/*
  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, 0, 7, -4, -2, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {25, 25, -w^4 + w^3 + 4*w^2 - 2*w + 1}>;

primesArray := [
[5, 5, w^4 - 2*w^3 - 4*w^2 + 6*w + 1],
[5, 5, w^4 - w^3 - 5*w^2 + 3*w + 1],
[7, 7, -w^4 + 2*w^3 + 4*w^2 - 6*w],
[7, 7, w^4 - w^3 - 5*w^2 + 3*w + 3],
[19, 19, -w^3 + w^2 + 4*w],
[23, 23, -w^2 + 3],
[32, 2, 2],
[37, 37, -w^4 + 2*w^3 + 3*w^2 - 7*w + 4],
[41, 41, w^4 - w^3 - 5*w^2 + 2*w + 4],
[43, 43, -w^4 + 2*w^3 + 4*w^2 - 5*w - 3],
[43, 43, -w^2 + w + 3],
[47, 47, 2*w^4 - 3*w^3 - 9*w^2 + 8*w + 3],
[61, 61, 2*w^4 - 3*w^3 - 9*w^2 + 10*w + 2],
[79, 79, w^2 - w - 5],
[79, 79, -w^3 + 2*w^2 + 3*w - 3],
[79, 79, -2*w^4 + 4*w^3 + 7*w^2 - 14*w + 2],
[97, 97, w^3 - 6*w],
[101, 101, -w^4 + 2*w^3 + 5*w^2 - 6*w - 4],
[101, 101, 2*w^4 - 3*w^3 - 9*w^2 + 10*w + 1],
[107, 107, 2*w^4 - 3*w^3 - 8*w^2 + 7*w + 2],
[121, 11, w^4 - w^3 - 6*w^2 + 3*w + 4],
[125, 5, 3*w^4 - 4*w^3 - 14*w^2 + 12*w + 3],
[131, 131, w^2 - 5],
[131, 131, w^4 - 6*w^2 - 3*w + 5],
[131, 131, -2*w^4 + 3*w^3 + 9*w^2 - 10*w + 1],
[137, 137, -2*w^4 + 3*w^3 + 8*w^2 - 8*w],
[137, 137, -2*w^4 + 3*w^3 + 8*w^2 - 9*w - 2],
[137, 137, -2*w^4 + 3*w^3 + 9*w^2 - 7*w - 3],
[139, 139, w^4 - 2*w^3 - 5*w^2 + 7*w],
[149, 149, 2*w^4 - w^3 - 11*w^2 + 3],
[149, 149, -2*w^4 + 3*w^3 + 8*w^2 - 9*w],
[149, 149, 2*w^4 - 2*w^3 - 10*w^2 + 5*w + 6],
[151, 151, -3*w^4 + 5*w^3 + 13*w^2 - 15*w - 3],
[157, 157, -w^4 + 2*w^3 + 3*w^2 - 7*w + 5],
[163, 163, -2*w^3 + 2*w^2 + 8*w - 5],
[167, 167, w^4 - 6*w^2 - w + 3],
[167, 167, -2*w^4 + 3*w^3 + 8*w^2 - 10*w + 2],
[173, 173, -w^3 + w^2 + 6*w - 4],
[173, 173, -w^3 + w^2 + 6*w - 2],
[179, 179, w^3 - 2*w^2 - 4*w + 4],
[181, 181, w^4 - 3*w^3 - 3*w^2 + 10*w - 2],
[199, 199, w^4 + w^3 - 7*w^2 - 6*w + 5],
[199, 199, -w^4 + 7*w^2 - w - 3],
[199, 199, -w^4 + 2*w^3 + 4*w^2 - 6*w + 3],
[211, 211, -w^4 - w^3 + 6*w^2 + 6*w - 3],
[223, 223, 3*w^4 - 5*w^3 - 12*w^2 + 15*w + 3],
[223, 223, w^4 - 3*w^3 - 3*w^2 + 11*w],
[229, 229, 2*w^4 - 3*w^3 - 10*w^2 + 10*w + 3],
[233, 233, -2*w^4 + 4*w^3 + 7*w^2 - 13*w + 1],
[239, 239, w^3 - 2*w^2 - 5*w + 2],
[243, 3, -3],
[251, 251, 2*w^4 - 3*w^3 - 8*w^2 + 8*w + 1],
[263, 263, -w^4 + 5*w^2 - 3],
[263, 263, 2*w^4 - 4*w^3 - 8*w^2 + 13*w + 3],
[277, 277, w^3 - w^2 - 6*w + 1],
[277, 277, -w^4 + w^3 + 6*w^2 - 4*w - 6],
[283, 283, -2*w^4 + 4*w^3 + 7*w^2 - 11*w - 2],
[293, 293, -2*w^4 + 4*w^3 + 8*w^2 - 12*w + 1],
[311, 311, 3*w^4 - w^3 - 16*w^2 - 2*w + 5],
[313, 313, -4*w^4 + 6*w^3 + 17*w^2 - 18*w + 1],
[331, 331, 2*w^4 - 2*w^3 - 9*w^2 + 6*w + 1],
[337, 337, 3*w^4 - 6*w^3 - 12*w^2 + 19*w - 1],
[337, 337, 3*w^4 - 3*w^3 - 13*w^2 + 5*w - 1],
[343, 7, -2*w^4 + 4*w^3 + 9*w^2 - 14*w - 1],
[349, 349, w^4 - w^3 - 3*w^2 - 3*w + 1],
[353, 353, -w^4 + 3*w^3 + 3*w^2 - 10*w + 1],
[367, 367, w - 4],
[379, 379, w^3 - 2*w^2 - 3*w + 8],
[379, 379, 2*w^4 - 2*w^3 - 11*w^2 + 6*w + 5],
[379, 379, -w^4 + 3*w^3 + 4*w^2 - 11*w],
[389, 389, 3*w^4 - 2*w^3 - 15*w^2 + 4],
[397, 397, 2*w^4 - w^3 - 10*w^2 + w],
[409, 409, 2*w^4 - 4*w^3 - 9*w^2 + 14*w + 2],
[409, 409, 3*w^4 - 2*w^3 - 16*w^2 + 3*w + 4],
[431, 431, -w^4 + w^3 + 7*w^2 - 5*w - 7],
[433, 433, -w^3 + 2*w^2 + 2*w - 5],
[433, 433, -w^4 + 2*w^3 + 3*w^2 - 5*w + 4],
[439, 439, w^4 + w^3 - 6*w^2 - 5*w + 4],
[439, 439, -3*w^4 + 5*w^3 + 12*w^2 - 17*w - 2],
[439, 439, -2*w^4 + 4*w^3 + 9*w^2 - 12*w],
[443, 443, w^4 - w^3 - 5*w^2 + 3*w - 2],
[457, 457, -w^4 + 6*w^2 + w + 1],
[457, 457, -2*w^4 + 4*w^3 + 8*w^2 - 15*w + 1],
[461, 461, 3*w^4 - 5*w^3 - 13*w^2 + 17*w + 6],
[461, 461, 2*w^3 - w^2 - 8*w - 2],
[461, 461, 3*w^4 - 3*w^3 - 13*w^2 + 7*w + 1],
[463, 463, 3*w^4 - 6*w^3 - 13*w^2 + 19*w + 3],
[467, 467, -w^4 + 4*w^3 + 2*w^2 - 15*w + 4],
[487, 487, -w^4 + 2*w^3 + 6*w^2 - 9*w - 3],
[487, 487, 3*w^4 - 3*w^3 - 16*w^2 + 8*w + 10],
[487, 487, -w^4 + 7*w^2 + 3*w - 7],
[491, 491, -w^4 + 3*w^3 + 3*w^2 - 9*w + 1],
[509, 509, 2*w^4 - w^3 - 11*w^2 + w + 5],
[521, 521, w^4 - 2*w^3 - 2*w^2 + 6*w - 1],
[521, 521, w^4 - 2*w^3 - 4*w^2 + 3*w + 3],
[529, 23, w^4 - 2*w^3 - 6*w^2 + 10*w + 3],
[529, 23, 3*w^4 - 2*w^3 - 16*w^2 + w + 4],
[541, 541, -w^4 + 2*w^3 + 4*w^2 - 9*w + 1],
[563, 563, 2*w^3 - w^2 - 8*w + 1],
[569, 569, -2*w^4 + 3*w^3 + 11*w^2 - 11*w - 7],
[571, 571, w^4 - w^3 - 7*w^2 + 4*w + 3],
[571, 571, -w^4 + 3*w^3 + 4*w^2 - 11*w + 1],
[587, 587, 3*w^4 - 5*w^3 - 11*w^2 + 14*w - 3],
[587, 587, 3*w^4 - 6*w^3 - 12*w^2 + 19*w + 2],
[587, 587, w^3 - 7*w],
[593, 593, -3*w^4 + 2*w^3 + 15*w^2 - 3*w - 5],
[607, 607, 3*w^4 - 4*w^3 - 14*w^2 + 11*w + 9],
[613, 613, -w^3 + w^2 + 4*w - 6],
[613, 613, -3*w^4 + 3*w^3 + 15*w^2 - 8*w - 8],
[617, 617, -w^4 + w^3 + 4*w^2 - 4],
[619, 619, -3*w^4 + 5*w^3 + 12*w^2 - 14*w + 1],
[631, 631, -w^3 + w^2 + 2*w - 4],
[641, 641, 2*w^4 - 2*w^3 - 9*w^2 + 4*w + 4],
[641, 641, -w^4 - w^3 + 6*w^2 + 7*w - 5],
[643, 643, 2*w^4 - 5*w^3 - 8*w^2 + 17*w - 2],
[643, 643, 2*w^4 - 2*w^3 - 10*w^2 + 7*w + 2],
[647, 647, 3*w^4 - 3*w^3 - 15*w^2 + 6*w + 2],
[653, 653, w^4 - 2*w^3 - 3*w^2 + 7*w + 1],
[653, 653, 3*w^4 - 4*w^3 - 13*w^2 + 9*w + 2],
[661, 661, -4*w^4 + 7*w^3 + 18*w^2 - 22*w - 4],
[673, 673, w^4 - 6*w^2 + 6],
[673, 673, -3*w - 2],
[673, 673, -2*w^4 + 2*w^3 + 11*w^2 - 7*w - 2],
[677, 677, 3*w^4 - 5*w^3 - 12*w^2 + 15*w + 2],
[683, 683, -2*w^4 + 3*w^3 + 10*w^2 - 12*w - 6],
[691, 691, -2*w^4 + 3*w^3 + 11*w^2 - 10*w - 4],
[701, 701, w^2 - 2*w - 4],
[709, 709, -w^3 - w^2 + 4*w + 5],
[709, 709, 2*w^4 - 2*w^3 - 10*w^2 + 5*w + 8],
[719, 719, 3*w^4 - 5*w^3 - 12*w^2 + 15*w + 1],
[719, 719, 2*w^3 - 2*w^2 - 7*w + 2],
[727, 727, 3*w^4 - 6*w^3 - 13*w^2 + 22*w + 2],
[727, 727, w^4 + w^3 - 7*w^2 - 8*w + 3],
[727, 727, -3*w^4 + 5*w^3 + 12*w^2 - 13*w - 2],
[733, 733, 2*w^4 - 3*w^3 - 7*w^2 + 8*w + 1],
[761, 761, 3*w^4 - 4*w^3 - 13*w^2 + 12*w],
[769, 769, -4*w^4 + 7*w^3 + 16*w^2 - 23*w + 2],
[787, 787, 2*w^4 - 5*w^3 - 7*w^2 + 17*w - 7],
[809, 809, -w^4 + 6*w^2 - 4],
[821, 821, w^3 - 5*w + 5],
[821, 821, -2*w^4 + 3*w^3 + 11*w^2 - 11*w - 8],
[827, 827, 2*w^4 - 2*w^3 - 10*w^2 + 3*w + 5],
[827, 827, -2*w^4 + 3*w^3 + 7*w^2 - 7*w + 3],
[827, 827, -2*w^4 + 3*w^3 + 9*w^2 - 12*w + 2],
[829, 829, 4*w^4 - 7*w^3 - 17*w^2 + 24*w + 3],
[829, 829, -w^4 + 2*w^3 + 6*w^2 - 9*w - 2],
[829, 829, -3*w^4 + 3*w^3 + 15*w^2 - 8*w - 7],
[839, 839, w^4 - w^3 - 3*w^2 - 4],
[841, 29, -6*w^4 + 10*w^3 + 27*w^2 - 31*w - 8],
[853, 853, 4*w^4 - 6*w^3 - 19*w^2 + 19*w + 6],
[853, 853, w^4 - w^3 - 3*w^2 + w - 6],
[881, 881, -w^4 + 2*w^3 + 4*w^2 - 9*w],
[883, 883, -w^4 + w^3 + 5*w^2 - 4],
[883, 883, 4*w^4 - 7*w^3 - 16*w^2 + 20*w + 2],
[887, 887, 2*w^4 - 5*w^3 - 6*w^2 + 15*w - 2],
[907, 907, 3*w^4 - 6*w^3 - 13*w^2 + 20*w],
[911, 911, w^4 - 5*w^2 - 3*w + 4],
[919, 919, 4*w^4 - 7*w^3 - 16*w^2 + 21*w + 3],
[929, 929, w^4 - 7*w^2 - 3*w + 4],
[937, 937, -2*w^4 + 2*w^3 + 9*w^2 - 7*w],
[937, 937, w^4 - 3*w^3 - 5*w^2 + 13*w + 4],
[947, 947, w^4 - 3*w^3 - w^2 + 11*w - 7],
[953, 953, 3*w^4 - 4*w^3 - 12*w^2 + 11*w - 1],
[977, 977, 2*w^4 - 2*w^3 - 11*w^2 + 6*w + 2],
[983, 983, -2*w^4 + 3*w^3 + 8*w^2 - 11*w + 4],
[983, 983, w^3 + w^2 - 3*w - 4],
[997, 997, -w^4 + 4*w^2 + 4*w - 1]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^4 + 5*x^3 - 2*x^2 - 24*x - 9;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, 0, 1/3*e^2 + 1/3*e - 1, 1/3*e^3 + 4/3*e^2 - 2*e - 6, -2/3*e^3 - 7/3*e^2 + 13/3*e + 8, -e - 3, -1/3*e^3 - 1/3*e^2 + 3*e - 3, 1/3*e^3 + 2/3*e^2 - 5/3*e - 1, 2/3*e^3 + 5/3*e^2 - 5*e - 3, 1/3*e^3 + e^2 - 10/3*e + 1, -e^2 - 3*e + 3, e^3 + 2*e^2 - 9*e - 9, -1/3*e^3 - 2/3*e^2 + 2/3*e + 1, 1/3*e^3 - e^2 - 19/3*e + 7, e^3 + 14/3*e^2 + 2/3*e - 17, -e^3 - 10/3*e^2 + 11/3*e + 4, 5/3*e^2 + 17/3*e - 2, -2*e^2 - e + 15, -2*e^3 - 8*e^2 + 9*e + 24, -2*e^3 - 6*e^2 + 10*e + 15, -4/3*e^3 - 4*e^2 + 31/3*e + 14, e^2 + e - 18, 3*e^2 + 5*e - 18, 1/3*e^3 + 1/3*e^2 + 3, -e^2 + 3*e + 15, 4/3*e^3 + 13/3*e^2 - 7*e - 15, -5/3*e^3 - 11/3*e^2 + 14*e + 12, e^3 + 3*e^2 - 9*e - 15, 5/3*e^3 + 17/3*e^2 - 7*e - 20, -1/3*e^3 + 5/3*e^2 + 9*e, e^3 + 4*e^2 - 7*e - 18, -e - 12, 1/3*e^3 + 4*e^2 + 5/3*e - 20, -2*e^3 - 25/3*e^2 + 20/3*e + 28, 1/3*e^3 + 1/3*e^2 + 9, 4/3*e^3 + 10/3*e^2 - 7*e - 9, 2/3*e^3 + 2/3*e^2 - 4*e + 9, -5/3*e^3 - 17/3*e^2 + 11*e + 21, -1/3*e^3 - 13/3*e^2 - 7*e + 18, 2*e^3 + 6*e^2 - 10*e - 18, 2/3*e^3 + 11/3*e^2 - 3*e - 18, -4/3*e^3 - 13/3*e^2 + 6*e + 12, -4/3*e^3 - 7/3*e^2 + 12*e, 4/3*e^3 + 4/3*e^2 - 14*e - 3, 2/3*e^3 + 5/3*e^2 - 3*e - 6, -4/3*e^3 - 22/3*e^2 + 21, -2/3*e^3 - 2*e^2 - 7/3*e - 5, 2/3*e^3 + 5/3*e^2 - e - 11, e^2 + 6*e - 9, -5/3*e^3 - 14/3*e^2 + 6*e, -2/3*e^3 - 14/3*e^2 - 3*e + 17, -2/3*e^3 + 1/3*e^2 + 11*e - 3, 1/3*e^3 - 2/3*e^2 - 9*e + 12, -2/3*e^3 - 11/3*e^2 - 2*e + 6, 2/3*e^3 + 8/3*e^2 - 9*e - 12, 2/3*e^3 + 5*e^2 - 5/3*e - 19, 2/3*e^3 + 10/3*e^2 - 22/3*e - 17, -e^3 - 3*e^2 + 4*e + 21, -5/3*e^3 - 26/3*e^2 + 3*e + 21, 4/3*e^3 + 14/3*e^2 - 2/3*e - 7, 4/3*e^3 + 2/3*e^2 - 41/3*e + 2, e^3 + 10/3*e^2 - 2/3*e - 10, -7/3*e^3 - 34/3*e^2 + 8*e + 34, 1/3*e^3 - 19/3*e - 5, -e^3 - 5*e^2 + 8*e + 21, 1/3*e^3 + 4/3*e^2 - 15, e^3 + 4*e^2 - 5*e - 33, -5/3*e^3 - 8*e^2 + 23/3*e + 40, -4/3*e^3 - 20/3*e^2 + 23/3*e + 28, 2*e^3 + 22/3*e^2 - 35/3*e - 31, -7/3*e^3 - 28/3*e^2 + 5*e + 33, e^3 + 17/3*e^2 + 5/3*e - 38, -1/3*e^3 - 8/3*e^2 - 1/3*e + 34, -2/3*e^3 - 10/3*e^2 + 10/3*e + 11, 5/3*e^3 + 14/3*e^2 - 3*e, 2/3*e^3 + e^2 - 5/3*e + 8, 7/3*e^3 + 26/3*e^2 - 35/3*e - 28, -e^3 - 4*e^2 + 9*e + 19, -7/3*e^3 - 7*e^2 + 34/3*e + 20, 3*e^3 + 8*e^2 - 18*e - 17, -2/3*e^3 + 7/3*e^2 + 13*e - 3, -7/3*e^3 - 19/3*e^2 + 20*e + 29, -2*e^2 - 3*e + 6, 7/3*e^3 + 25/3*e^2 - 16*e - 33, 1/3*e^3 + 1/3*e^2 - 10*e - 3, 2/3*e^3 + 11/3*e^2 - 30, -3*e^3 - 12*e^2 + 10*e + 33, -10/3*e^3 - 28/3*e^2 + 25*e + 27, 4/3*e^3 + 11/3*e^2 - 47/3*e - 31, 7/3*e^3 + 19/3*e^2 - 11*e - 14, -3*e^3 - 10*e^2 + 9*e + 29, e^3 - e^2 - 16*e - 12, e^3 + 4*e^2 + e - 3, 7/3*e^3 + 22/3*e^2 - 16*e - 45, -e^2 - 2*e - 18, 7/3*e^3 + 14/3*e^2 - 77/3*e - 31, e^3 + 22/3*e^2 + 46/3*e - 31, -1/3*e^3 + 11/3*e^2 + 14*e - 12, 4/3*e^3 + 22/3*e^2 - 8*e - 30, -2*e^3 - 4*e^2 + 20*e + 3, 4/3*e^3 + 6*e^2 + 2/3*e - 5, e^3 + 8/3*e^2 - 13/3*e - 11, 2*e^3 + 5*e^2 - 14*e - 24, -2*e^3 - 5*e^2 + 18*e + 9, 2*e^3 + 11*e^2 + 8*e - 42, -2*e^3 - 5*e^2 + 9*e - 12, -4*e^3 - 43/3*e^2 + 80/3*e + 55, -7/3*e^3 - 22/3*e^2 + 11*e + 19, 1/3*e^3 + 7/3*e^2 - 2*e - 37, -4/3*e^3 - 13/3*e^2 + e + 18, -4/3*e^3 - 20/3*e^2 - 16/3*e + 37, -2*e^3 - 28/3*e^2 + 11/3*e + 40, -2*e^2 - e + 36, -2*e^2 - e + 18, -e^3 - 2/3*e^2 + 49/3*e + 17, 1/3*e^3 + 11/3*e^2 - 11/3*e - 28, 2*e^3 + 9*e^2 - 13*e - 48, -2*e^3 - 3*e^2 + 15*e - 12, -3*e^3 - 9*e^2 + 15*e + 27, -2*e^2 + 3*e + 22, 13/3*e^3 + 37/3*e^2 - 25*e - 45, -7/3*e^3 - 5/3*e^2 + 95/3*e + 7, 10/3*e^3 + 22/3*e^2 - 32*e - 36, 1/3*e^3 - 11/3*e^2 - 22*e + 12, -1/3*e^3 - 1/3*e^2 - e, -1/3*e^3 - 4/3*e^2 + 4*e - 15, 4*e^3 + 14*e^2 - 21*e - 30, 1/3*e^3 + 2/3*e^2 + 19/3*e + 2, -2*e^3 - 25/3*e^2 + 26/3*e + 40, 7/3*e^3 + 28/3*e^2 + e - 18, e^3 + 5*e^2 - 2*e - 33, 7/3*e^3 + 7*e^2 - 31/3*e - 5, -2/3*e^3 - e^2 + 17/3*e + 19, -1/3*e^3 - 3*e^2 + 19/3*e + 44, -14/3*e^3 - 50/3*e^2 + 13*e + 47, 5/3*e^3 + 23/3*e^2 + 3*e - 36, -4/3*e^3 + e^2 + 64/3*e - 10, 3*e^3 + 11*e^2 - 18*e - 40, -2*e^3 - 3*e^2 + 14*e - 12, -2*e^3 - 8*e^2 + 10*e - 3, -5/3*e^3 - 14/3*e^2 + 23*e + 30, 2*e^3 + 4*e^2 - 15*e, 7/3*e^3 + 16/3*e^2 - 33*e - 27, -5/3*e^3 - 23/3*e^2 + 13*e + 63, 7/3*e^3 + 40/3*e^2 - 6*e - 42, -8/3*e^3 - 19/3*e^2 + 79/3*e + 38, 7/3*e^3 + 16/3*e^2 - 18*e - 9, 5/3*e^3 + 5/3*e^2 - 21*e + 9, -2/3*e^3 - 4*e^2 + 26/3*e + 37, -5/3*e^3 - 2/3*e^2 + 25*e + 5, 4/3*e^3 + 1/3*e^2 - 26*e + 7, -4/3*e^3 - 22/3*e^2 + 5*e + 27, -2*e^3 - 8*e^2 + 21*e + 38, e^3 + 14/3*e^2 - 31/3*e - 20, -e^3 - 5*e^2 + 2*e + 12, 7/3*e^3 + 25/3*e^2 - 6*e - 6, -7/3*e^3 - 22/3*e^2 + 19*e + 66, 3*e^3 + 10*e^2 - 23*e - 57, 4/3*e^3 + 10/3*e^2 - 10*e - 45, 3*e^3 + 14/3*e^2 - 103/3*e - 20, -10/3*e^3 - 38/3*e^2 + 65/3*e + 61, 4/3*e^3 - 2/3*e^2 - 14*e + 21, -8/3*e^3 - 20/3*e^2 + 21*e + 12, -7/3*e^3 - 13/3*e^2 + 24*e + 15, 4/3*e^3 + 31/3*e^2 + 2*e - 24, 4/3*e^3 + 1/3*e^2 - 24*e - 15, -e^3 - 8/3*e^2 + 25/3*e + 8];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

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