/*
  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, 7, 11, -2, -7, 0, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {29, 29, w^5 - w^4 - 7*w^3 + 4*w^2 + 11*w}>;

primesArray := [
[5, 5, -2*w^5 + w^4 + 13*w^3 - 2*w^2 - 19*w - 5],
[9, 3, 2*w^5 - w^4 - 14*w^3 + 2*w^2 + 23*w + 6],
[11, 11, w - 1],
[25, 5, w^3 + w^2 - 4*w - 3],
[29, 29, w^5 - w^4 - 7*w^3 + 4*w^2 + 11*w],
[41, 41, w^4 - w^3 - 5*w^2 + 3*w + 3],
[49, 7, w^5 - w^4 - 7*w^3 + 4*w^2 + 11*w + 1],
[59, 59, 2*w^5 - w^4 - 14*w^3 + 2*w^2 + 24*w + 7],
[59, 59, -w^5 + 8*w^3 + 2*w^2 - 15*w - 8],
[61, 61, w^5 - 7*w^3 - 2*w^2 + 12*w + 4],
[61, 61, -w^5 + 7*w^3 - 11*w - 1],
[64, 2, 2],
[71, 71, w^3 + w^2 - 5*w - 3],
[71, 71, -3*w^5 + w^4 + 21*w^3 - w^2 - 33*w - 9],
[79, 79, -2*w^5 + w^4 + 13*w^3 - 3*w^2 - 19*w - 5],
[81, 3, 2*w^5 - w^4 - 13*w^3 + w^2 + 19*w + 8],
[89, 89, 2*w^5 - w^4 - 13*w^3 + 2*w^2 + 20*w + 7],
[89, 89, -w^5 + 8*w^3 + w^2 - 16*w - 5],
[89, 89, -3*w^5 + w^4 + 20*w^3 - 30*w - 11],
[89, 89, -w^5 + 7*w^3 + 2*w^2 - 11*w - 4],
[89, 89, -2*w^5 + 14*w^3 + 3*w^2 - 22*w - 8],
[89, 89, 2*w^5 - w^4 - 14*w^3 + 3*w^2 + 22*w + 6],
[101, 101, 3*w^5 - 2*w^4 - 18*w^3 + 6*w^2 + 22*w + 6],
[101, 101, w^2 - w - 4],
[101, 101, -3*w^5 + w^4 + 20*w^3 - 2*w^2 - 29*w - 7],
[109, 109, w^4 - w^3 - 5*w^2 + 4*w + 5],
[121, 11, w^5 - 7*w^3 - 2*w^2 + 10*w + 6],
[131, 131, w^5 - 2*w^4 - 5*w^3 + 9*w^2 + 4*w - 3],
[131, 131, -2*w^5 + w^4 + 13*w^3 - 3*w^2 - 20*w - 6],
[131, 131, 3*w^5 - w^4 - 21*w^3 + w^2 + 35*w + 11],
[131, 131, -w^5 + w^4 + 6*w^3 - 4*w^2 - 7*w + 2],
[139, 139, -w^5 + 6*w^3 + w^2 - 8*w - 1],
[151, 151, -w^3 + w^2 + 4*w - 2],
[179, 179, w^4 - 6*w^2 + 6],
[181, 181, w^3 + w^2 - 3*w],
[191, 191, 3*w^5 - w^4 - 20*w^3 + 2*w^2 + 28*w + 5],
[199, 199, -2*w^5 + w^4 + 12*w^3 - 3*w^2 - 15*w - 2],
[199, 199, 5*w^5 - w^4 - 34*w^3 - w^2 + 52*w + 15],
[211, 211, 3*w^5 - w^4 - 19*w^3 + w^2 + 26*w + 9],
[229, 229, -w^5 + w^4 + 7*w^3 - 3*w^2 - 11*w],
[229, 229, -w^4 + 2*w^3 + 6*w^2 - 7*w - 6],
[239, 239, -w^5 + w^4 + 7*w^3 - 3*w^2 - 13*w - 6],
[239, 239, 3*w^5 - w^4 - 21*w^3 + w^2 + 35*w + 10],
[239, 239, w^3 - w^2 - 4*w + 1],
[239, 239, 5*w^5 - 2*w^4 - 34*w^3 + 2*w^2 + 53*w + 17],
[241, 241, -w^5 + 7*w^3 + w^2 - 13*w - 4],
[241, 241, -3*w^5 + w^4 + 21*w^3 - 35*w - 10],
[241, 241, -w^5 + 7*w^3 - 11*w],
[251, 251, 3*w^5 - w^4 - 21*w^3 + 35*w + 11],
[251, 251, w^5 - 2*w^4 - 6*w^3 + 9*w^2 + 8*w - 3],
[269, 269, -w^5 + 8*w^3 + 3*w^2 - 16*w - 8],
[269, 269, -2*w^5 + w^4 + 13*w^3 - w^2 - 21*w - 9],
[269, 269, 3*w^5 - 21*w^3 - 3*w^2 + 32*w + 12],
[271, 271, -5*w^5 + 3*w^4 + 34*w^3 - 8*w^2 - 53*w - 14],
[281, 281, -w^5 + 8*w^3 + 3*w^2 - 16*w - 11],
[281, 281, 4*w^5 - w^4 - 29*w^3 + 49*w + 13],
[281, 281, w^5 - w^4 - 7*w^3 + 3*w^2 + 11*w + 5],
[281, 281, -5*w^5 + 3*w^4 + 33*w^3 - 8*w^2 - 50*w - 12],
[311, 311, 2*w^5 - w^4 - 14*w^3 + w^2 + 21*w + 9],
[311, 311, -3*w^5 + 20*w^3 + 3*w^2 - 30*w - 10],
[311, 311, 5*w^5 - 2*w^4 - 35*w^3 + 2*w^2 + 57*w + 19],
[331, 331, -w^5 - w^4 + 7*w^3 + 6*w^2 - 11*w - 7],
[349, 349, -6*w^5 + 3*w^4 + 41*w^3 - 7*w^2 - 64*w - 17],
[349, 349, 4*w^5 - w^4 - 29*w^3 - w^2 + 50*w + 17],
[349, 349, 5*w^5 - w^4 - 34*w^3 - w^2 + 52*w + 17],
[359, 359, w^5 - 6*w^3 - w^2 + 5*w + 3],
[379, 379, -w^5 + w^4 + 8*w^3 - 3*w^2 - 17*w - 3],
[379, 379, 2*w^5 - 14*w^3 - 4*w^2 + 22*w + 11],
[379, 379, -5*w^5 + 3*w^4 + 34*w^3 - 8*w^2 - 53*w - 13],
[379, 379, 4*w^5 - 2*w^4 - 27*w^3 + 3*w^2 + 43*w + 14],
[389, 389, 4*w^5 - 2*w^4 - 28*w^3 + 4*w^2 + 44*w + 13],
[401, 401, -2*w^5 + 2*w^4 + 13*w^3 - 6*w^2 - 20*w - 6],
[409, 409, -w^5 + 2*w^4 + 6*w^3 - 8*w^2 - 9*w],
[419, 419, -5*w^5 + w^4 + 34*w^3 + 2*w^2 - 53*w - 19],
[419, 419, 2*w^5 - w^4 - 14*w^3 + 4*w^2 + 23*w + 3],
[421, 421, -w^5 - w^4 + 8*w^3 + 6*w^2 - 13*w - 7],
[421, 421, 2*w^5 - 2*w^4 - 13*w^3 + 8*w^2 + 18*w + 2],
[421, 421, 4*w^5 - w^4 - 26*w^3 + 37*w + 11],
[431, 431, 2*w^5 - w^4 - 13*w^3 + 2*w^2 + 20*w + 9],
[431, 431, w^5 - 8*w^3 - w^2 + 14*w + 2],
[449, 449, 4*w^5 - 2*w^4 - 26*w^3 + 4*w^2 + 38*w + 9],
[449, 449, -3*w^5 + w^4 + 20*w^3 - 31*w - 14],
[461, 461, 3*w^5 - 2*w^4 - 18*w^3 + 6*w^2 + 23*w + 4],
[461, 461, -4*w^5 + 2*w^4 + 26*w^3 - 5*w^2 - 36*w - 9],
[461, 461, -4*w^5 + 2*w^4 + 27*w^3 - 3*w^2 - 42*w - 15],
[491, 491, -3*w^5 + w^4 + 20*w^3 - 29*w - 10],
[491, 491, -5*w^5 + w^4 + 35*w^3 + 2*w^2 - 56*w - 18],
[491, 491, 2*w^5 - 14*w^3 - 4*w^2 + 24*w + 13],
[491, 491, -3*w^5 + w^4 + 21*w^3 - 2*w^2 - 33*w - 10],
[499, 499, 6*w^5 - 3*w^4 - 40*w^3 + 6*w^2 + 61*w + 18],
[499, 499, -4*w^5 + w^4 + 29*w^3 + w^2 - 49*w - 17],
[499, 499, -w^5 - w^4 + 8*w^3 + 5*w^2 - 14*w - 4],
[499, 499, -3*w^5 + w^4 + 19*w^3 - 2*w^2 - 26*w - 8],
[509, 509, -4*w^5 + 2*w^4 + 27*w^3 - 5*w^2 - 43*w - 11],
[509, 509, -2*w^5 + 16*w^3 + 2*w^2 - 30*w - 9],
[521, 521, 3*w^5 - 21*w^3 - 3*w^2 + 34*w + 10],
[521, 521, -3*w^5 + w^4 + 19*w^3 - 26*w - 9],
[521, 521, 4*w^5 - w^4 - 27*w^3 + 40*w + 15],
[521, 521, -3*w^5 + 22*w^3 + 3*w^2 - 37*w - 11],
[541, 541, -4*w^5 + w^4 + 29*w^3 + w^2 - 49*w - 18],
[569, 569, -6*w^5 + 2*w^4 + 40*w^3 - 2*w^2 - 60*w - 19],
[569, 569, -w^5 + w^4 + 7*w^3 - 2*w^2 - 12*w - 6],
[571, 571, 4*w^5 - w^4 - 28*w^3 + 43*w + 14],
[571, 571, -4*w^5 + 2*w^4 + 28*w^3 - 4*w^2 - 45*w - 14],
[599, 599, -w^5 + w^4 + 7*w^3 - 2*w^2 - 13*w - 7],
[601, 601, 4*w^5 - w^4 - 26*w^3 + 38*w + 13],
[601, 601, 4*w^5 - 2*w^4 - 28*w^3 + 5*w^2 + 44*w + 11],
[601, 601, 3*w^5 - w^4 - 21*w^3 + 2*w^2 + 35*w + 9],
[619, 619, 4*w^5 - 2*w^4 - 27*w^3 + 4*w^2 + 40*w + 13],
[619, 619, -4*w^5 + w^4 + 26*w^3 - 36*w - 12],
[619, 619, w^4 - w^3 - 6*w^2 + 4*w + 4],
[631, 631, -3*w^5 + w^4 + 20*w^3 - 30*w - 9],
[631, 631, -5*w^5 + 2*w^4 + 35*w^3 - 3*w^2 - 57*w - 15],
[641, 641, 5*w^5 - w^4 - 34*w^3 - w^2 + 52*w + 16],
[641, 641, -5*w^5 + 2*w^4 + 34*w^3 - 2*w^2 - 52*w - 18],
[659, 659, 3*w^5 - w^4 - 20*w^3 + w^2 + 32*w + 9],
[659, 659, -4*w^5 + w^4 + 27*w^3 - w^2 - 40*w - 11],
[661, 661, 4*w^5 - w^4 - 26*w^3 + 38*w + 12],
[661, 661, -w^5 + 6*w^3 - 8*w + 1],
[691, 691, 3*w^5 - 20*w^3 - 3*w^2 + 29*w + 9],
[691, 691, w^5 + w^4 - 8*w^3 - 7*w^2 + 14*w + 9],
[691, 691, -w^5 + w^4 + 6*w^3 - 3*w^2 - 7*w + 2],
[701, 701, -2*w^5 + w^4 + 13*w^3 - 2*w^2 - 19*w - 9],
[701, 701, 5*w^5 - 2*w^4 - 33*w^3 + 3*w^2 + 48*w + 14],
[701, 701, w^4 + 3*w^3 - 3*w^2 - 10*w - 1],
[701, 701, -2*w^5 + w^4 + 14*w^3 - 3*w^2 - 24*w - 3],
[709, 709, -2*w^5 + 2*w^4 + 14*w^3 - 7*w^2 - 22*w - 5],
[709, 709, -3*w^5 + 21*w^3 + 4*w^2 - 34*w - 11],
[719, 719, 5*w^5 - 2*w^4 - 35*w^3 + 3*w^2 + 58*w + 17],
[719, 719, 4*w^5 - 2*w^4 - 26*w^3 + 4*w^2 + 37*w + 11],
[719, 719, 4*w^5 - 2*w^4 - 27*w^3 + 4*w^2 + 40*w + 12],
[719, 719, 2*w^5 - 16*w^3 - 3*w^2 + 30*w + 10],
[739, 739, -5*w^5 + w^4 + 34*w^3 + w^2 - 51*w - 17],
[751, 751, -w^5 - w^4 + 9*w^3 + 7*w^2 - 18*w - 10],
[761, 761, 4*w^5 - 2*w^4 - 28*w^3 + 3*w^2 + 46*w + 16],
[761, 761, -4*w^5 + 2*w^4 + 27*w^3 - 5*w^2 - 43*w - 12],
[769, 769, -4*w^5 + w^4 + 28*w^3 - 44*w - 12],
[769, 769, -w^4 + w^3 + 3*w^2 - 3*w + 1],
[809, 809, 2*w^5 - 2*w^4 - 13*w^3 + 7*w^2 + 20*w + 1],
[811, 811, -2*w^5 + 15*w^3 + 2*w^2 - 24*w - 8],
[811, 811, -w^5 + 9*w^3 + w^2 - 18*w - 6],
[821, 821, -w^5 + 8*w^3 + 3*w^2 - 17*w - 9],
[821, 821, w^5 + w^4 - 7*w^3 - 6*w^2 + 11*w + 8],
[821, 821, 4*w^5 - 2*w^4 - 28*w^3 + 4*w^2 + 46*w + 11],
[821, 821, 5*w^5 - 2*w^4 - 34*w^3 + 4*w^2 + 51*w + 13],
[829, 829, -2*w^5 + w^4 + 15*w^3 - w^2 - 28*w - 10],
[829, 829, -6*w^5 + 3*w^4 + 42*w^3 - 7*w^2 - 68*w - 21],
[829, 829, 4*w^5 - 2*w^4 - 26*w^3 + 5*w^2 + 38*w + 12],
[829, 829, w^5 - 7*w^3 - 3*w^2 + 11*w + 6],
[839, 839, 3*w^5 - 2*w^4 - 20*w^3 + 5*w^2 + 32*w + 11],
[839, 839, -w^4 + w^3 + 6*w^2 - 3*w - 4],
[841, 29, -3*w^5 + w^4 + 22*w^3 - 38*w - 11],
[911, 911, 3*w^5 - w^4 - 21*w^3 + 36*w + 12],
[919, 919, -w^5 - w^4 + 7*w^3 + 6*w^2 - 12*w - 5],
[941, 941, -w^5 + w^4 + 7*w^3 - 5*w^2 - 11*w],
[941, 941, 4*w^5 - w^4 - 29*w^3 + 48*w + 15],
[971, 971, -4*w^5 + w^4 + 27*w^3 + w^2 - 41*w - 13],
[991, 991, w^5 - w^4 - 8*w^3 + 3*w^2 + 16*w + 2],
[991, 991, -4*w^5 + 3*w^4 + 27*w^3 - 9*w^2 - 42*w - 10]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^5 + 2*x^4 - 9*x^3 - 18*x^2 + 13*x + 27;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, -e^4 - e^3 + 9*e^2 + 6*e - 13, -e^4 + 10*e^2 - e - 18, e^2 + e - 8, 1, e^4 + e^3 - 10*e^2 - 6*e + 15, 3*e^4 - 29*e^2 - 2*e + 49, 5*e^4 + 3*e^3 - 45*e^2 - 18*e + 69, e^4 + e^3 - 8*e^2 - 7*e + 6, -2*e^4 + 19*e^2 - e - 41, -2*e^4 - e^3 + 21*e^2 + 3*e - 47, -3*e^2 - e + 11, -2*e^4 + 18*e^2 + e - 30, 3*e^4 - 28*e^2 - 2*e + 42, -5*e^4 - e^3 + 47*e^2 + 9*e - 83, e^3 - 2*e^2 - 6*e + 2, 3*e^4 + 2*e^3 - 26*e^2 - 13*e + 39, -e^3 - e^2 + 5*e - 3, 3*e^4 + e^3 - 32*e^2 - 7*e + 63, 2*e^4 + 2*e^3 - 19*e^2 - 13*e + 30, -2*e^4 + 18*e^2 + e - 27, -2*e^4 - e^3 + 20*e^2 + 5*e - 39, e^4 + e^3 - 7*e^2 - 5*e - 3, -4*e^4 + 35*e^2 - 54, 5*e^4 + e^3 - 47*e^2 - 11*e + 81, 5*e^4 - 49*e^2 - 2*e + 88, 2*e^4 - 21*e^2 + 2*e + 41, e^4 - 6*e^2 + 3*e + 6, -5*e^4 + 46*e^2 + 2*e - 78, -e^3 + 4*e - 6, 2*e^4 - e^3 - 18*e^2 + 9*e + 30, 4*e^4 + 2*e^3 - 37*e^2 - 12*e + 49, -3*e^4 + 30*e^2 + 3*e - 56, e^4 + 2*e^3 - 14*e^2 - 14*e + 33, 8*e^4 + 2*e^3 - 76*e^2 - 15*e + 133, -3*e^4 - e^3 + 28*e^2 + 10*e - 48, 4*e^4 - 40*e^2 + 2*e + 80, -2*e^4 - e^3 + 22*e^2 + 9*e - 44, -6*e^4 - e^3 + 55*e^2 + 6*e - 100, e^4 + 3*e^3 - 11*e^2 - 15*e + 25, e^4 - 7*e^2 + 8, -9*e^4 - 2*e^3 + 83*e^2 + 16*e - 132, -4*e^4 - e^3 + 39*e^2 + 8*e - 66, 2*e^4 + e^3 - 13*e^2 - 4*e + 6, e^4 - e^3 - 12*e^2 + 4*e + 24, 4*e^4 - 38*e^2 + 3*e + 73, 2*e^4 - 20*e^2 + e + 31, 3*e^4 + 3*e^3 - 34*e^2 - 20*e + 68, 8*e^4 + 3*e^3 - 79*e^2 - 24*e + 132, 5*e^4 + 3*e^3 - 49*e^2 - 17*e + 81, 4*e^4 + e^3 - 39*e^2 - 3*e + 63, -e^4 - 2*e^3 + 10*e^2 + 13*e - 33, 5*e^4 + 2*e^3 - 46*e^2 - 11*e + 87, -2*e^4 - 2*e^3 + 22*e^2 + 14*e - 38, -2*e^3 + 2*e^2 + 17*e - 15, 5*e^4 - 49*e^2 - 4*e + 93, e^4 + e^3 - 17*e^2 - 2*e + 57, 5*e^4 - 41*e^2 - 3*e + 51, e^4 - e^3 - 8*e^2 + 4*e + 9, 4*e^4 + e^3 - 38*e^2 - 7*e + 63, -e^4 + e^3 + 8*e^2 - 9*e - 15, 5*e^2 + 3*e - 25, -2*e^4 + 2*e^3 + 18*e^2 - 9*e - 29, -e^4 - 2*e^3 + 7*e^2 + 10*e + 10, 8*e^4 + 3*e^3 - 71*e^2 - 21*e + 107, 3*e^4 + e^3 - 26*e^2 - 4*e + 42, -3*e^4 - e^3 + 31*e^2 + 11*e - 73, 6*e^4 + 2*e^3 - 63*e^2 - 10*e + 133, 4*e^4 - 37*e^2 - 8*e + 61, -e^4 - 2*e^3 + 11*e^2 + 16*e - 22, -e^4 + e^3 + 13*e^2 - 4*e - 39, 2*e^4 - 18*e^2 - 10*e + 15, -2*e^4 - 3*e^3 + 14*e^2 + 16*e - 22, -e^4 + 14*e^2 - 4*e - 39, 7*e^4 + 6*e^3 - 64*e^2 - 39*e + 102, 2*e^4 - 3*e^3 - 20*e^2 + 19*e + 28, -12*e^4 - 3*e^3 + 110*e^2 + 26*e - 178, 2*e^4 - 20*e^2 + 2*e + 37, 10*e^4 + 3*e^3 - 93*e^2 - 25*e + 153, -3*e^4 - 3*e^3 + 20*e^2 + 17*e - 3, -6*e^4 + e^3 + 60*e^2 - 6*e - 111, -9*e^4 - 2*e^3 + 83*e^2 + 15*e - 123, -8*e^4 - 3*e^3 + 80*e^2 + 15*e - 150, e^3 + 6*e^2 - 8*e - 9, -2*e^4 - 4*e^3 + 14*e^2 + 19*e - 12, 10*e^4 + 3*e^3 - 91*e^2 - 27*e + 132, 2*e^4 - 5*e^3 - 21*e^2 + 28*e + 27, -11*e^4 - 5*e^3 + 107*e^2 + 35*e - 186, 8*e^4 + 2*e^3 - 80*e^2 - 7*e + 162, 11*e^4 + 4*e^3 - 97*e^2 - 25*e + 152, -12*e^4 - 3*e^3 + 118*e^2 + 15*e - 214, 8*e^4 + e^3 - 80*e^2 - 6*e + 143, e^4 - e^3 - 6*e^2 + 7*e + 1, e^4 + 5*e^3 - 8*e^2 - 29*e - 3, 2*e^4 + 5*e^3 - 20*e^2 - 23*e + 48, -12*e^4 - 4*e^3 + 110*e^2 + 22*e - 186, 2*e^4 - 2*e^3 - 16*e^2 + 10*e + 30, -10*e^4 - 2*e^3 + 96*e^2 + 25*e - 177, -14*e^4 - 3*e^3 + 134*e^2 + 30*e - 234, -5*e^4 + 50*e^2 - 4*e - 101, -10*e^4 - 2*e^3 + 91*e^2 + 10*e - 156, 18*e^4 + 5*e^3 - 168*e^2 - 36*e + 285, 7*e^4 + 5*e^3 - 64*e^2 - 37*e + 101, -6*e^4 - 3*e^3 + 53*e^2 + 21*e - 85, -e^4 + 20*e^2 + e - 60, -2*e^4 - e^3 + 10*e^2 + 8*e + 23, 13*e^4 + 7*e^3 - 114*e^2 - 44*e + 167, 2*e^4 - 2*e^3 - 13*e^2 + 10*e + 2, 3*e^4 - 28*e^2 + 5*e + 61, -3*e^4 + 2*e^3 + 30*e^2 - 12*e - 59, -e^4 - 4*e^3 + 10*e^2 + 24*e - 1, -18*e^4 - 7*e^3 + 176*e^2 + 42*e - 304, -6*e^4 - 4*e^3 + 54*e^2 + 27*e - 74, e^4 + 3*e^3 - 9*e^2 - 9*e + 24, -2*e^4 - 2*e^3 + 22*e^2 + 12*e - 36, 3*e^4 - e^3 - 20*e^2 + 8*e + 6, -2*e^4 + 2*e^3 + 9*e^2 - 19*e + 12, -5*e^4 + 46*e^2 + 11*e - 86, 8*e^4 + e^3 - 82*e^2 - 6*e + 151, 10*e^4 + 5*e^3 - 94*e^2 - 32*e + 169, -9*e^4 - 5*e^3 + 85*e^2 + 24*e - 152, -12*e^4 - 6*e^3 + 113*e^2 + 45*e - 169, 2*e^4 - 5*e^3 - 14*e^2 + 24*e + 3, -10*e^4 - e^3 + 96*e^2 + 15*e - 165, -18*e^4 - 10*e^3 + 167*e^2 + 56*e - 273, 12*e^4 + e^3 - 118*e^2 - 4*e + 219, 5*e^4 + 4*e^3 - 50*e^2 - 21*e + 104, 9*e^4 - e^3 - 84*e^2 - 5*e + 137, 2*e^4 + e^3 - 15*e^2 - 11*e + 15, 14*e^4 + e^3 - 136*e^2 - 9*e + 258, 5*e^4 + 4*e^3 - 43*e^2 - 24*e + 48, 2*e^4 + 2*e^3 - 21*e^2 - 21*e + 39, -3*e^4 - e^3 + 35*e^2 - 7*e - 92, 14*e^4 + 4*e^3 - 131*e^2 - 37*e + 200, -9*e^4 - 6*e^3 + 90*e^2 + 34*e - 156, -15*e^4 - e^3 + 147*e^2 + 8*e - 264, -e^4 - 7*e^3 + 8*e^2 + 45*e - 23, 5*e^4 + 6*e^3 - 47*e^2 - 27*e + 92, -e^3 - 3*e^2 + 12*e + 36, -15*e^4 - 8*e^3 + 135*e^2 + 50*e - 218, 7*e^4 + 2*e^3 - 69*e^2 - 4*e + 113, -6*e^4 - 2*e^3 + 63*e^2 + 14*e - 117, -12*e^4 + 114*e^2 + 6*e - 192, 10*e^4 + 2*e^3 - 105*e^2 - 10*e + 198, -17*e^4 - 2*e^3 + 159*e^2 + 19*e - 279, -13*e^4 - 4*e^3 + 109*e^2 + 27*e - 151, -4*e^4 - 3*e^3 + 46*e^2 + 17*e - 92, -13*e^4 - 2*e^3 + 133*e^2 + 17*e - 260, 8*e^4 + 4*e^3 - 80*e^2 - 25*e + 169, -16*e^4 - 6*e^3 + 153*e^2 + 49*e - 258, -e^4 - 2*e^3 + 4*e^2 + 2*e + 24, -16*e^4 - 6*e^3 + 143*e^2 + 38*e - 226, -13*e^4 - 4*e^3 + 119*e^2 + 20*e - 189, 4*e^4 + e^3 - 34*e^2 - 7*e + 32, 12*e^4 + 7*e^3 - 115*e^2 - 56*e + 174, -5*e^4 - 4*e^3 + 48*e^2 + 17*e - 96, 11*e^4 + 8*e^3 - 101*e^2 - 49*e + 141, -15*e^4 - 6*e^3 + 143*e^2 + 49*e - 241, -7*e^4 - 5*e^3 + 76*e^2 + 31*e - 143];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {29, 29, w^5 - w^4 - 7*w^3 + 4*w^2 + 11*w}>] := -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;