/*
  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![-1, -12, 12, 7, -7, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {41, 41, -w^5 + 7*w^3 + w^2 - 12*w - 2}>;

primesArray := [
[4, 2, -w^5 + 6*w^3 + w^2 - 8*w - 2],
[11, 11, w + 1],
[16, 2, -w^5 + 6*w^3 + w^2 - 7*w - 2],
[19, 19, -w^3 + w^2 + 4*w - 3],
[29, 29, 2*w^5 - 13*w^3 + 19*w - 2],
[31, 31, -2*w^5 + 12*w^3 + w^2 - 16*w - 2],
[41, 41, -w^5 + 7*w^3 + w^2 - 12*w - 2],
[41, 41, 2*w^5 + w^4 - 13*w^3 - 5*w^2 + 19*w + 4],
[59, 59, -w^5 + 7*w^3 + w^2 - 11*w],
[61, 61, w^5 - 7*w^3 + 10*w],
[61, 61, -2*w^5 + 12*w^3 + w^2 - 16*w - 3],
[71, 71, -2*w^5 - w^4 + 12*w^3 + 5*w^2 - 16*w - 4],
[71, 71, -3*w^5 - w^4 + 20*w^3 + 6*w^2 - 31*w - 5],
[71, 71, -w^5 + 7*w^3 - w^2 - 12*w + 2],
[79, 79, w^5 + w^4 - 7*w^3 - 5*w^2 + 10*w + 4],
[79, 79, -w^5 - w^4 + 8*w^3 + 4*w^2 - 15*w + 1],
[79, 79, 2*w^5 + w^4 - 13*w^3 - 5*w^2 + 20*w + 3],
[89, 89, 2*w^5 + w^4 - 14*w^3 - 5*w^2 + 23*w + 1],
[89, 89, -2*w^5 - w^4 + 12*w^3 + 5*w^2 - 15*w - 4],
[89, 89, 2*w^5 + w^4 - 11*w^3 - 5*w^2 + 13*w + 3],
[101, 101, w^5 + w^4 - 7*w^3 - 3*w^2 + 11*w - 2],
[109, 109, w^5 + w^4 - 8*w^3 - 4*w^2 + 14*w],
[121, 11, -w^5 - w^4 + 8*w^3 + 4*w^2 - 13*w - 2],
[125, 5, -w^5 + 6*w^3 - 8*w - 2],
[131, 131, -w^4 + 5*w^2 - w - 3],
[131, 131, -w^5 - w^4 + 6*w^3 + 5*w^2 - 9*w - 3],
[139, 139, 2*w^5 - w^4 - 11*w^3 + 4*w^2 + 12*w - 2],
[149, 149, w^5 + w^4 - 7*w^3 - 5*w^2 + 9*w + 4],
[151, 151, w^5 - 8*w^3 + 2*w^2 + 15*w - 6],
[151, 151, -w^5 + 7*w^3 - 12*w + 3],
[151, 151, w^5 - 5*w^3 + 4*w - 3],
[151, 151, -w^4 + w^3 + 5*w^2 - 4*w - 5],
[179, 179, -3*w^5 - w^4 + 20*w^3 + 4*w^2 - 29*w - 1],
[179, 179, -3*w^5 - 2*w^4 + 20*w^3 + 10*w^2 - 31*w - 6],
[179, 179, -w^4 + 3*w^2 - 2*w + 2],
[179, 179, 3*w^5 + w^4 - 18*w^3 - 4*w^2 + 23*w + 1],
[181, 181, -2*w^5 - w^4 + 13*w^3 + 4*w^2 - 20*w + 1],
[181, 181, w^5 - 8*w^3 + 14*w - 3],
[191, 191, w^5 + w^4 - 7*w^3 - 6*w^2 + 11*w + 5],
[199, 199, -2*w^5 - 2*w^4 + 14*w^3 + 10*w^2 - 23*w - 5],
[199, 199, w^5 + w^4 - 8*w^3 - 6*w^2 + 14*w + 4],
[199, 199, -w^5 + 6*w^3 + 2*w^2 - 8*w - 2],
[199, 199, w^5 - w^4 - 6*w^3 + 4*w^2 + 8*w - 2],
[211, 211, -3*w^5 - 2*w^4 + 20*w^3 + 9*w^2 - 30*w - 5],
[229, 229, 2*w^5 + w^4 - 14*w^3 - 4*w^2 + 22*w - 1],
[229, 229, 4*w^5 + w^4 - 25*w^3 - 5*w^2 + 35*w + 1],
[239, 239, 2*w^4 - w^3 - 8*w^2 + 5*w + 2],
[241, 241, w^4 - 3*w^2 + w - 2],
[251, 251, -3*w^5 - 2*w^4 + 21*w^3 + 9*w^2 - 33*w - 3],
[251, 251, -3*w^5 - w^4 + 21*w^3 + 4*w^2 - 34*w + 1],
[251, 251, -2*w^5 - 2*w^4 + 15*w^3 + 9*w^2 - 26*w - 4],
[251, 251, 2*w^5 - 11*w^3 + w^2 + 12*w - 3],
[269, 269, -2*w^5 + 11*w^3 - w^2 - 12*w + 1],
[271, 271, -w^5 + 5*w^3 - w^2 - 5*w + 3],
[281, 281, w^4 - w^3 - 5*w^2 + 5*w + 5],
[289, 17, -2*w^5 - w^4 + 11*w^3 + 4*w^2 - 12*w - 2],
[311, 311, -3*w^5 + 19*w^3 - w^2 - 26*w + 4],
[311, 311, -w^4 + 3*w^2 - w + 3],
[311, 311, 3*w^5 - 19*w^3 + w^2 + 27*w - 4],
[311, 311, w^5 + 2*w^4 - 7*w^3 - 10*w^2 + 12*w + 8],
[331, 331, -3*w^5 - w^4 + 19*w^3 + 4*w^2 - 27*w - 1],
[349, 349, -2*w^5 - 2*w^4 + 14*w^3 + 8*w^2 - 22*w - 3],
[349, 349, -2*w^5 - w^4 + 14*w^3 + 4*w^2 - 22*w + 2],
[349, 349, 3*w^5 + w^4 - 19*w^3 - 6*w^2 + 25*w + 4],
[359, 359, -4*w^5 - w^4 + 25*w^3 + 4*w^2 - 35*w + 2],
[359, 359, 4*w^5 + w^4 - 25*w^3 - 6*w^2 + 34*w + 4],
[361, 19, 2*w^5 - w^4 - 12*w^3 + 5*w^2 + 15*w - 6],
[379, 379, 2*w^5 + 2*w^4 - 14*w^3 - 8*w^2 + 23*w + 1],
[379, 379, -2*w^5 - 2*w^4 + 14*w^3 + 10*w^2 - 23*w - 7],
[389, 389, w^5 - w^4 - 4*w^3 + 5*w^2 + w - 3],
[401, 401, -w^5 + w^4 + 7*w^3 - 4*w^2 - 12*w + 1],
[401, 401, -3*w^5 - 2*w^4 + 20*w^3 + 8*w^2 - 31*w - 2],
[401, 401, 2*w^5 + w^4 - 11*w^3 - 6*w^2 + 12*w + 5],
[401, 401, -4*w^5 - w^4 + 26*w^3 + 4*w^2 - 38*w + 1],
[409, 409, -w^5 + w^4 + 7*w^3 - 4*w^2 - 11*w + 3],
[409, 409, 3*w^5 - 17*w^3 - w^2 + 20*w + 2],
[419, 419, -3*w^5 - w^4 + 21*w^3 + 4*w^2 - 33*w + 1],
[419, 419, -2*w^5 + w^4 + 12*w^3 - 5*w^2 - 16*w + 4],
[419, 419, -2*w^5 - 2*w^4 + 15*w^3 + 9*w^2 - 26*w - 3],
[419, 419, w^2 + 2*w - 2],
[421, 421, 3*w^5 + w^4 - 21*w^3 - 5*w^2 + 34*w],
[421, 421, w^4 - 4*w^2 - 2*w + 2],
[421, 421, -3*w^5 + 18*w^3 + w^2 - 24*w - 3],
[421, 421, -2*w^5 + 14*w^3 + w^2 - 22*w + 1],
[431, 431, 2*w^5 + w^4 - 13*w^3 - 3*w^2 + 18*w - 1],
[431, 431, 3*w^5 + 2*w^4 - 20*w^3 - 8*w^2 + 30*w + 1],
[431, 431, 2*w^5 + w^4 - 11*w^3 - 6*w^2 + 12*w + 7],
[439, 439, -3*w^5 + 19*w^3 + 2*w^2 - 27*w - 3],
[461, 461, -3*w^5 + 20*w^3 - w^2 - 30*w + 5],
[461, 461, w^5 - 5*w^3 + 5*w - 3],
[491, 491, -w^5 - w^4 + 9*w^3 + 4*w^2 - 18*w + 1],
[491, 491, 2*w^5 - w^4 - 12*w^3 + 3*w^2 + 15*w],
[499, 499, -3*w^5 - w^4 + 20*w^3 + 5*w^2 - 29*w - 3],
[499, 499, -2*w^5 - w^4 + 12*w^3 + 4*w^2 - 16*w - 3],
[509, 509, -w^5 + w^4 + 6*w^3 - 4*w^2 - 9*w],
[509, 509, 2*w^5 - 14*w^3 + 24*w - 3],
[509, 509, -2*w^5 - w^4 + 14*w^3 + 5*w^2 - 24*w],
[509, 509, -2*w^5 - 2*w^4 + 14*w^3 + 9*w^2 - 22*w - 5],
[569, 569, -2*w^5 + w^4 + 12*w^3 - 3*w^2 - 17*w + 3],
[571, 571, -w^5 + w^4 + 5*w^3 - 4*w^2 - 5*w - 1],
[571, 571, -w^5 - w^4 + 8*w^3 + 3*w^2 - 16*w + 2],
[571, 571, -2*w^5 + 13*w^3 - 21*w + 2],
[571, 571, 2*w^5 + w^4 - 12*w^3 - 7*w^2 + 15*w + 8],
[571, 571, 2*w^5 + 2*w^4 - 13*w^3 - 10*w^2 + 19*w + 8],
[571, 571, 2*w^5 - 12*w^3 + 15*w - 4],
[601, 601, 3*w^5 + 2*w^4 - 21*w^3 - 9*w^2 + 33*w + 2],
[601, 601, 3*w^5 + w^4 - 19*w^3 - 3*w^2 + 25*w - 4],
[601, 601, -2*w^5 - 2*w^4 + 14*w^3 + 10*w^2 - 22*w - 5],
[601, 601, -3*w^5 - w^4 + 18*w^3 + 6*w^2 - 23*w - 6],
[619, 619, w^5 + w^4 - 7*w^3 - 5*w^2 + 13*w + 4],
[631, 631, 2*w^5 + w^4 - 15*w^3 - 4*w^2 + 26*w - 1],
[631, 631, -2*w^5 + 12*w^3 + w^2 - 14*w - 4],
[641, 641, -w^5 + w^4 + 6*w^3 - 6*w^2 - 9*w + 7],
[641, 641, -2*w^5 - w^4 + 13*w^3 + 7*w^2 - 19*w - 8],
[659, 659, 2*w^5 - w^4 - 12*w^3 + 4*w^2 + 16*w - 4],
[659, 659, 2*w^5 - 13*w^3 + 20*w - 4],
[661, 661, -2*w^5 - w^4 + 13*w^3 + 3*w^2 - 19*w],
[661, 661, -w^5 - w^4 + 6*w^3 + 7*w^2 - 8*w - 8],
[661, 661, -3*w^5 - 2*w^4 + 21*w^3 + 10*w^2 - 34*w - 5],
[691, 691, 3*w^5 + w^4 - 20*w^3 - 3*w^2 + 31*w - 3],
[691, 691, 5*w^5 + 2*w^4 - 34*w^3 - 9*w^2 + 54*w + 3],
[691, 691, -2*w^5 - w^4 + 13*w^3 + 5*w^2 - 22*w - 1],
[691, 691, 3*w^5 + w^4 - 19*w^3 - 5*w^2 + 28*w],
[691, 691, w^5 + w^4 - 8*w^3 - 3*w^2 + 14*w - 4],
[691, 691, -w^5 - 2*w^4 + 8*w^3 + 9*w^2 - 15*w - 5],
[701, 701, -3*w^5 - w^4 + 19*w^3 + 7*w^2 - 26*w - 11],
[709, 709, -w^5 + 7*w^3 - w^2 - 13*w + 4],
[709, 709, -w^5 + 4*w^3 - 2*w - 4],
[709, 709, 2*w^5 + 2*w^4 - 15*w^3 - 8*w^2 + 27*w + 1],
[709, 709, -w^5 - w^4 + 6*w^3 + 6*w^2 - 7*w - 4],
[719, 719, -w^4 + w^3 + 4*w^2 - 3*w + 2],
[719, 719, -3*w^5 - w^4 + 20*w^3 + 6*w^2 - 32*w - 2],
[729, 3, -3],
[739, 739, w^5 + 2*w^4 - 7*w^3 - 9*w^2 + 11*w + 7],
[739, 739, -3*w^5 - 2*w^4 + 21*w^3 + 10*w^2 - 34*w - 6],
[739, 739, w^5 + w^4 - 6*w^3 - 3*w^2 + 9*w - 3],
[739, 739, w^4 + w^3 - 4*w^2 - 2*w + 1],
[739, 739, -3*w^5 + 18*w^3 + w^2 - 23*w + 1],
[739, 739, 3*w^5 - 18*w^3 - 2*w^2 + 24*w + 5],
[751, 751, -w^5 + w^4 + 5*w^3 - 5*w^2 - 5*w + 2],
[761, 761, -w^5 - 2*w^4 + 8*w^3 + 8*w^2 - 15*w - 4],
[761, 761, -w^5 + 6*w^3 - 6*w + 2],
[769, 769, w^3 - 2*w^2 - 3*w + 3],
[811, 811, -3*w^5 + 19*w^3 + w^2 - 25*w - 1],
[829, 829, -2*w^5 + w^4 + 13*w^3 - 4*w^2 - 19*w + 2],
[829, 829, 3*w^5 + w^4 - 20*w^3 - 4*w^2 + 29*w - 1],
[839, 839, -2*w^5 - w^4 + 15*w^3 + 5*w^2 - 25*w - 1],
[839, 839, -w^5 + 6*w^3 + 2*w^2 - 6*w - 6],
[841, 29, 4*w^5 + 2*w^4 - 27*w^3 - 10*w^2 + 41*w + 5],
[859, 859, -2*w^5 + 13*w^3 - 17*w + 2],
[881, 881, -2*w^5 - 2*w^4 + 14*w^3 + 8*w^2 - 23*w - 3],
[881, 881, w^5 - 7*w^3 + w^2 + 11*w],
[911, 911, 4*w^5 - 25*w^3 + 35*w - 3],
[911, 911, 6*w^5 + 2*w^4 - 39*w^3 - 11*w^2 + 58*w + 5],
[919, 919, w^4 + w^3 - 6*w^2 - 3*w + 6],
[919, 919, -3*w^5 + 17*w^3 - 20*w + 1],
[929, 929, -w^3 + 2*w^2 + 3*w - 8],
[941, 941, 2*w^5 + 2*w^4 - 15*w^3 - 8*w^2 + 27*w],
[941, 941, 2*w^5 + w^4 - 15*w^3 - 3*w^2 + 25*w - 3],
[961, 31, -3*w^5 - w^4 + 18*w^3 + 6*w^2 - 21*w - 6],
[971, 971, -w^5 + 4*w^3 + 2*w^2 - 5],
[991, 991, -2*w^5 - 3*w^4 + 14*w^3 + 13*w^2 - 22*w - 6],
[1009, 1009, -3*w^5 + 17*w^3 - 20*w],
[1009, 1009, 4*w^5 - 25*w^3 + 35*w - 1],
[1019, 1019, -3*w^5 + 17*w^3 + 2*w^2 - 20*w - 4],
[1019, 1019, -w^5 + 5*w^3 - w^2 - 3*w + 1],
[1019, 1019, -w^3 + 2*w^2 + 5*w - 5],
[1021, 1021, 2*w^5 + w^4 - 13*w^3 - 4*w^2 + 18*w + 3],
[1031, 1031, 2*w^5 - 11*w^3 - 2*w^2 + 12*w + 6],
[1031, 1031, 2*w^5 - 14*w^3 + w^2 + 23*w - 2],
[1039, 1039, w^4 - 2*w^3 - 5*w^2 + 7*w + 5],
[1039, 1039, 2*w^5 - w^4 - 11*w^3 + 5*w^2 + 13*w - 5],
[1039, 1039, -w^5 - w^4 + 7*w^3 + 6*w^2 - 13*w - 6],
[1039, 1039, 4*w^5 + w^4 - 23*w^3 - 5*w^2 + 28*w + 3],
[1049, 1049, -w^5 + w^4 + 6*w^3 - 6*w^2 - 9*w + 6],
[1049, 1049, -3*w^5 + 19*w^3 + w^2 - 25*w + 2],
[1051, 1051, w^5 + w^4 - 7*w^3 - 2*w^2 + 12*w - 5],
[1069, 1069, w^5 + 2*w^4 - 7*w^3 - 9*w^2 + 12*w + 6],
[1091, 1091, -4*w^5 + w^4 + 24*w^3 - 4*w^2 - 30*w + 2],
[1109, 1109, w^5 + w^4 - 5*w^3 - 5*w^2 + 6*w + 3],
[1129, 1129, -3*w^5 + 19*w^3 + 2*w^2 - 26*w - 3],
[1129, 1129, -2*w^5 - w^4 + 14*w^3 + 2*w^2 - 22*w + 2],
[1129, 1129, 2*w^5 - 11*w^3 - 2*w^2 + 11*w + 3],
[1129, 1129, -4*w^5 - 2*w^4 + 25*w^3 + 11*w^2 - 35*w - 11],
[1151, 1151, 2*w^5 + w^4 - 14*w^3 - 5*w^2 + 21*w + 4],
[1181, 1181, w^5 - 7*w^3 + w^2 + 12*w],
[1201, 1201, w^5 + 2*w^4 - 6*w^3 - 9*w^2 + 8*w + 5],
[1229, 1229, w^5 + 2*w^4 - 8*w^3 - 10*w^2 + 14*w + 6],
[1229, 1229, 2*w^5 + w^4 - 13*w^3 - 5*w^2 + 17*w + 4],
[1231, 1231, -2*w^5 - 2*w^4 + 15*w^3 + 9*w^2 - 25*w - 3],
[1249, 1249, -2*w^5 - 2*w^4 + 14*w^3 + 10*w^2 - 21*w - 7],
[1249, 1249, -w^5 - w^4 + 7*w^3 + 7*w^2 - 11*w - 8],
[1259, 1259, w^3 + w^2 - 3*w - 5],
[1259, 1259, -w^5 - w^4 + 9*w^3 + 3*w^2 - 19*w + 2],
[1259, 1259, -w^4 + w^3 + 3*w^2 - 3*w + 4],
[1279, 1279, -5*w^5 - 2*w^4 + 35*w^3 + 9*w^2 - 57*w - 2],
[1279, 1279, 3*w^5 + 2*w^4 - 20*w^3 - 9*w^2 + 31*w + 5],
[1289, 1289, -w^5 + w^4 + 5*w^3 - 5*w^2 - 6*w + 4],
[1291, 1291, w^5 + w^4 - 7*w^3 - 3*w^2 + 10*w - 4],
[1291, 1291, -w^5 - w^4 + 9*w^3 + 3*w^2 - 18*w + 4],
[1291, 1291, 2*w^5 - 13*w^3 + 17*w],
[1291, 1291, -5*w^5 - 3*w^4 + 32*w^3 + 15*w^2 - 46*w - 8],
[1321, 1321, w^4 - w^3 - 4*w^2 + 2*w - 1],
[1321, 1321, 2*w^5 - 2*w^4 - 12*w^3 + 8*w^2 + 14*w - 3],
[1321, 1321, 3*w^5 - 19*w^3 - 2*w^2 + 26*w + 4],
[1321, 1321, -2*w^5 - w^4 + 12*w^3 + 3*w^2 - 15*w + 1],
[1331, 11, -3*w^5 + 18*w^3 - 24*w + 1],
[1361, 1361, -4*w^5 - w^4 + 25*w^3 + 6*w^2 - 35*w - 6],
[1381, 1381, -2*w^5 - 3*w^4 + 14*w^3 + 14*w^2 - 24*w - 6],
[1381, 1381, -3*w^5 - w^4 + 19*w^3 + 4*w^2 - 28*w + 3],
[1381, 1381, 2*w^5 + w^4 - 12*w^3 - 3*w^2 + 15*w - 3],
[1399, 1399, 4*w^5 + 2*w^4 - 27*w^3 - 10*w^2 + 44*w + 4],
[1409, 1409, 2*w^5 - 14*w^3 + 21*w - 1],
[1429, 1429, -3*w^5 - w^4 + 20*w^3 + 6*w^2 - 29*w - 5],
[1429, 1429, -3*w^5 + 18*w^3 - w^2 - 23*w + 3],
[1429, 1429, w^4 - 2*w^2 - 2*w - 5],
[1439, 1439, 3*w^5 + 3*w^4 - 19*w^3 - 14*w^2 + 27*w + 5],
[1451, 1451, w^5 + 2*w^4 - 10*w^3 - 6*w^2 + 22*w - 4],
[1459, 1459, -w^5 + w^4 + 7*w^3 - 5*w^2 - 10*w + 4],
[1471, 1471, -w^5 - 2*w^4 + 9*w^3 + 9*w^2 - 17*w - 6],
[1481, 1481, 2*w^5 + w^4 - 14*w^3 - 5*w^2 + 21*w + 5],
[1489, 1489, 3*w^5 - w^4 - 17*w^3 + 3*w^2 + 22*w - 2],
[1499, 1499, -2*w^4 + w^3 + 8*w^2 - 3*w - 4],
[1511, 1511, 3*w^5 - 18*w^3 - 2*w^2 + 23*w + 4],
[1511, 1511, w^5 + 2*w^4 - 8*w^3 - 10*w^2 + 14*w + 8],
[1511, 1511, w^5 + 2*w^4 - 10*w^3 - 7*w^2 + 22*w - 3],
[1511, 1511, 5*w^5 + w^4 - 29*w^3 - 5*w^2 + 34*w + 4],
[1531, 1531, 2*w^4 - 3*w^3 - 8*w^2 + 8*w + 2],
[1549, 1549, -4*w^5 + 27*w^3 - 41*w + 6],
[1549, 1549, w^5 + w^4 - 7*w^3 - 4*w^2 + 13*w + 5],
[1559, 1559, 2*w^5 - 11*w^3 + w^2 + 12*w - 4],
[1559, 1559, -w^5 + 6*w^3 + w^2 - 10*w - 3],
[1571, 1571, w^3 + 2*w^2 - 4*w - 6],
[1579, 1579, 2*w^5 + 2*w^4 - 15*w^3 - 11*w^2 + 26*w + 8],
[1601, 1601, -w^5 + 8*w^3 - 2*w^2 - 16*w + 3],
[1601, 1601, 3*w^5 + 2*w^4 - 19*w^3 - 7*w^2 + 27*w - 2],
[1609, 1609, -5*w^5 - 2*w^4 + 32*w^3 + 10*w^2 - 47*w - 2],
[1619, 1619, 4*w^5 + 3*w^4 - 25*w^3 - 14*w^2 + 35*w + 8],
[1619, 1619, -2*w^5 + 13*w^3 - 17*w + 1],
[1621, 1621, -5*w^5 - 3*w^4 + 34*w^3 + 15*w^2 - 53*w - 7],
[1621, 1621, 4*w^5 + 2*w^4 - 26*w^3 - 7*w^2 + 38*w - 5],
[1669, 1669, w^5 + w^4 - 9*w^3 - 4*w^2 + 17*w - 1],
[1681, 41, 4*w^5 - 24*w^3 - 2*w^2 + 33*w + 5],
[1681, 41, 2*w^4 - 2*w^3 - 9*w^2 + 6*w + 4],
[1699, 1699, -3*w^5 - 2*w^4 + 18*w^3 + 10*w^2 - 24*w - 3],
[1699, 1699, -2*w^5 + w^4 + 13*w^3 - 3*w^2 - 19*w + 4],
[1699, 1699, 4*w^5 - w^4 - 23*w^3 + 3*w^2 + 28*w - 1],
[1699, 1699, 2*w^4 - w^3 - 8*w^2 + 3*w + 2],
[1709, 1709, -4*w^5 - 2*w^4 + 26*w^3 + 11*w^2 - 38*w - 7],
[1709, 1709, 3*w^5 - 17*w^3 + w^2 + 20*w - 4],
[1721, 1721, 2*w^5 + 2*w^4 - 14*w^3 - 9*w^2 + 22*w - 1],
[1721, 1721, -2*w^5 + w^4 + 12*w^3 - 5*w^2 - 17*w + 5],
[1721, 1721, w^5 - w^4 - 7*w^3 + 7*w^2 + 10*w - 12],
[1721, 1721, -2*w^5 - w^4 + 16*w^3 + 4*w^2 - 30*w + 4],
[1741, 1741, -2*w^5 + w^4 + 12*w^3 - 3*w^2 - 15*w + 1],
[1741, 1741, 3*w^5 + 2*w^4 - 19*w^3 - 10*w^2 + 26*w + 8],
[1759, 1759, 3*w^5 + w^4 - 19*w^3 - 6*w^2 + 25*w + 5],
[1789, 1789, w^4 + w^3 - 4*w^2 - 5*w + 2],
[1789, 1789, w^5 - 3*w^3 - 3*w + 3],
[1789, 1789, -3*w^5 - w^4 + 19*w^3 + 7*w^2 - 26*w - 8],
[1789, 1789, 2*w^5 - 13*w^3 + w^2 + 20*w - 1],
[1801, 1801, 6*w^5 + 2*w^4 - 39*w^3 - 9*w^2 + 56*w + 3],
[1811, 1811, 2*w^5 + w^4 - 15*w^3 - 4*w^2 + 26*w - 3],
[1811, 1811, -4*w^5 - w^4 + 26*w^3 + 5*w^2 - 37*w - 5],
[1811, 1811, -4*w^5 + w^4 + 23*w^3 - 3*w^2 - 27*w],
[1831, 1831, 5*w^5 + w^4 - 30*w^3 - 4*w^2 + 39*w],
[1831, 1831, 2*w^5 - 12*w^3 - w^2 + 16*w + 6],
[1831, 1831, 2*w^5 + 2*w^4 - 14*w^3 - 12*w^2 + 22*w + 13],
[1831, 1831, w^5 - 7*w^3 + 11*w + 3],
[1871, 1871, 3*w^5 + w^4 - 21*w^3 - 3*w^2 + 32*w - 1],
[1871, 1871, 3*w^5 + w^4 - 17*w^3 - 7*w^2 + 22*w + 8],
[1871, 1871, w - 4],
[1879, 1879, -3*w^5 - 2*w^4 + 21*w^3 + 9*w^2 - 32*w - 6],
[1901, 1901, 3*w^5 + 2*w^4 - 20*w^3 - 7*w^2 + 31*w - 5],
[1931, 1931, -4*w^5 - 3*w^4 + 27*w^3 + 15*w^2 - 41*w - 10],
[1931, 1931, -3*w^5 - w^4 + 19*w^3 + 7*w^2 - 26*w - 7],
[1949, 1949, -2*w^5 - 3*w^4 + 13*w^3 + 13*w^2 - 19*w - 8],
[1949, 1949, -5*w^5 - 2*w^4 + 33*w^3 + 8*w^2 - 50*w - 2],
[1949, 1949, -4*w^5 - 2*w^4 + 28*w^3 + 11*w^2 - 45*w - 6],
[1951, 1951, 5*w^5 + w^4 - 31*w^3 - 3*w^2 + 42*w - 5],
[1951, 1951, 3*w^5 + 2*w^4 - 21*w^3 - 11*w^2 + 35*w + 8],
[1979, 1979, 5*w^5 + w^4 - 31*w^3 - 3*w^2 + 43*w - 6],
[1979, 1979, -5*w^5 - w^4 + 31*w^3 + 7*w^2 - 42*w - 9],
[1999, 1999, -3*w^5 - w^4 + 21*w^3 + 4*w^2 - 33*w + 2],
[1999, 1999, 3*w^5 + 2*w^4 - 17*w^3 - 10*w^2 + 20*w + 7],
[1999, 1999, w^5 + w^4 - 9*w^3 - 5*w^2 + 19*w + 6]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x;
K := Rationals(); e := 1;

heckeEigenvaluesArray := [-1, -4, -5, 4, -6, 0, 1, -6, 4, 10, 14, -8, 8, 0, -8, -8, -8, 10, -10, -10, -6, -18, 10, -18, 12, -20, 4, 10, 0, 16, 8, -8, 4, -4, 12, 20, 18, 2, -24, 16, 24, 16, -8, 12, -22, 10, 8, -2, -4, -12, 28, -4, 10, 8, 14, 14, 16, -16, -24, 32, -12, -10, -30, 10, 0, 8, -10, -4, 20, -30, -6, -30, -30, 22, 2, 38, -12, 4, 4, 28, -26, 26, -6, -30, -8, 24, 16, 16, -14, 18, -36, -28, 4, -12, 34, -42, 26, -2, -2, 28, -28, -4, 20, 36, 20, -18, -10, -38, 2, -12, 16, -8, 10, -30, 12, -44, -22, 6, -6, -20, 12, -4, -36, 28, -44, 34, 6, 6, 46, -10, 32, -40, -14, 44, 4, -20, 28, 20, -4, 32, -42, -42, -14, 28, 2, -50, -40, 24, 30, 12, -22, -18, -48, 32, -48, -8, 14, -14, -50, -50, 12, -16, 46, -18, 60, 4, 12, 10, -56, 0, -48, -56, 8, -8, 42, 6, -36, -14, -52, 22, -54, -6, -46, -22, -48, 6, 66, 58, 58, -8, -30, -14, 60, 28, 28, -16, 56, 22, -12, 52, -60, -44, 38, -26, 18, -10, 60, -34, 30, 62, 54, 40, -14, -42, -54, -22, -48, -12, -36, -8, 42, 50, -36, -24, 0, -56, 40, -44, 14, 50, -64, -72, 60, -60, -54, -18, -58, 20, 12, -38, -10, -58, 2, -14, 20, 44, 28, -76, 50, -78, -22, -42, -58, 38, 42, -18, 0, 30, 62, -2, -34, -38, 20, -20, -60, -16, 16, 8, -8, -80, 48, -48, -24, 14, -12, 12, -30, 62, -30, 32, -32, -84, 36, 8, 24, -40];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {41, 41, -w^5 + 7*w^3 + w^2 - 12*w - 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;