*e^3 + 12467/459*e, -8/153*e^11 + 230/153*e^9 - 139/9*e^7 + 10339/153*e^5 - 17653/153*e^3 + 9403/153*e, 10/27*e^10 - 244/27*e^8 + 1960/27*e^6 - 5861/27*e^4 + 5885/27*e^2 - 494/27, 25/27*e^10 - 610/27*e^8 + 4927/27*e^6 - 15125/27*e^4 + 16805/27*e^2 - 3098/27, 11/306*e^11 - 329/306*e^9 + 107/9*e^7 - 18175/306*e^5 + 38929/306*e^3 - 12065/153*e, -203/918*e^11 + 4829/918*e^9 - 1094/27*e^7 + 103213/918*e^5 - 94585/918*e^3 + 1685/459*e, -13/918*e^11 + 157/918*e^9 + 32/27*e^7 - 18217/918*e^5 + 53239/918*e^3 - 20171/459*e, 181/918*e^11 - 4069/918*e^9 + 820/27*e^7 - 56153/918*e^5 + 12239/918*e^3 + 10715/459*e, -428/459*e^11 + 10571/459*e^9 - 5131/27*e^7 + 278731/459*e^5 - 331033/459*e^3 + 82642/459*e, 71/459*e^11 - 1799/459*e^9 + 916/27*e^7 - 54943/459*e^5 + 80011/459*e^3 - 37864/459*e, 37/27*e^10 - 1829/54*e^8 + 7540/27*e^6 - 23897/27*e^4 + 54169/54*e^2 - 4850/27, 58/27*e^10 - 2879/54*e^8 + 11962/27*e^6 - 38519/27*e^4 + 90109/54*e^2 - 9140/27, -11/27*e^10 + 281/27*e^8 - 2435/27*e^6 + 8302/27*e^4 - 9871/27*e^2 + 1258/27, 67/27*e^10 - 1642/27*e^8 + 13357/27*e^6 - 41453/27*e^4 + 46445/27*e^2 - 8798/27, -11/18*e^10 + 136/9*e^8 - 1123/9*e^6 + 7141/18*e^4 - 4058/9*e^2 + 764/9, 113/54*e^10 - 1393/27*e^8 + 11443/27*e^6 - 72145/54*e^4 + 40754/27*e^2 - 7112/27, -37/54*e^10 + 937/54*e^8 - 4022/27*e^6 + 27353/54*e^4 - 34163/54*e^2 + 3469/27, -17/54*e^10 + 431/54*e^8 - 1855/27*e^6 + 12607/54*e^4 - 15121/54*e^2 + 1157/27, -1/18*e^10 + 25/18*e^8 - 104/9*e^6 + 677/18*e^4 - 965/18*e^2 + 376/9, 119/54*e^10 - 2927/54*e^8 + 11977/27*e^6 - 75127/54*e^4 + 85201/54*e^2 - 8378/27, 115/54*e^10 - 2851/54*e^8 + 11828/27*e^6 - 76109/54*e^4 + 89543/54*e^2 - 9181/27, 23/54*e^10 - 545/54*e^8 + 2092/27*e^6 - 11647/54*e^4 + 11227/54*e^2 - 1181/27, 91/54*e^10 - 2233/54*e^8 + 9098/27*e^6 - 56621/54*e^4 + 63485/54*e^2 - 5683/27, -59/54*e^10 + 1463/54*e^8 - 6070/27*e^6 + 39043/54*e^4 - 45733/54*e^2 + 4277/27, 101/153*e^11 - 5017/306*e^9 + 1228/9*e^7 - 67513/153*e^5 + 160151/306*e^3 - 16171/153*e, 1/459*e^11 - 83/918*e^9 + 38/27*e^7 - 4601/459*e^5 + 28753/918*e^3 - 14633/459*e, 26/27*e^10 - 1321/54*e^8 + 5699/27*e^6 - 19510/27*e^4 + 48737/54*e^2 - 4915/27, 4/9*e^10 - 209/18*e^8 + 940/9*e^6 - 3419/9*e^4 + 9097/18*e^2 - 1109/9, -13/18*e^11 + 313/18*e^9 - 1232/9*e^7 + 7145/18*e^5 - 7031/18*e^3 + 337/9*e, 403/918*e^11 - 9457/918*e^9 + 2086/27*e^7 - 185279/918*e^5 + 152543/918*e^3 - 6283/459*e, -853/918*e^11 + 20941/918*e^9 - 5029/27*e^7 + 536315/918*e^5 - 623603/918*e^3 + 75457/459*e, 89/306*e^11 - 2189/306*e^9 + 527/9*e^7 - 56365/306*e^5 + 65581/306*e^3 - 8237/153*e, -199/459*e^11 + 4969/459*e^9 - 2450/27*e^7 + 135299/459*e^5 - 153818/459*e^3 + 14453/459*e, -92/459*e^11 + 2135/459*e^9 - 922/27*e^7 + 39109/459*e^5 - 30859/459*e^3 + 5191/459*e, 19/54*e^10 - 487/54*e^8 + 2123/27*e^6 - 14681/54*e^4 + 18359/54*e^2 - 1480/27, -47/27*e^10 + 2317/54*e^8 - 9527/27*e^6 + 30244/27*e^4 - 70745/54*e^2 + 7504/27, -11/54*e^10 + 263/54*e^8 - 1024/27*e^6 + 5737/54*e^4 - 4597/54*e^2 - 352/27, 58/27*e^10 - 2879/54*e^8 + 11962/27*e^6 - 38492/27*e^4 + 89407/54*e^2 - 8168/27, 43/459*e^11 - 1249/459*e^9 + 761/27*e^7 - 56012/459*e^5 + 86285/459*e^3 - 24410/459*e, 190/459*e^11 - 4672/459*e^9 + 2252/27*e^7 - 120971/459*e^5 + 141857/459*e^3 - 31778/459*e, 329/918*e^11 - 8069/918*e^9 + 1940/27*e^7 - 209251/918*e^5 + 258367/918*e^3 - 44732/459*e, -883/918*e^11 + 22033/918*e^9 - 5428/27*e^7 + 599987/918*e^5 - 692441/918*e^3 + 47704/459*e, -793/918*e^11 + 19675/918*e^9 - 4798/27*e^7 + 520049/918*e^5 - 578645/918*e^3 + 29983/459*e, 395/918*e^11 - 10043/918*e^9 + 2555/27*e^7 - 299941/918*e^5 + 389125/918*e^3 - 52403/459*e, -4/3*e^10 + 197/6*e^8 - 808/3*e^6 + 2540/3*e^4 - 5695/6*e^2 + 545/3, -32/27*e^10 + 1585/54*e^8 - 6560/27*e^6 + 20980/27*e^4 - 48851/54*e^2 + 5305/27, 137/306*e^11 - 3365/306*e^9 + 806/9*e^7 - 84433/306*e^5 + 89695/306*e^3 - 6203/153*e, 263/918*e^11 - 6401/918*e^9 + 1505/27*e^7 - 149773/918*e^5 + 129139/918*e^3 + 10981/459*e, 22/9*e^10 - 538/9*e^8 + 4363/9*e^6 - 13481/9*e^4 + 15068/9*e^2 - 2906/9, 4/3*e^10 - 100/3*e^8 + 841/3*e^6 - 2765/3*e^4 + 3326/3*e^2 - 662/3, -121/306*e^11 + 2905/306*e^9 - 667/9*e^7 + 63755/306*e^5 - 53777/306*e^3 - 4730/153*e, -389/918*e^11 + 9641/918*e^9 - 2351/27*e^7 + 255811/918*e^5 - 289249/918*e^3 + 16460/459*e, -623/918*e^11 + 7840/459*e^9 - 3935/27*e^7 + 456673/918*e^5 - 306008/459*e^3 + 99530/459*e, 17/18*e^11 - 208/9*e^9 + 1687/9*e^7 - 10393/18*e^5 + 5687/9*e^3 - 902/9*e, 37/54*e^10 - 883/54*e^8 + 3428/27*e^6 - 19523/54*e^4 + 20015/54*e^2 - 2443/27, -5/54*e^10 + 95/54*e^8 - 193/27*e^6 - 917/54*e^4 + 3443/54*e^2 + 461/27, -341/459*e^11 + 8414/459*e^9 - 4075/27*e^7 + 220399/459*e^5 - 260290/459*e^3 + 63865/459*e, -143/459*e^11 + 3716/459*e^9 - 1960/27*e^7 + 122749/459*e^5 - 178453/459*e^3 + 63127/459*e, 7/9*e^10 - 335/18*e^8 + 1306/9*e^6 - 3731/9*e^4 + 7297/18*e^2 - 347/9, 19/27*e^10 - 911/54*e^8 + 3562/27*e^6 - 10280/27*e^4 + 21391/54*e^2 - 2321/27, -167/459*e^11 + 8047/918*e^9 - 1873/27*e^7 + 95014/459*e^5 - 217565/918*e^3 + 33349/459*e, 55/153*e^11 - 2729/306*e^9 + 668/9*e^7 - 36866/153*e^5 + 87829/306*e^3 - 6155/153*e, 925/918*e^11 - 22399/918*e^9 + 5254/27*e^7 - 537107/918*e^5 + 595361/918*e^3 - 74548/459*e, -607/918*e^11 + 15169/918*e^9 - 3748/27*e^7 + 417941/918*e^5 - 501179/918*e^3 + 55294/459*e, -235/918*e^11 + 5851/918*e^9 - 1441/27*e^7 + 160481/918*e^5 - 192329/918*e^3 + 16717/459*e, 17/54*e^11 - 431/54*e^9 + 1855/27*e^7 - 12769/54*e^5 + 17227/54*e^3 - 3533/27*e, 35/54*e^10 - 827/54*e^8 + 3160/27*e^6 - 17503/54*e^4 + 17317/54*e^2 - 1931/27, 5/6*e^10 - 41/2*e^8 + 503/3*e^6 - 3133/6*e^4 + 1135/2*e^2 - 280/3, -55/54*e^10 + 1369/54*e^8 - 5714/27*e^6 + 36893/54*e^4 - 41669/54*e^2 + 2965/27, 19/18*e^10 - 463/18*e^8 + 1865/9*e^6 - 11369/18*e^4 + 12371/18*e^2 - 1144/9, -251/459*e^11 + 6056/459*e^9 - 2815/27*e^7 + 140002/459*e^5 - 141904/459*e^3 + 24292/459*e, 266/459*e^11 - 6296/459*e^9 + 2827/27*e^7 - 130987/459*e^5 + 118336/459*e^3 - 15970/459*e, 179/459*e^11 - 4292/459*e^9 + 1978/27*e^7 - 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18271/27*e^6 + 116017/54*e^4 - 131707/54*e^2 + 12308/27, 47/54*e^10 - 1127/54*e^8 + 4408/27*e^6 - 25249/54*e^4 + 24145/54*e^2 - 1556/27, 185/54*e^10 - 4577/54*e^8 + 18922/27*e^6 - 120889/54*e^4 + 139873/54*e^2 - 12782/27, -67/54*e^10 + 1633/54*e^8 - 6602/27*e^6 + 40805/54*e^4 - 46103/54*e^2 + 4516/27, -545/459*e^11 + 27181/918*e^9 - 6715/27*e^7 + 378244/459*e^5 - 964637/918*e^3 + 152860/459*e, -235/153*e^11 + 11651/306*e^9 - 2843/9*e^7 + 155585/153*e^5 - 370633/306*e^3 + 48020/153*e, -5/54*e^10 + 167/54*e^8 - 994/27*e^6 + 9829/54*e^4 - 16897/54*e^2 + 2279/27, 265/54*e^10 - 6493/54*e^8 + 26402/27*e^6 - 163835/54*e^4 + 183317/54*e^2 - 16657/27, -1043/918*e^11 + 25613/918*e^9 - 6128/27*e^7 + 638467/918*e^5 - 651169/918*e^3 - 454/459*e, 185/918*e^11 - 4541/918*e^9 + 1076/27*e^7 - 104851/918*e^5 + 54751/918*e^3 + 62080/459*e, -47/306*e^11 + 1211/306*e^9 - 314/9*e^7 + 37237/306*e^5 - 41485/306*e^3 - 4378/153*e, 601/918*e^11 - 14461/918*e^9 + 3337/27*e^7 - 324239/918*e^5 + 302465/918*e^3 - 17362/459*e, -85/54*e^10 + 2173/54*e^8 - 9455/27*e^6 + 65411/54*e^4 - 81743/54*e^2 + 8251/27, 199/54*e^10 - 4897/54*e^8 + 20051/27*e^6 - 125903/54*e^4 + 143063/54*e^2 - 13414/27, -301/54*e^10 + 7393/54*e^8 - 30173/27*e^6 + 188315/54*e^4 - 212549/54*e^2 + 22444/27, 97/54*e^10 - 2383/54*e^8 + 9749/27*e^6 - 61169/54*e^4 + 68411/54*e^2 - 5077/27, 59/54*e^10 - 1427/54*e^8 + 5683/27*e^6 - 34129/54*e^4 + 37345/54*e^2 - 3179/27, -73/18*e^10 + 1789/18*e^8 - 7277/9*e^6 + 45263/18*e^4 - 51527/18*e^2 + 5551/9, -43/18*e^10 + 1039/18*e^8 - 4121/9*e^6 + 24305/18*e^4 - 24683/18*e^2 + 1732/9, -41/18*e^10 + 1019/18*e^8 - 4240/9*e^6 + 27325/18*e^4 - 31903/18*e^2 + 3380/9, -1535/918*e^11 + 37157/918*e^9 - 8690/27*e^7 + 875215/918*e^5 - 901525/918*e^3 + 63740/459*e, 971/918*e^11 - 23237/918*e^9 + 5336/27*e^7 - 523843/918*e^5 + 557317/918*e^3 - 79862/459*e, -211/54*e^10 + 5161/54*e^8 - 20912/27*e^6 + 128573/54*e^4 - 140045/54*e^2 + 12481/27, 31/18*e^10 - 751/18*e^8 + 2993/9*e^6 - 17855/18*e^4 + 18743/18*e^2 - 1717/9, 112/459*e^11 - 2353/459*e^9 + 827/27*e^7 - 20051/459*e^5 + 20651/459*e^3 - 66362/459*e, 155/459*e^11 - 4214/459*e^9 + 2362/27*e^7 - 158224/459*e^5 + 227959/459*e^3 - 51910/459*e];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {4, 2, 1/6*w^3 - 7/3*w - 17/6}>] := 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;