/*
  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^4 + 4*x^3 - 39*x^2 - 176*x - 139;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [0, 0, 0, 0, 0, e, -8/61*e^3 - 13/61*e^2 + 320/61*e + 709/61, -3/61*e^3 + 18/61*e^2 + 59/61*e - 428/61, 17/61*e^3 + 20/61*e^2 - 741/61*e - 1560/61, 0, 0, 6/61*e^3 + 25/61*e^2 - 362/61*e - 852/61, -6/61*e^3 - 25/61*e^2 + 118/61*e + 1096/61, 18/61*e^3 + 14/61*e^2 - 781/61*e - 1458/61, 18/61*e^3 + 14/61*e^2 - 659/61*e - 726/61, -4/61*e^3 - 37/61*e^2 + 282/61*e + 1239/61, -18/61*e^3 - 14/61*e^2 + 659/61*e + 1214/61, 0, 0, 0, 0, 27/61*e^3 + 82/61*e^2 - 1141/61*e - 2492/61, -11/61*e^3 - 56/61*e^2 + 501/61*e + 1440/61, 0, 0, 2/61*e^3 - 12/61*e^2 - 80/61*e - 284/61, -7/61*e^3 + 42/61*e^2 + 219/61*e - 836/61, 22/61*e^3 + 51/61*e^2 - 1002/61*e - 2392/61, 0, 0, 0, 16/61*e^3 - 35/61*e^2 - 640/61*e + 290/61, -21/61*e^3 + 4/61*e^2 + 779/61*e + 908/61, 0, 0, 36/61*e^3 + 89/61*e^2 - 1562/61*e - 3221/61, 28/61*e^3 + 15/61*e^2 - 1120/61*e - 1963/61, -16/61*e^3 + 35/61*e^2 + 396/61*e - 1266/61, 6/61*e^3 + 25/61*e^2 - 118/61*e - 1828/61, -19/61*e^3 - 8/61*e^2 + 577/61*e + 380/61, 13/61*e^3 + 44/61*e^2 - 337/61*e - 1724/61, 0, 0, -28/61*e^3 - 76/61*e^2 + 1120/61*e + 2512/61, 36/61*e^3 + 28/61*e^2 - 1440/61*e - 2428/61, 0, 0, 0, 0, 0, 0, 33/61*e^3 + 46/61*e^2 - 1137/61*e - 2368/61, -47/61*e^3 - 84/61*e^2 + 2063/61*e + 4112/61, 28/61*e^3 + 76/61*e^2 - 1120/61*e - 3244/61, -36/61*e^3 - 28/61*e^2 + 1440/61*e + 1696/61, 6/61*e^3 - 36/61*e^2 + 4/61*e + 1100/61, 2/61*e^3 - 12/61*e^2 - 324/61*e + 204/61, -20/61*e^3 - 63/61*e^2 + 1044/61*e + 2169/61, 12/61*e^3 - 11/61*e^2 - 358/61*e - 423/61, 40/61*e^3 + 65/61*e^2 - 1600/61*e - 3728/61, 29/61*e^3 + 70/61*e^2 - 977/61*e - 2776/61, -46/61*e^3 - 90/61*e^2 + 2023/61*e + 4214/61, -56/61*e^3 - 91/61*e^2 + 1996/61*e + 4353/61, -4/61*e^3 - 98/61*e^2 + 221/61*e + 2154/61, -36/61*e^3 - 89/61*e^2 + 1562/61*e + 2489/61, 0, 0, 60/61*e^3 + 128/61*e^2 - 2339/61*e - 4860/61, -20/61*e^3 - 2/61*e^2 + 617/61*e + 278/61, 0, 0, 0, 0, 0, 0, -6/61*e^3 - 25/61*e^2 + 118/61*e - 246/61, -69/61*e^3 - 74/61*e^2 + 2699/61*e + 4796/61, 50/61*e^3 + 127/61*e^2 - 2366/61*e - 5026/61, -5/61*e^3 + 30/61*e^2 - 105/61*e - 1608/61, 0, 0, 0, 0, 0, 0, -20/61*e^3 - 2/61*e^2 + 861/61*e - 210/61, -24/61*e^3 - 100/61*e^2 + 1387/61*e + 4140/61, 0, 0, 44/61*e^3 - 20/61*e^2 - 1760/61*e - 1612/61, -20/61*e^3 - 124/61*e^2 + 800/61*e + 3328/61, 0, 0, 0, 0, 16/61*e^3 + 87/61*e^2 - 762/61*e - 1479/61, 20/61*e^3 + 63/61*e^2 - 678/61*e - 3267/61, 0, 0, -4/61*e^3 + 24/61*e^2 + 343/61*e - 2116/61, -2*e^2 + e + 42, 0, 0, 0, 0, 0, 0, -24/61*e^3 - 100/61*e^2 + 777/61*e + 2920/61, 67/61*e^3 + 86/61*e^2 - 2741/61*e - 5976/61, -60/61*e^3 - 67/61*e^2 + 2766/61*e + 4555/61, -62/61*e^3 - 55/61*e^2 + 2602/61*e + 5754/61, 0, 0, 0, 0, 54/61*e^3 + 164/61*e^2 - 2099/61*e - 5472/61, 34/61*e^3 + 40/61*e^2 - 1177/61*e - 436/61, 36/61*e^3 + 89/61*e^2 - 1440/61*e - 1574/61, -32/61*e^3 - 52/61*e^2 + 1768/61*e + 2592/61, 24/61*e^3 + 100/61*e^2 - 1448/61*e - 4140/61, -27/61*e^3 - 82/61*e^2 + 1141/61*e + 540/61, 38/61*e^3 + 16/61*e^2 - 1337/61*e - 28/61, -10/61*e^3 + 60/61*e^2 + 705/61*e - 1508/61, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -18/61*e^3 - 136/61*e^2 + 1147/61*e + 4752/61, -40/61*e^3 + 57/61*e^2 + 1234/61*e - 1091/61, -24/61*e^3 + 83/61*e^2 + 838/61*e - 2265/61, -66/61*e^3 - 92/61*e^2 + 2396/61*e + 3516/61, -4/61*e^3 - 159/61*e^2 + 282/61*e + 3801/61, 66/61*e^3 + 92/61*e^2 - 2396/61*e - 5468/61, 0, 0, 0, 0, -74/61*e^3 - 105/61*e^2 + 2594/61*e + 4774/61, -96/61*e^3 - 95/61*e^2 + 3962/61*e + 7715/61, 0, 0, 0, 0, -6/61*e^3 + 36/61*e^2 - 65/61*e - 2808/61, 30/61*e^3 - 58/61*e^2 - 773/61*e + 2694/61, 84/61*e^3 + 45/61*e^2 - 3482/61*e - 5401/61, -56/61*e^3 - 91/61*e^2 + 2240/61*e + 5329/61, -54/61*e^3 + 80/61*e^2 + 2221/61*e + 836/61, 24/61*e^3 + 100/61*e^2 - 1387/61*e - 5116/61, 0, 0, 0, 0, 0, 0, 0, 0, 22/61*e^3 - 132/61*e^2 - 880/61*e + 1756/61, -22/61*e^3 + 132/61*e^2 + 880/61*e - 2732/61, 24/61*e^3 + 100/61*e^2 - 472/61*e - 4384/61, -48/61*e^3 + 44/61*e^2 + 1432/61*e - 1236/61, 82/61*e^3 + 179/61*e^2 - 3158/61*e - 7008/61, -65/61*e^3 - 220/61*e^2 + 2905/61*e + 6912/61, 0, 0, -33/61*e^3 - 46/61*e^2 + 1747/61*e + 1392/61, -31/61*e^3 + 64/61*e^2 + 1301/61*e - 1332/61, -e^2 - 8*e + 42, -26/61*e^3 - 27/61*e^2 + 674/61*e + 154/61, -9/61*e^3 + 54/61*e^2 + 665/61*e - 2504/61, -24/61*e^3 + 83/61*e^2 + 1204/61*e - 984/61, 0, 0, 0, 0, -86/61*e^3 - 216/61*e^2 + 3989/61*e + 8308/61, -64/61*e^3 - 165/61*e^2 + 2926/61*e + 5611/61, -54/61*e^3 - 225/61*e^2 + 2282/61*e + 6570/61, -24/61*e^3 - 39/61*e^2 + 1204/61*e + 4506/61, 75/61*e^3 + 38/61*e^2 - 2939/61*e - 5160/61, 58/61*e^3 + 201/61*e^2 - 2442/61*e - 5308/61, 0, 0, 60/61*e^3 + 67/61*e^2 - 1912/61*e - 3640/61, -89/61*e^3 - 198/61*e^2 + 4353/61*e + 8856/61, 117/61*e^3 + 152/61*e^2 - 4253/61*e - 7464/61, -68/61*e^3 - 141/61*e^2 + 3452/61*e + 6484/61, -28/61*e^3 + 107/61*e^2 + 1242/61*e - 1209/61, 20/61*e^3 - 59/61*e^2 - 434/61*e + 3809/61, 0, 0, 0, 0, 10/61*e^3 + 123/61*e^2 - 34/61*e - 3860/61, 75/61*e^3 + 160/61*e^2 - 2695/61*e - 6868/61, -96/61*e^3 - 217/61*e^2 + 4084/61*e + 7959/61, -82/61*e^3 - 240/61*e^2 + 3463/61*e + 6520/61, 0, 0, 0, 0, 0, 0, -104/61*e^3 - 108/61*e^2 + 4160/61*e + 8668/61, 88/61*e^3 + 204/61*e^2 - 3520/61*e - 6152/61, 40/61*e^3 + 65/61*e^2 - 1600/61*e - 68/61, 76/61*e^3 - 29/61*e^2 - 2796/61*e - 1764/61, 102/61*e^3 + 59/61*e^2 - 4202/61*e - 6920/61, -68/61*e^3 - 141/61*e^2 + 2354/61*e + 6911/61, -68/61*e^3 - 141/61*e^2 + 3452/61*e + 6179/61, -108/61*e^3 - 145/61*e^2 + 4564/61*e + 8260/61, 112/61*e^3 + 60/61*e^2 - 4480/61*e - 6754/61, -80/61*e^3 - 252/61*e^2 + 3200/61*e + 8066/61, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -84/61*e^3 - 45/61*e^2 + 2872/61*e + 3510/61, 21/61*e^3 - 4/61*e^2 - 1511/61*e + 556/61, -104/61*e^3 - 108/61*e^2 + 4160/61*e + 8668/61, 88/61*e^3 + 204/61*e^2 - 3520/61*e - 6152/61, 0, 0, 0, 0, -8/61*e^3 + 48/61*e^2 + 15/61*e - 3012/61, 36/61*e^3 + 28/61*e^2 - 769/61*e - 1452/61, 0, 0, 0, 0, 0, 0, 0, 0, -72/61*e^3 - 117/61*e^2 + 2880/61*e + 3209/61, -116/61*e^3 - 219/61*e^2 + 4274/61*e + 9091/61, -8/61*e^3 + 48/61*e^2 + 320/61*e + 526/61, 0, 0, 0, 0, -45/61*e^3 - 96/61*e^2 + 1739/61*e + 6024/61, 60/61*e^3 + 128/61*e^2 - 2705/61*e - 7544/61, 96/61*e^3 + 156/61*e^2 - 4633/61*e - 8264/61, 15/61*e^3 - 212/61*e^2 - 661/61*e + 4580/61, -14/61*e^3 - 99/61*e^2 + 194/61*e + 4306/61, -84/61*e^3 - 106/61*e^2 + 3421/61*e + 4242/61, 96/61*e^3 + 95/61*e^2 - 4328/61*e - 7410/61, -76/61*e^3 - 276/61*e^2 + 3223/61*e + 7132/61, 0, 0, 0, 0, -2*e^3 - 3*e^2 + 74*e + 164, 56/61*e^3 + 213/61*e^2 - 2972/61*e - 8440/61, -15/61*e^3 - 32/61*e^2 + 173/61*e - 676/61, -158/61*e^3 - 211/61*e^2 + 5954/61*e + 10846/61, -112/61*e^3 - 243/61*e^2 + 5212/61*e + 10292/61, 89/61*e^3 + 198/61*e^2 - 3377/61*e - 6172/61, 0, 0, 96/61*e^3 + 95/61*e^2 - 4328/61*e - 7715/61, 100/61*e^3 + 10/61*e^2 - 4305/61*e - 5538/61, 0, 0, 0, 0, 120/61*e^3 + 195/61*e^2 - 4678/61*e - 7829/61, 24/61*e^3 - 83/61*e^2 - 472/61*e + 4644/61, 36/61*e^3 - 33/61*e^2 - 1196/61*e - 2428/61, 24/61*e^3 + 161/61*e^2 - 1814/61*e - 7007/61, -8/61*e^3 - 196/61*e^2 + 137/61*e + 4552/61, 40/61*e^3 + 187/61*e^2 - 1356/61*e - 6351/61, -15/61*e^3 + 212/61*e^2 + 661/61*e - 4092/61, -2/61*e^3 - 171/61*e^2 - 42/61*e + 2846/61, 0, 0, 72/61*e^3 + 56/61*e^2 - 3185/61*e - 6320/61, 156/61*e^3 + 223/61*e^2 - 5874/61*e - 10867/61, 0, 0];
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;