/*
  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 | {45, 15, 2*w^5 - 14*w^3 - 3*w^2 + 23*w + 9}>;

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^6 - 4*x^5 - 22*x^4 + 80*x^3 + 32*x^2 - 160*x + 64;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [1, 1, e, 1/48*e^5 - 1/8*e^4 - 11/24*e^3 + 37/12*e^2 + e - 16/3, -1/2*e^2 + 8, -1/6*e^5 + 1/2*e^4 + 25/6*e^3 - 55/6*e^2 - 13*e + 32/3, 1/48*e^5 - 1/8*e^4 - 11/24*e^3 + 31/12*e^2 + e - 10/3, -1/8*e^5 + 1/2*e^4 + 11/4*e^3 - 9*e^2 - 5*e + 10, -5/16*e^5 + 7/8*e^4 + 63/8*e^3 - 65/4*e^2 - 26*e + 28, 1/24*e^5 - 1/4*e^4 - 11/12*e^3 + 17/3*e^2 + e - 32/3, 1/24*e^5 - 17/12*e^3 - 1/3*e^2 + 10*e - 8/3, 1/48*e^5 - 11/24*e^3 - 1/6*e^2 - e + 5/3, -1/4*e^5 + 3/4*e^4 + 13/2*e^3 - 29/2*e^2 - 27*e + 26, 1/3*e^5 - e^4 - 47/6*e^3 + 107/6*e^2 + 20*e - 64/3, 1/6*e^5 - 1/2*e^4 - 25/6*e^3 + 26/3*e^2 + 13*e - 14/3, 5/48*e^5 - 3/8*e^4 - 55/24*e^3 + 83/12*e^2 + 6*e - 14/3, -5/24*e^5 + 1/2*e^4 + 61/12*e^3 - 47/6*e^2 - 16*e + 22/3, 1/12*e^5 - 1/4*e^4 - 7/3*e^3 + 29/6*e^2 + 14*e - 34/3, -e + 8, 1/48*e^5 + 1/8*e^4 - 23/24*e^3 - 35/12*e^2 + 7*e + 26/3, 1/8*e^5 - 1/2*e^4 - 11/4*e^3 + 19/2*e^2 + 4*e - 14, 1/4*e^5 - 3/4*e^4 - 13/2*e^3 + 29/2*e^2 + 25*e - 28, -1/16*e^5 + 1/8*e^4 + 15/8*e^3 - 11/4*e^2 - 11*e + 10, -1/24*e^5 + 1/4*e^4 + 5/12*e^3 - 31/6*e^2 + 8*e + 38/3, 3/8*e^5 - 5/4*e^4 - 37/4*e^3 + 24*e^2 + 30*e - 38, -11/24*e^5 + 3/2*e^4 + 127/12*e^3 - 82/3*e^2 - 25*e + 112/3, 1/24*e^5 - 1/4*e^4 - 11/12*e^3 + 31/6*e^2 + 4*e - 26/3, 5/12*e^5 - 5/4*e^4 - 61/6*e^3 + 133/6*e^2 + 29*e - 74/3, 17/48*e^5 - 9/8*e^4 - 199/24*e^3 + 245/12*e^2 + 22*e - 80/3, -7/48*e^5 + 3/8*e^4 + 89/24*e^3 - 73/12*e^2 - 15*e + 34/3, 17/48*e^5 - 9/8*e^4 - 199/24*e^3 + 257/12*e^2 + 18*e - 86/3, -19/48*e^5 + 11/8*e^4 + 233/24*e^3 - 307/12*e^2 - 34*e + 112/3, -5/16*e^5 + 9/8*e^4 + 59/8*e^3 - 87/4*e^2 - 23*e + 34, 1/8*e^5 - 1/4*e^4 - 15/4*e^3 + 4*e^2 + 24*e - 8, -3/16*e^5 + 3/8*e^4 + 41/8*e^3 - 23/4*e^2 - 23*e + 8, 3/16*e^5 - 5/8*e^4 - 37/8*e^3 + 49/4*e^2 + 16*e - 20, -7/24*e^5 + 3/4*e^4 + 89/12*e^3 - 41/3*e^2 - 25*e + 62/3, -3/16*e^5 + 5/8*e^4 + 37/8*e^3 - 49/4*e^2 - 15*e + 24, 1/6*e^5 - 1/4*e^4 - 14/3*e^3 + 19/6*e^2 + 20*e - 20/3, -17/48*e^5 + 11/8*e^4 + 199/24*e^3 - 317/12*e^2 - 24*e + 122/3, 1/6*e^5 - 1/2*e^4 - 25/6*e^3 + 29/3*e^2 + 15*e - 68/3, 11/24*e^5 - 3/2*e^4 - 127/12*e^3 + 167/6*e^2 + 25*e - 124/3, -1/12*e^5 + 17/6*e^3 + 7/6*e^2 - 19*e + 10/3, 13/24*e^5 - 7/4*e^4 - 155/12*e^3 + 98/3*e^2 + 39*e - 146/3, 1/6*e^5 - 1/2*e^4 - 14/3*e^3 + 29/3*e^2 + 25*e - 62/3, 7/48*e^5 - 3/8*e^4 - 77/24*e^3 + 73/12*e^2 + 5*e - 16/3, 7/48*e^5 - 5/8*e^4 - 89/24*e^3 + 163/12*e^2 + 14*e - 100/3, 1/8*e^5 - 1/4*e^4 - 13/4*e^3 + 7/2*e^2 + 10*e - 6, 3/8*e^5 - 5/4*e^4 - 35/4*e^3 + 47/2*e^2 + 23*e - 32, -17/48*e^5 + 11/8*e^4 + 199/24*e^3 - 317/12*e^2 - 22*e + 116/3, -13/48*e^5 + 9/8*e^4 + 143/24*e^3 - 259/12*e^2 - 14*e + 82/3, -7/16*e^5 + 11/8*e^4 + 85/8*e^3 - 103/4*e^2 - 28*e + 36, 5/16*e^5 - 11/8*e^4 - 55/8*e^3 + 107/4*e^2 + 14*e - 40, 13/24*e^5 - 3/2*e^4 - 167/12*e^3 + 169/6*e^2 + 51*e - 164/3, -1/2*e^5 + 5/4*e^4 + 13*e^3 - 45/2*e^2 - 47*e + 30, -1/6*e^5 + 1/2*e^4 + 25/6*e^3 - 29/3*e^2 - 11*e + 50/3, -1/6*e^5 + 1/2*e^4 + 14/3*e^3 - 61/6*e^2 - 23*e + 80/3, 3/8*e^5 - 3/2*e^4 - 33/4*e^3 + 29*e^2 + 15*e - 44, -1/24*e^5 + 17/12*e^3 - 7/6*e^2 - 9*e + 50/3, -1/2*e^2 + 8, 1/8*e^5 - 3/4*e^4 - 9/4*e^3 + 31/2*e^2 - 2*e - 24, -7/12*e^5 + 2*e^4 + 43/3*e^3 - 227/6*e^2 - 50*e + 196/3, 13/48*e^5 - 7/8*e^4 - 155/24*e^3 + 181/12*e^2 + 17*e - 22/3, -5/24*e^5 + 1/2*e^4 + 67/12*e^3 - 25/3*e^2 - 23*e + 88/3, -1/12*e^5 + 1/2*e^4 + 4/3*e^3 - 34/3*e^2 + 7*e + 88/3, -1/6*e^5 + 1/2*e^4 + 25/6*e^3 - 61/6*e^2 - 14*e + 44/3, 2/3*e^5 - 9/4*e^4 - 97/6*e^3 + 253/6*e^2 + 48*e - 164/3, 1/8*e^5 - 1/2*e^4 - 11/4*e^3 + 11*e^2 + 4*e - 34, 35/48*e^5 - 21/8*e^4 - 409/24*e^3 + 599/12*e^2 + 45*e - 200/3, 1/8*e^5 - 1/2*e^4 - 13/4*e^3 + 10*e^2 + 16*e - 16, -5/12*e^5 + 3/2*e^4 + 61/6*e^3 - 89/3*e^2 - 40*e + 158/3, -3/16*e^5 + 5/8*e^4 + 37/8*e^3 - 45/4*e^2 - 17*e + 12, -1/2*e^3 + 3*e^2 + 10*e - 30, 3/8*e^5 - e^4 - 41/4*e^3 + 35/2*e^2 + 49*e - 24, 7/48*e^5 - 1/8*e^4 - 101/24*e^3 + 7/12*e^2 + 17*e - 10/3, 5/12*e^5 - 3/2*e^4 - 61/6*e^3 + 175/6*e^2 + 34*e - 110/3, 11/24*e^5 - 5/4*e^4 - 145/12*e^3 + 143/6*e^2 + 46*e - 178/3, 7/48*e^5 - 5/8*e^4 - 65/24*e^3 + 139/12*e^2 - 6*e - 58/3, -1/24*e^5 + 1/2*e^4 + 5/12*e^3 - 32/3*e^2 + 6*e + 50/3, -1/6*e^5 + 3/4*e^4 + 11/3*e^3 - 91/6*e^2 - 4*e + 98/3, 5/24*e^5 - 3/4*e^4 - 67/12*e^3 + 46/3*e^2 + 31*e - 112/3, -23/48*e^5 + 13/8*e^4 + 265/24*e^3 - 371/12*e^2 - 21*e + 134/3, 1/2*e^4 - e^3 - 12*e^2 + 17*e + 20, -5/24*e^5 + 3/4*e^4 + 67/12*e^3 - 89/6*e^2 - 30*e + 58/3, 7/16*e^5 - 9/8*e^4 - 85/8*e^3 + 73/4*e^2 + 26*e, 7/16*e^5 - 9/8*e^4 - 93/8*e^3 + 85/4*e^2 + 45*e - 30, -5/24*e^5 + 3/4*e^4 + 61/12*e^3 - 95/6*e^2 - 15*e + 94/3, -1/12*e^5 + 17/6*e^3 - 1/3*e^2 - 20*e + 40/3, 7/12*e^5 - 3/2*e^4 - 89/6*e^3 + 161/6*e^2 + 52*e - 124/3, -1/6*e^5 + 1/4*e^4 + 14/3*e^3 - 19/6*e^2 - 22*e - 28/3, 1/12*e^5 - 1/2*e^4 - 5/6*e^3 + 28/3*e^2 - 17*e - 34/3, -5/12*e^5 + 3/2*e^4 + 29/3*e^3 - 169/6*e^2 - 22*e + 92/3, 3/16*e^5 - 1/8*e^4 - 45/8*e^3 - 5/4*e^2 + 29*e + 16, 5/12*e^5 - 5/4*e^4 - 32/3*e^3 + 139/6*e^2 + 36*e - 134/3, -35/48*e^5 + 19/8*e^4 + 445/24*e^3 - 533/12*e^2 - 72*e + 218/3, -11/24*e^5 + e^4 + 145/12*e^3 - 101/6*e^2 - 47*e + 64/3, -1/4*e^5 + 3/4*e^4 + 13/2*e^3 - 27/2*e^2 - 32*e + 16, 1/6*e^5 - 1/4*e^4 - 31/6*e^3 + 8/3*e^2 + 28*e + 46/3, 5/12*e^5 - 5/4*e^4 - 32/3*e^3 + 145/6*e^2 + 42*e - 182/3, -7/48*e^5 + 5/8*e^4 + 89/24*e^3 - 139/12*e^2 - 18*e + 40/3, 1/8*e^5 - 3/4*e^4 - 9/4*e^3 + 31/2*e^2 - 3*e - 36, -1/24*e^5 + 11/12*e^3 - 1/6*e^2 + 2*e + 50/3, 7/24*e^5 - 1/2*e^4 - 101/12*e^3 + 23/3*e^2 + 36*e + 4/3, -5/24*e^5 + e^4 + 49/12*e^3 - 125/6*e^2 + 6*e + 112/3, 1/4*e^5 - 3/4*e^4 - 7*e^3 + 14*e^2 + 44*e - 32, -1/24*e^5 + 3/4*e^4 - 1/12*e^3 - 97/6*e^2 + 16*e + 50/3, 5/24*e^5 - 3/4*e^4 - 55/12*e^3 + 83/6*e^2 + 2*e - 46/3, 7/24*e^5 - 3/4*e^4 - 83/12*e^3 + 41/3*e^2 + 12*e - 62/3, -7/24*e^5 + 3/4*e^4 + 89/12*e^3 - 73/6*e^2 - 29*e - 28/3, -25/48*e^5 + 15/8*e^4 + 275/24*e^3 - 409/12*e^2 - 15*e + 148/3, 1/8*e^5 - e^4 - 7/4*e^3 + 22*e^2 - 6*e - 44, 1/6*e^5 - 1/4*e^4 - 25/6*e^3 + 7/6*e^2 + 20*e + 76/3, 7/48*e^5 - 1/8*e^4 - 101/24*e^3 - 5/12*e^2 + 21*e + 68/3, 1/3*e^5 - 5/4*e^4 - 25/3*e^3 + 70/3*e^2 + 36*e - 82/3, 1/8*e^5 - 1/4*e^4 - 13/4*e^3 + 7/2*e^2 + 12*e - 6, -1/6*e^5 + 1/4*e^4 + 31/6*e^3 - 31/6*e^2 - 28*e + 68/3, -3/8*e^5 + 5/4*e^4 + 35/4*e^3 - 20*e^2 - 25*e - 6, 1/3*e^5 - 5/4*e^4 - 47/6*e^3 + 143/6*e^2 + 15*e - 100/3, -e^5 + 3*e^4 + 25*e^3 - 57*e^2 - 85*e + 94, -5/48*e^5 + 3/8*e^4 + 43/24*e^3 - 71/12*e^2 + 8*e + 44/3, 11/48*e^5 - 7/8*e^4 - 133/24*e^3 + 221/12*e^2 + 26*e - 164/3, 17/24*e^5 - 2*e^4 - 223/12*e^3 + 112/3*e^2 + 72*e - 172/3, -1/3*e^5 + 5/4*e^4 + 22/3*e^3 - 149/6*e^2 - 14*e + 130/3, 3/8*e^5 - e^4 - 35/4*e^3 + 15*e^2 + 17*e + 8, -5/8*e^5 + 7/4*e^4 + 65/4*e^3 - 67/2*e^2 - 64*e + 62, -11/24*e^5 + e^4 + 145/12*e^3 - 49/3*e^2 - 54*e + 76/3, -25/24*e^5 + 3*e^4 + 305/12*e^3 - 331/6*e^2 - 76*e + 242/3, -1/3*e^5 + 3/4*e^4 + 28/3*e^3 - 77/6*e^2 - 50*e + 82/3, -7/48*e^5 + 5/8*e^4 + 77/24*e^3 - 175/12*e^2 + 148/3, -7/12*e^5 + 3/2*e^4 + 46/3*e^3 - 161/6*e^2 - 63*e + 154/3, -1/24*e^5 + 5/12*e^3 + 13/3*e^2 + 6*e - 124/3, -9/16*e^5 + 15/8*e^4 + 107/8*e^3 - 145/4*e^2 - 38*e + 54, -1/2*e^5 + 2*e^4 + 12*e^3 - 41*e^2 - 34*e + 76, -29/24*e^5 + 15/4*e^4 + 349/12*e^3 - 205/3*e^2 - 80*e + 280/3, 1/8*e^5 - 1/2*e^4 - 9/4*e^3 + 21/2*e^2 - 6*e - 42, -3/8*e^5 + 5/4*e^4 + 37/4*e^3 - 23*e^2 - 34*e + 54, 5/48*e^5 - 3/8*e^4 - 55/24*e^3 + 71/12*e^2 + 4*e + 16/3, 1/4*e^5 - 1/4*e^4 - 7*e^3 - 1/2*e^2 + 29*e + 28, -3/8*e^5 + 5/4*e^4 + 37/4*e^3 - 49/2*e^2 - 26*e + 34, 17/48*e^5 - 5/8*e^4 - 235/24*e^3 + 119/12*e^2 + 50*e - 74/3, -7/12*e^5 + 2*e^4 + 43/3*e^3 - 239/6*e^2 - 48*e + 172/3, 1/8*e^5 - 1/2*e^4 - 11/4*e^3 + 11*e^2 + 9*e - 28, -17/24*e^5 + 2*e^4 + 223/12*e^3 - 115/3*e^2 - 75*e + 178/3, 7/12*e^5 - 7/4*e^4 - 89/6*e^3 + 97/3*e^2 + 54*e - 130/3, 7/24*e^5 - 3/4*e^4 - 95/12*e^3 + 41/3*e^2 + 34*e - 86/3, 3/4*e^4 - e^3 - 37/2*e^2 + 12*e + 54, -31/48*e^5 + 17/8*e^4 + 389/24*e^3 - 481/12*e^2 - 64*e + 178/3, -1/24*e^5 + 1/4*e^4 + 5/12*e^3 - 20/3*e^2 + 8*e + 50/3, -11/12*e^5 + 3*e^4 + 65/3*e^3 - 343/6*e^2 - 58*e + 266/3, 5/24*e^5 - e^4 - 55/12*e^3 + 49/3*e^2 + 14*e + 44/3, -3/8*e^5 + 3/2*e^4 + 37/4*e^3 - 32*e^2 - 31*e + 62, -35/24*e^5 + 19/4*e^4 + 415/12*e^3 - 262/3*e^2 - 96*e + 334/3, 1/48*e^5 + 5/8*e^4 - 59/24*e^3 - 191/12*e^2 + 27*e + 134/3, -13/12*e^5 + 7/2*e^4 + 161/6*e^3 - 401/6*e^2 - 88*e + 316/3, -35/48*e^5 + 21/8*e^4 + 421/24*e^3 - 617/12*e^2 - 52*e + 218/3, 4/3*e^5 - 4*e^4 - 100/3*e^3 + 455/6*e^2 + 114*e - 400/3, -1/48*e^5 - 3/8*e^4 + 59/24*e^3 + 95/12*e^2 - 28*e - 56/3, -17/24*e^5 + 5/2*e^4 + 211/12*e^3 - 142/3*e^2 - 62*e + 184/3, 5/48*e^5 - 3/8*e^4 - 19/24*e^3 + 65/12*e^2 - 23*e + 16/3];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {5, 5, -2*w^5 + w^4 + 13*w^3 - 2*w^2 - 19*w - 5}>] := -1;
ALEigenvalues[ideal<ZF | {9, 3, 2*w^5 - w^4 - 14*w^3 + 2*w^2 + 23*w + 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;