/*
  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 | {16, 2, -w^5 + 6*w^3 + w^2 - 7*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^6 - 6*x^5 + x^4 + 38*x^3 - 25*x^2 - 57*x + 37;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, 1/7*e^5 - 2/7*e^4 - 2*e^3 + 24/7*e^2 + 36/7*e - 39/7, -1, 1/7*e^5 - 2/7*e^4 - 2*e^3 + 17/7*e^2 + 57/7*e - 32/7, -3/7*e^5 + 13/7*e^4 - 37/7*e^2 + 32/7*e - 2/7, -2/7*e^5 + 4/7*e^4 + 4*e^3 - 48/7*e^2 - 72/7*e + 113/7, 1/7*e^5 - 9/7*e^4 + e^3 + 59/7*e^2 - 27/7*e - 67/7, -2/7*e^5 + 11/7*e^4 - 2*e^3 - 20/7*e^2 + 75/7*e - 20/7, -5/7*e^5 + 31/7*e^4 - 2*e^3 - 141/7*e^2 + 65/7*e + 90/7, -e^4 + 3*e^3 + 5*e^2 - 8*e - 11, -5/7*e^5 + 17/7*e^4 + 4*e^3 - 92/7*e^2 - 33/7*e + 132/7, -1/7*e^5 + 2/7*e^4 + 4*e^3 - 73/7*e^2 - 71/7*e + 158/7, 3/7*e^5 - 13/7*e^4 + 37/7*e^2 - 25/7*e - 33/7, 8/7*e^5 - 37/7*e^4 - e^3 + 150/7*e^2 - 34/7*e - 81/7, 1/7*e^5 - 2/7*e^4 - e^3 + 10/7*e^2 - 6/7*e + 3/7, 3/7*e^5 - 6/7*e^4 - 5*e^3 + 30/7*e^2 + 108/7*e + 9/7, -5/7*e^5 + 24/7*e^4 + 3*e^3 - 155/7*e^2 - 19/7*e + 167/7, -3/7*e^5 - 1/7*e^4 + 6*e^3 + 26/7*e^2 - 87/7*e - 37/7, 4/7*e^5 - 15/7*e^4 - 3*e^3 + 82/7*e^2 + 11/7*e - 37/7, -e^4 + 3*e^3 + 3*e^2 - 5*e + 1, -4/7*e^5 + 22/7*e^4 - e^3 - 89/7*e^2 + 52/7*e + 9/7, -e^4 + 4*e^3 + 5*e^2 - 18*e - 4, 1/7*e^5 + 5/7*e^4 - 7*e^3 + 10/7*e^2 + 183/7*e - 11/7, -5/7*e^5 + 17/7*e^4 + 7*e^3 - 141/7*e^2 - 152/7*e + 251/7, -5/7*e^5 + 38/7*e^4 - 5*e^3 - 169/7*e^2 + 121/7*e + 97/7, -1/7*e^5 + 16/7*e^4 - 6*e^3 - 38/7*e^2 + 125/7*e - 87/7, 1/7*e^5 - 2/7*e^4 - 6*e^3 + 108/7*e^2 + 127/7*e - 221/7, 1/7*e^5 - 23/7*e^4 + 7*e^3 + 115/7*e^2 - 118/7*e - 88/7, -2*e^4 + 3*e^3 + 16*e^2 + e - 24, -3*e^3 + 9*e^2 + 12*e - 13, 1/7*e^5 - 2/7*e^4 - 25/7*e^2 + 1/7*e + 115/7, 2/7*e^5 - 25/7*e^4 + 8*e^3 + 69/7*e^2 - 166/7*e + 62/7, 6/7*e^5 - 12/7*e^4 - 10*e^3 + 109/7*e^2 + 132/7*e - 171/7, 5/7*e^5 - 17/7*e^4 - 5*e^3 + 106/7*e^2 + 47/7*e - 125/7, 8/7*e^5 - 44/7*e^4 + 5*e^3 + 108/7*e^2 - 167/7*e + 87/7, -4/7*e^5 + 22/7*e^4 + 2*e^3 - 159/7*e^2 - 32/7*e + 219/7, -6/7*e^5 + 12/7*e^4 + 9*e^3 - 81/7*e^2 - 146/7*e + 213/7, 8/7*e^5 - 23/7*e^4 - 10*e^3 + 136/7*e^2 + 155/7*e - 67/7, 1/7*e^5 - 2/7*e^4 + e^3 - 32/7*e^2 - 41/7*e + 24/7, -4/7*e^5 - 6/7*e^4 + 16*e^3 - 75/7*e^2 - 305/7*e + 233/7, 3/7*e^5 - 20/7*e^4 + 6*e^3 - 5/7*e^2 - 165/7*e + 191/7, 4/7*e^5 - 1/7*e^4 - 10*e^3 + 12/7*e^2 + 221/7*e - 2/7, 2/7*e^5 - 11/7*e^4 + 4*e^3 - 22/7*e^2 - 117/7*e + 153/7, 10/7*e^5 - 34/7*e^4 - 9*e^3 + 191/7*e^2 + 101/7*e - 243/7, -1/7*e^5 + 9/7*e^4 - 66/7*e^2 - 36/7*e + 158/7, -5/7*e^5 + 24/7*e^4 + e^3 - 120/7*e^2 + 51/7*e + 76/7, 6/7*e^5 - 5/7*e^4 - 15*e^3 + 109/7*e^2 + 258/7*e - 262/7, -1/7*e^5 + 2/7*e^4 - 2*e^3 + 67/7*e^2 + 34/7*e - 164/7, 13/7*e^5 - 47/7*e^4 - 9*e^3 + 235/7*e^2 + 62/7*e - 248/7, -2/7*e^5 + 4/7*e^4 + 7*e^3 - 111/7*e^2 - 156/7*e + 190/7, 4/7*e^5 - 15/7*e^4 - 2*e^3 + 96/7*e^2 - 101/7*e - 163/7, 6/7*e^5 - 19/7*e^4 - 10*e^3 + 193/7*e^2 + 202/7*e - 360/7, -9/7*e^5 + 39/7*e^4 + e^3 - 139/7*e^2 + 61/7*e + 64/7, -e^5 + 6*e^4 - e^3 - 31*e^2 + 14*e + 18, -2/7*e^5 + 4/7*e^4 + 5*e^3 - 69/7*e^2 - 37/7*e + 148/7, 15/7*e^5 - 65/7*e^4 - 5*e^3 + 262/7*e^2 + 36/7*e - 81/7, -8/7*e^5 + 37/7*e^4 - 3*e^3 - 66/7*e^2 + 132/7*e - 129/7, -8/7*e^5 + 58/7*e^4 - 7*e^3 - 276/7*e^2 + 195/7*e + 179/7, 8/7*e^5 - 23/7*e^4 - 8*e^3 + 115/7*e^2 + 85/7*e - 151/7, -6/7*e^5 + 33/7*e^4 + 3*e^3 - 242/7*e^2 - 34/7*e + 283/7, -1/7*e^5 + 16/7*e^4 - 3*e^3 - 108/7*e^2 + 48/7*e + 228/7, 1/7*e^5 + 26/7*e^4 - 15*e^3 - 67/7*e^2 + 274/7*e - 95/7, -11/7*e^5 + 43/7*e^4 + 9*e^3 - 243/7*e^2 - 158/7*e + 338/7, -2/7*e^5 - 3/7*e^4 + 9*e^3 - 55/7*e^2 - 191/7*e + 253/7, 8/7*e^5 - 16/7*e^4 - 16*e^3 + 157/7*e^2 + 309/7*e - 193/7, 10/7*e^5 - 20/7*e^4 - 17*e^3 + 184/7*e^2 + 220/7*e - 306/7, 13/7*e^5 - 61/7*e^4 - e^3 + 228/7*e^2 - 43/7*e - 122/7, -2/7*e^5 - 10/7*e^4 + 3*e^3 + 169/7*e^2 - 23/7*e - 279/7, 2/7*e^5 - 25/7*e^4 + 9*e^3 + 41/7*e^2 - 201/7*e + 111/7, -13/7*e^5 + 47/7*e^4 + 8*e^3 - 172/7*e^2 - 146/7*e + 59/7, 1/7*e^5 - 9/7*e^4 - 3*e^3 + 157/7*e^2 + 78/7*e - 354/7, -11/7*e^5 + 50/7*e^4 + 5*e^3 - 257/7*e^2 - 53/7*e + 128/7, 3/7*e^5 + 8/7*e^4 - 10*e^3 - 40/7*e^2 + 164/7*e - 47/7, 2/7*e^5 - 11/7*e^4 + 3*e^3 - 36/7*e^2 - 68/7*e + 286/7, -9/7*e^5 + 46/7*e^4 - 6*e^3 - 62/7*e^2 + 208/7*e - 223/7, -2/7*e^5 - 3/7*e^4 + 7*e^3 + 1/7*e^2 - 219/7*e + 92/7, 9/7*e^5 - 39/7*e^4 - 7*e^3 + 244/7*e^2 + 149/7*e - 323/7, -3/7*e^5 + 6/7*e^4 + 8*e^3 - 121/7*e^2 - 143/7*e + 222/7, -2/7*e^5 - 10/7*e^4 + 14*e^3 - 34/7*e^2 - 366/7*e + 141/7, 3*e^3 - 6*e^2 - 25*e + 14, -9/7*e^5 + 25/7*e^4 + 9*e^3 - 111/7*e^2 - 58/7*e + 36/7, -2/7*e^5 - 10/7*e^4 + 11*e^3 + 43/7*e^2 - 268/7*e - 111/7, 12/7*e^5 - 31/7*e^4 - 13*e^3 + 134/7*e^2 + 166/7*e - 20/7, e^5 - 3*e^4 - 7*e^3 + 9*e^2 + 22*e + 18, -6/7*e^5 + 19/7*e^4 + 3*e^3 - 4/7*e^2 - 97/7*e - 165/7, 2/7*e^5 - 4/7*e^4 + 3*e^3 - 106/7*e^2 - 138/7*e + 244/7, -8/7*e^5 + 30/7*e^4 + 2*e^3 - 87/7*e^2 + 90/7*e - 101/7, 15/7*e^5 - 51/7*e^4 - 13*e^3 + 276/7*e^2 + 120/7*e - 242/7, 5/7*e^5 - 24/7*e^4 + 3*e^3 + 43/7*e^2 - 128/7*e + 8/7, -5/7*e^5 - 4/7*e^4 + 9*e^3 + 90/7*e^2 - 124/7*e - 29/7, 8/7*e^5 - 30/7*e^4 - 9*e^3 + 241/7*e^2 + 85/7*e - 361/7, -2/7*e^5 + 32/7*e^4 - 14*e^3 - 83/7*e^2 + 355/7*e + 85/7, -5/7*e^5 + 31/7*e^4 - 2*e^3 - 169/7*e^2 + 79/7*e + 230/7, -12/7*e^5 + 66/7*e^4 - 5*e^3 - 246/7*e^2 + 226/7*e + 237/7, 13/7*e^5 - 54/7*e^4 - e^3 + 116/7*e^2 - 64/7*e + 158/7, 4/7*e^5 - 8/7*e^4 - 9*e^3 + 103/7*e^2 + 200/7*e - 191/7, 9/7*e^5 - 39/7*e^4 - 2*e^3 + 132/7*e^2 + 16/7*e - 8/7, -2/7*e^5 + 25/7*e^4 - 13*e^3 + 29/7*e^2 + 278/7*e - 251/7, 3/7*e^5 + 22/7*e^4 - 19*e^3 - 26/7*e^2 + 346/7*e - 54/7, -26/7*e^5 + 115/7*e^4 + 12*e^3 - 596/7*e^2 - 61/7*e + 489/7, -3/7*e^5 + 6/7*e^4 + 9*e^3 - 149/7*e^2 - 143/7*e + 397/7, 2/7*e^5 + 10/7*e^4 - 17*e^3 + 153/7*e^2 + 324/7*e - 477/7, -3/7*e^5 + 41/7*e^4 - 11*e^3 - 198/7*e^2 + 207/7*e + 257/7, 16/7*e^5 - 81/7*e^4 - 2*e^3 + 412/7*e^2 - 82/7*e - 400/7, -8/7*e^5 + 37/7*e^4 + 8*e^3 - 262/7*e^2 - 197/7*e + 396/7, -12/7*e^5 + 66/7*e^4 - 3*e^3 - 260/7*e^2 + 135/7*e + 244/7, -10/7*e^5 + 48/7*e^4 + 5*e^3 - 289/7*e^2 - 73/7*e + 383/7, -9/7*e^5 + 39/7*e^4 + 6*e^3 - 230/7*e^2 - 58/7*e + 169/7, -11/7*e^5 + 36/7*e^4 + 11*e^3 - 187/7*e^2 - 221/7*e + 191/7, -6/7*e^5 + 19/7*e^4 + 2*e^3 - 11/7*e^2 + 78/7*e - 221/7, 4/7*e^5 - 36/7*e^4 + 12*e^3 + 68/7*e^2 - 374/7*e + 82/7, 10/7*e^5 - 48/7*e^4 + 219/7*e^2 - 186/7*e - 215/7, 8/7*e^5 - 51/7*e^4 + e^3 + 297/7*e^2 - 76/7*e - 291/7, 13/7*e^5 - 47/7*e^4 - 10*e^3 + 249/7*e^2 + 83/7*e - 311/7, -e^5 + 6*e^4 + e^3 - 35*e^2 - 7*e + 38, -2*e^5 + 8*e^4 + 9*e^3 - 44*e^2 - 9*e + 60, -6/7*e^5 + 5/7*e^4 + 15*e^3 - 137/7*e^2 - 216/7*e + 283/7, -8/7*e^5 + 51/7*e^4 - 4*e^3 - 213/7*e^2 + 27/7*e + 74/7, 5/7*e^5 - 38/7*e^4 + 6*e^3 + 162/7*e^2 - 114/7*e - 34/7, e^5 - 3*e^4 - 5*e^3 + 9*e^2 + 8*e - 11, 3/7*e^5 - 20/7*e^4 + 7*e^3 - 54/7*e^2 - 165/7*e + 415/7, 1/7*e^5 + 12/7*e^4 - 10*e^3 - 18/7*e^2 + 274/7*e - 144/7, -13/7*e^5 + 61/7*e^4 + 3*e^3 - 270/7*e^2 - 55/7*e + 150/7, 12/7*e^5 - 66/7*e^4 + 3*e^3 + 260/7*e^2 - 163/7*e - 90/7, -6/7*e^5 + 40/7*e^4 - 6*e^3 - 165/7*e^2 + 176/7*e + 136/7, 5/7*e^5 - 3/7*e^4 - 10*e^3 + 1/7*e^2 + 117/7*e + 239/7, -12/7*e^5 + 66/7*e^4 - 2*e^3 - 344/7*e^2 + 205/7*e + 328/7, 3/7*e^5 - 20/7*e^4 - 3*e^3 + 184/7*e^2 + 115/7*e - 320/7, 1/7*e^5 - 16/7*e^4 + 5*e^3 + 52/7*e^2 - 132/7*e + 115/7, 19/7*e^5 - 73/7*e^4 - 17*e^3 + 463/7*e^2 + 271/7*e - 664/7, 11/7*e^5 - 8/7*e^4 - 25*e^3 + 145/7*e^2 + 375/7*e - 310/7, -8/7*e^5 + 30/7*e^4 - 2*e^3 + 53/7*e^2 + 83/7*e - 430/7, 5/7*e^5 - 17/7*e^4 - 6*e^3 + 92/7*e^2 + 96/7*e + 64/7, -22/7*e^5 + 107/7*e^4 + 6*e^3 - 556/7*e^2 - 8/7*e + 564/7, 10/7*e^5 - 69/7*e^4 + 9*e^3 + 296/7*e^2 - 263/7*e - 152/7, -12/7*e^5 + 31/7*e^4 + 17*e^3 - 162/7*e^2 - 390/7*e + 167/7, 11/7*e^5 - 22/7*e^4 - 12*e^3 + 40/7*e^2 + 130/7*e + 103/7, 1/7*e^5 + 26/7*e^4 - 19*e^3 + 17/7*e^2 + 386/7*e - 214/7, 3/7*e^5 - 27/7*e^4 + 6*e^3 + 107/7*e^2 - 144/7*e - 33/7, -15/7*e^5 + 72/7*e^4 + 6*e^3 - 367/7*e^2 - 99/7*e + 305/7, -17/7*e^5 + 69/7*e^4 + 9*e^3 - 324/7*e^2 - 87/7*e + 201/7, -8/7*e^5 + 16/7*e^4 + 11*e^3 - 31/7*e^2 - 302/7*e - 45/7, -3/7*e^5 + 20/7*e^4 - 9*e^3 + 33/7*e^2 + 298/7*e - 233/7, -13/7*e^5 + 47/7*e^4 + 15*e^3 - 333/7*e^2 - 279/7*e + 388/7, 1/7*e^5 + 5/7*e^4 - 151/7*e^2 + 22/7*e + 318/7, 17/7*e^5 - 62/7*e^4 - 7*e^3 + 261/7*e^2 - 186/7*e - 215/7, 5*e^3 - 19*e^2 - 19*e + 54, 17/7*e^5 - 97/7*e^4 + 8*e^3 + 352/7*e^2 - 277/7*e - 117/7, 3/7*e^5 + 22/7*e^4 - 21*e^3 - 40/7*e^2 + 535/7*e + 30/7, -6/7*e^5 + 40/7*e^4 - 8*e^3 - 130/7*e^2 + 365/7*e + 52/7, 9/7*e^5 - 25/7*e^4 - 13*e^3 + 167/7*e^2 + 226/7*e - 239/7, -13/7*e^5 + 26/7*e^4 + 13*e^3 - 67/7*e^2 - 27/7*e + 108/7, -5/7*e^5 + 24/7*e^4 + 10*e^3 - 316/7*e^2 - 173/7*e + 496/7, -3/7*e^5 + 6/7*e^4 + 3*e^3 - 51/7*e^2 + 25/7*e + 96/7, -1/7*e^5 + 23/7*e^4 - 9*e^3 - 59/7*e^2 + 118/7*e - 45/7, 8/7*e^5 - 30/7*e^4 + e^3 - 11/7*e^2 - 76/7*e + 276/7, -11/7*e^5 + 71/7*e^4 - 13*e^3 - 131/7*e^2 + 339/7*e - 208/7, 8/7*e^5 - 44/7*e^4 - e^3 + 199/7*e^2 - 13/7*e + 129/7, -11/7*e^5 + 71/7*e^4 - 7*e^3 - 306/7*e^2 + 150/7*e + 303/7, 7*e^3 - 17*e^2 - 31*e + 21, -e^5 + 3*e^4 + 6*e^3 - 18*e^2 + 6*e + 39, -3/7*e^5 + 6/7*e^4 + 2*e^3 + 54/7*e^2 - 73/7*e - 247/7, 3/7*e^5 - 20/7*e^4 + e^3 + 107/7*e^2 - 53/7*e - 26/7, 4/7*e^5 - 36/7*e^4 + 7*e^3 + 243/7*e^2 - 367/7*e - 282/7, 5/7*e^5 - 17/7*e^4 - 19*e^3 + 379/7*e^2 + 425/7*e - 706/7, 16/7*e^5 - 74/7*e^4 - 10*e^3 + 426/7*e^2 + 212/7*e - 505/7, -11/7*e^5 + 85/7*e^4 - 16*e^3 - 341/7*e^2 + 472/7*e + 191/7, -20/7*e^5 + 103/7*e^4 + e^3 - 501/7*e^2 + 99/7*e + 437/7, 22/7*e^5 - 100/7*e^4 - 7*e^3 + 437/7*e^2 + 85/7*e - 256/7, e^5 - 4*e^4 - 5*e^3 + 22*e^2 + 12*e - 37, 18/7*e^5 - 99/7*e^4 + 2*e^3 + 474/7*e^2 - 115/7*e - 443/7, -10/7*e^5 + 48/7*e^4 + 10*e^3 - 401/7*e^2 - 192/7*e + 565/7, -2*e^5 + 12*e^4 - 4*e^3 - 53*e^2 + 24*e + 33, -3/7*e^5 - 1/7*e^4 + 13*e^3 - 107/7*e^2 - 325/7*e + 243/7, -1/7*e^5 + 16/7*e^4 - 2*e^3 - 115/7*e^2 + 69/7*e - 73/7, -6/7*e^5 + 19/7*e^4 - 9*e^3 + 213/7*e^2 + 316/7*e - 564/7, -24/7*e^5 + 125/7*e^4 - 2*e^3 - 492/7*e^2 + 130/7*e + 180/7, 27/7*e^5 - 110/7*e^4 - 12*e^3 + 494/7*e^2 + 6/7*e - 325/7, -10/7*e^5 + 34/7*e^4 + 11*e^3 - 233/7*e^2 - 192/7*e + 201/7, 12/7*e^5 - 59/7*e^4 + 3*e^3 + 155/7*e^2 - 86/7*e + 225/7, -6/7*e^5 + 5/7*e^4 + 10*e^3 + 17/7*e^2 - 188/7*e - 102/7, 13/7*e^5 - 54/7*e^4 - 16*e^3 + 501/7*e^2 + 237/7*e - 766/7, -2*e^5 + 11*e^4 - 59*e^2 + 17*e + 59, 20/7*e^5 - 89/7*e^4 - 4*e^3 + 361/7*e^2 - 15/7*e - 199/7, 1/7*e^5 + 12/7*e^4 - 4*e^3 - 158/7*e^2 + 141/7*e + 325/7, 5/7*e^5 - 31/7*e^4 + 5*e^3 + 92/7*e^2 - 177/7*e - 90/7, 2/7*e^5 + 45/7*e^4 - 31*e^3 - 15/7*e^2 + 590/7*e - 323/7, -8/7*e^5 + 30/7*e^4 - e^3 - 80/7*e^2 + 195/7*e + 151/7, -3/7*e^5 + 13/7*e^4 + 5*e^3 - 191/7*e^2 - 31/7*e + 334/7, -15/7*e^5 + 23/7*e^4 + 33*e^3 - 360/7*e^2 - 533/7*e + 879/7, -5/7*e^5 + 38/7*e^4 - 11*e^3 - 43/7*e^2 + 254/7*e - 400/7, 12/7*e^5 - 66/7*e^4 - 3*e^3 + 435/7*e^2 - 149/7*e - 496/7, 26/7*e^5 - 101/7*e^4 - 20*e^3 + 589/7*e^2 + 208/7*e - 685/7, -3/7*e^5 + 41/7*e^4 - 4*e^3 - 303/7*e^2 + 74/7*e + 215/7, 1/7*e^5 - 9/7*e^4 - 6*e^3 + 199/7*e^2 + 71/7*e - 235/7, 5/7*e^5 + 18/7*e^4 - 16*e^3 - 153/7*e^2 + 222/7*e + 330/7, -13/7*e^5 + 26/7*e^4 + 35*e^3 - 508/7*e^2 - 706/7*e + 1039/7, 6/7*e^5 - 47/7*e^4 + 8*e^3 + 144/7*e^2 - 176/7*e + 109/7, -1/7*e^5 - 5/7*e^4 + e^3 + 137/7*e^2 - 36/7*e - 318/7, 3*e^5 - 13*e^4 - 12*e^3 + 72*e^2 + 18*e - 76, 4/7*e^5 - 8/7*e^4 - 7*e^3 + 33/7*e^2 + 67/7*e + 236/7, -17/7*e^5 + 69/7*e^4 + 10*e^3 - 387/7*e^2 - 66/7*e + 460/7, 10/7*e^5 - 20/7*e^4 - 14*e^3 + 37/7*e^2 + 367/7*e - 40/7, -1/7*e^5 + 37/7*e^4 - 16*e^3 - 143/7*e^2 + 342/7*e + 172/7, -19/7*e^5 + 66/7*e^4 + 11*e^3 - 239/7*e^2 - 89/7*e + 272/7, -10/7*e^5 + 27/7*e^4 + 12*e^3 - 226/7*e^2 - 52/7*e + 509/7, -6/7*e^5 - 2/7*e^4 + 22*e^3 - 137/7*e^2 - 419/7*e + 430/7, 10/7*e^5 - 34/7*e^4 - 9*e^3 + 156/7*e^2 + 129/7*e - 26/7, -20/7*e^5 + 82/7*e^4 + 10*e^3 - 375/7*e^2 - 83/7*e + 255/7, -2*e^5 + 6*e^4 + 15*e^3 - 31*e^2 - 29*e + 37, 12/7*e^5 - 73/7*e^4 + 4*e^3 + 358/7*e^2 - 142/7*e - 265/7, -6/7*e^5 + 82/7*e^4 - 24*e^3 - 312/7*e^2 + 484/7*e + 178/7, 1/7*e^5 - 30/7*e^4 + 11*e^3 + 136/7*e^2 - 356/7*e + 38/7, -3*e^5 + 16*e^4 - 3*e^3 - 58*e^2 + 19*e + 19, -3/7*e^5 + 6/7*e^4 + 2*e^3 + 75/7*e^2 - 171/7*e - 191/7, -18/7*e^5 + 99/7*e^4 - 10*e^3 - 278/7*e^2 + 381/7*e - 180/7, -25/7*e^5 + 92/7*e^4 + 30*e^3 - 656/7*e^2 - 641/7*e + 1059/7, -10/7*e^5 + 48/7*e^4 - e^3 - 191/7*e^2 + 186/7*e - 135/7, 2/7*e^5 - 53/7*e^4 + 20*e^3 + 230/7*e^2 - 439/7*e - 127/7, 5/7*e^5 - 24/7*e^4 + 6*e^3 + 29/7*e^2 - 268/7*e - 209/7, -13/7*e^5 + 33/7*e^4 + 7*e^3 + 45/7*e^2 - 34/7*e - 319/7, 8/7*e^5 + 5/7*e^4 - 24*e^3 + 101/7*e^2 + 400/7*e - 557/7, 6/7*e^5 - 47/7*e^4 + 16*e^3 + 25/7*e^2 - 449/7*e + 228/7, 2/7*e^5 - 95/7*e^4 + 36*e^3 + 412/7*e^2 - 649/7*e - 176/7, -31/7*e^5 + 139/7*e^4 + 15*e^3 - 744/7*e^2 - 213/7*e + 859/7, -5/7*e^5 + 38/7*e^4 - 6*e^3 - 141/7*e^2 + 177/7*e - 106/7, -5/7*e^5 + 10/7*e^4 + e^3 + 6/7*e^2 + 205/7*e - 162/7, 1/7*e^5 + 47/7*e^4 - 26*e^3 - 151/7*e^2 + 561/7*e - 11/7, -24/7*e^5 + 62/7*e^4 + 37*e^3 - 534/7*e^2 - 626/7*e + 1013/7, -4/7*e^5 - 6/7*e^4 + 18*e^3 - 159/7*e^2 - 193/7*e + 548/7, 9/7*e^5 - 46/7*e^4 + 14*e^3 - 78/7*e^2 - 488/7*e + 370/7, -15/7*e^5 + 51/7*e^4 + 28*e^3 - 591/7*e^2 - 512/7*e + 956/7, -16/7*e^5 + 60/7*e^4 + 12*e^3 - 321/7*e^2 - 79/7*e + 106/7, -e^5 + 7*e^3 + 20*e^2 + 8*e - 25, 27/7*e^5 - 96/7*e^4 - 16*e^3 + 459/7*e^2 - 71/7*e - 591/7, 39/7*e^5 - 169/7*e^4 - 14*e^3 + 747/7*e^2 + 39/7*e - 436/7, 24/7*e^5 - 118/7*e^4 - 7*e^3 + 618/7*e^2 + 136/7*e - 614/7, -34/7*e^5 + 117/7*e^4 + 32*e^3 - 732/7*e^2 - 314/7*e + 941/7, 15/7*e^5 - 114/7*e^4 + 12*e^3 + 584/7*e^2 - 286/7*e - 501/7, -3*e^4 + 10*e^3 + 9*e^2 - 41*e + 34, 9/7*e^5 - 46/7*e^4 + 13*e^3 - 57/7*e^2 - 460/7*e + 538/7, 12/7*e^5 - 45/7*e^4 - 3*e^3 + 148/7*e^2 - 79/7*e - 167/7, 32/7*e^5 - 134/7*e^4 - 12*e^3 + 635/7*e^2 - 66/7*e - 681/7, 16/7*e^5 - 88/7*e^4 - 2*e^3 + 475/7*e^2 - 47/7*e - 239/7, 5/7*e^5 - 17/7*e^4 - 11*e^3 + 281/7*e^2 + 89/7*e - 601/7, -5/7*e^5 + 10/7*e^4 - 4*e^3 + 195/7*e^2 + 198/7*e - 708/7, -16/7*e^5 + 46/7*e^4 + 21*e^3 - 293/7*e^2 - 261/7*e + 64/7, -1/7*e^5 + 9/7*e^4 + 6*e^3 - 164/7*e^2 - 295/7*e + 137/7, -18/7*e^5 + 99/7*e^4 - e^3 - 495/7*e^2 + 3/7*e + 625/7, 16/7*e^5 - 116/7*e^4 + 19*e^3 + 440/7*e^2 - 404/7*e - 162/7, 6/7*e^5 - 47/7*e^4 + 7*e^3 + 221/7*e^2 - 253/7*e - 150/7, 12/7*e^5 - 66/7*e^4 + 365/7*e^2 - 93/7*e - 489/7, 5/7*e^5 - 52/7*e^4 + 18*e^3 + 106/7*e^2 - 541/7*e + 127/7, -18/7*e^5 + 29/7*e^4 + 42*e^3 - 481/7*e^2 - 725/7*e + 961/7, -e^5 - 2*e^4 + 22*e^3 + 6*e^2 - 36*e - 20, -11/7*e^5 + 43/7*e^4 + 17*e^3 - 474/7*e^2 - 256/7*e + 660/7, -15/7*e^5 + 30/7*e^4 + 21*e^3 - 136/7*e^2 - 379/7*e + 403/7, -26/7*e^5 + 87/7*e^4 + 23*e^3 - 442/7*e^2 - 334/7*e + 482/7, -26/7*e^5 + 94/7*e^4 + 31*e^3 - 673/7*e^2 - 635/7*e + 1147/7, 6/7*e^5 - 19/7*e^4 - 3*e^3 - 38/7*e^2 + 139/7*e + 375/7, 19/7*e^5 - 66/7*e^4 - 25*e^3 + 554/7*e^2 + 320/7*e - 734/7, -5/7*e^5 + 24/7*e^4 + 17*e^3 - 449/7*e^2 - 495/7*e + 895/7, 5/7*e^5 - 24/7*e^4 + 120/7*e^2 + 40/7*e - 321/7, 11/7*e^5 - 29/7*e^4 - 11*e^3 + 166/7*e^2 - 10/7*e - 149/7, -2*e^5 + 12*e^4 - 13*e^3 - 25*e^2 + 55*e - 47, -16/7*e^5 + 60/7*e^4 + 15*e^3 - 377/7*e^2 - 219/7*e + 449/7, 16/7*e^5 - 39/7*e^4 - 12*e^3 + 6/7*e^2 + 156/7*e + 237/7, 20/7*e^5 - 75/7*e^4 - 20*e^3 + 487/7*e^2 + 356/7*e - 752/7, -2/7*e^5 - 3/7*e^4 + 5*e^3 + 92/7*e^2 - 275/7*e - 76/7, -27/7*e^5 + 103/7*e^4 + 31*e^3 - 809/7*e^2 - 496/7*e + 1144/7, -9/7*e^5 + 32/7*e^4 + e^3 - 13/7*e^2 + 47/7*e - 517/7, 3*e^5 - 13*e^4 - 14*e^3 + 76*e^2 + 39*e - 80, -19/7*e^5 + 66/7*e^4 + 22*e^3 - 554/7*e^2 - 222/7*e + 832/7, 9/7*e^5 - 74/7*e^4 + 7*e^3 + 405/7*e^2 - 131/7*e - 442/7, -2*e^5 + 7*e^4 + 6*e^3 - 26*e^2 + 18*e + 13, 18/7*e^5 - 106/7*e^4 + 12*e^3 + 334/7*e^2 - 402/7*e + 54/7, -15/7*e^5 + 65/7*e^4 + 4*e^3 - 234/7*e^2 - 43/7*e + 25/7, -15/7*e^5 + 16/7*e^4 + 46*e^3 - 514/7*e^2 - 883/7*e + 1292/7, 22/7*e^5 - 107/7*e^4 - 4*e^3 + 458/7*e^2 + 127/7*e - 193/7, -4/7*e^5 - 20/7*e^4 + 3*e^3 + 380/7*e^2 + 143/7*e - 628/7, -1/7*e^5 + 2/7*e^4 + 11*e^3 - 199/7*e^2 - 330/7*e + 571/7, 26/7*e^5 - 38/7*e^4 - 54*e^3 + 449/7*e^2 + 929/7*e - 734/7, -3*e^5 + 12*e^4 - e^3 - 22*e^2 + 36*e - 36, 2*e^5 - 10*e^4 - 5*e^3 + 55*e^2 - 10*e - 9, 4/7*e^5 - 85/7*e^4 + 31*e^3 + 348/7*e^2 - 710/7*e - 107/7, 18/7*e^5 - 78/7*e^4 - 23*e^3 + 712/7*e^2 + 494/7*e - 1108/7];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {16, 2, -w^5 + 6*w^3 + w^2 - 7*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;