/*
  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![25, 5, -11, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {25,5,1/5*w^3 - 1/5*w^2 - 11/5*w + 1}>;

primesArray := [
[5, 5, 1/5*w^3 + 4/5*w^2 - 11/5*w - 7],
[5, 5, 1/5*w^3 - 6/5*w^2 - 1/5*w + 4],
[11, 11, w + 1],
[11, 11, 1/5*w^3 - 1/5*w^2 - 11/5*w + 2],
[16, 2, 2],
[19, 19, -1/5*w^3 + 1/5*w^2 + 1/5*w + 1],
[19, 19, 1/5*w^3 - 1/5*w^2 - 11/5*w],
[19, 19, -2/5*w^3 + 2/5*w^2 + 17/5*w],
[19, 19, w - 1],
[29, 29, w^2 - w - 8],
[29, 29, 2/5*w^3 - 2/5*w^2 - 7/5*w + 2],
[31, 31, 1/5*w^3 - 1/5*w^2 - 1/5*w + 2],
[31, 31, 2/5*w^3 - 2/5*w^2 - 17/5*w + 3],
[61, 61, -2/5*w^3 - 3/5*w^2 + 12/5*w + 5],
[61, 61, -4/5*w^3 - 6/5*w^2 + 39/5*w + 15],
[61, 61, 3/5*w^3 + 2/5*w^2 - 18/5*w - 3],
[61, 61, 7/5*w^3 + 3/5*w^2 - 62/5*w - 13],
[71, 71, 1/5*w^3 - 6/5*w^2 - 6/5*w + 4],
[71, 71, w^2 - 8],
[79, 79, 3/5*w^3 + 12/5*w^2 - 33/5*w - 22],
[79, 79, 1/5*w^3 + 4/5*w^2 - 11/5*w - 9],
[81, 3, -3],
[101, 101, 3/5*w^3 + 7/5*w^2 - 28/5*w - 15],
[101, 101, 1/5*w^3 + 9/5*w^2 - 16/5*w - 16],
[121, 11, 3/5*w^3 - 3/5*w^2 - 18/5*w + 1],
[131, 131, 1/5*w^3 + 4/5*w^2 - 16/5*w - 8],
[131, 131, w^2 - 2*w - 2],
[149, 149, 4/5*w^3 + 1/5*w^2 - 29/5*w - 5],
[149, 149, 4/5*w^3 - 9/5*w^2 - 19/5*w + 6],
[151, 151, 2/5*w^3 + 3/5*w^2 - 17/5*w - 3],
[151, 151, -2/5*w^3 + 7/5*w^2 + 7/5*w - 8],
[151, 151, 4/5*w^3 + 1/5*w^2 - 29/5*w - 2],
[151, 151, -4/5*w^3 + 9/5*w^2 + 19/5*w - 9],
[179, 179, -3/5*w^3 + 8/5*w^2 + 18/5*w - 6],
[179, 179, 2/5*w^3 + 3/5*w^2 - 12/5*w - 6],
[211, 211, 1/5*w^3 + 4/5*w^2 - 6/5*w - 10],
[211, 211, 2/5*w^3 - 7/5*w^2 - 12/5*w + 2],
[229, 229, 6/5*w^3 - 11/5*w^2 - 41/5*w + 11],
[229, 229, 2*w^2 - 2*w - 9],
[229, 229, -4/5*w^3 - 1/5*w^2 + 19/5*w + 2],
[229, 229, -3/5*w^3 + 3/5*w^2 + 28/5*w - 3],
[239, 239, 11/5*w^3 + 4/5*w^2 - 96/5*w - 19],
[239, 239, -6/5*w^3 + 1/5*w^2 + 51/5*w + 3],
[241, 241, 1/5*w^3 + 4/5*w^2 - 1/5*w - 6],
[241, 241, 4/5*w^3 - 4/5*w^2 - 29/5*w - 1],
[251, 251, -1/5*w^3 + 6/5*w^2 + 11/5*w - 12],
[251, 251, -1/5*w^3 + 11/5*w^2 + 1/5*w - 15],
[269, 269, 2/5*w^3 + 3/5*w^2 - 17/5*w - 2],
[269, 269, -2/5*w^3 + 7/5*w^2 + 7/5*w - 9],
[271, 271, 1/5*w^3 - 1/5*w^2 + 4/5*w],
[271, 271, 3/5*w^3 - 3/5*w^2 - 28/5*w + 2],
[281, 281, w^3 - 3*w^2 - 4*w + 13],
[281, 281, -3/5*w^3 + 3/5*w^2 + 28/5*w - 6],
[281, 281, -6/5*w^3 + 26/5*w^2 + 11/5*w - 16],
[281, 281, w^3 + w^2 - 8*w - 9],
[289, 17, 2/5*w^3 + 3/5*w^2 - 7/5*w - 5],
[289, 17, 4/5*w^3 - 9/5*w^2 - 29/5*w + 8],
[311, 311, -w^2 + 2*w + 11],
[311, 311, -1/5*w^3 + 6/5*w^2 + 1/5*w - 1],
[311, 311, 1/5*w^3 + 4/5*w^2 - 16/5*w + 1],
[311, 311, -1/5*w^3 - 4/5*w^2 + 11/5*w + 10],
[349, 349, -1/5*w^3 - 9/5*w^2 + 11/5*w + 16],
[349, 349, 2/5*w^3 - 12/5*w^2 - 7/5*w + 7],
[359, 359, -4/5*w^3 + 4/5*w^2 + 19/5*w - 1],
[359, 359, -w^3 + 3*w^2 + 5*w - 12],
[359, 359, w^3 - w^2 - 7*w + 2],
[359, 359, 4/5*w^3 + 6/5*w^2 - 29/5*w - 11],
[379, 379, 4/5*w^3 + 6/5*w^2 - 34/5*w - 9],
[379, 379, 3/5*w^3 - 3/5*w^2 - 28/5*w + 7],
[389, 389, w^2 - 3*w - 8],
[389, 389, -4/5*w^3 + 14/5*w^2 + 19/5*w - 16],
[401, 401, -3*w^2 + 2*w + 23],
[401, 401, 1/5*w^3 - 16/5*w^2 + 4/5*w + 11],
[409, 409, -w^3 + 2*w^2 + 3*w - 7],
[409, 409, 7/5*w^3 - 2/5*w^2 - 57/5*w - 2],
[419, 419, -1/5*w^3 + 1/5*w^2 + 16/5*w - 2],
[419, 419, 1/5*w^3 - 1/5*w^2 - 16/5*w],
[421, 421, 2/5*w^3 + 3/5*w^2 - 27/5*w - 3],
[421, 421, 1/5*w^3 + 4/5*w^2 + 4/5*w - 3],
[439, 439, -w^3 + 3*w^2 + 5*w - 16],
[439, 439, -4/5*w^3 - 6/5*w^2 + 29/5*w + 7],
[439, 439, -3/5*w^3 + 13/5*w^2 + 3/5*w - 11],
[439, 439, 4/5*w^3 + 6/5*w^2 - 39/5*w - 10],
[449, 449, 2/5*w^3 - 12/5*w^2 - 2/5*w + 5],
[449, 449, 4/5*w^3 - 24/5*w^2 - 4/5*w + 15],
[461, 461, -1/5*w^3 - 4/5*w^2 - 4/5*w + 4],
[461, 461, 2*w^2 - 2*w - 11],
[461, 461, 3/5*w^3 + 2/5*w^2 - 33/5*w - 4],
[479, 479, -w^3 + 3*w^2 + 6*w - 14],
[479, 479, -3/5*w^3 + 3/5*w^2 + 28/5*w + 3],
[499, 499, 2/5*w^3 + 8/5*w^2 - 17/5*w - 6],
[499, 499, -3/5*w^3 + 13/5*w^2 + 13/5*w - 17],
[509, 509, -w^3 + 2*w^2 + 4*w - 6],
[509, 509, 6/5*w^3 - 1/5*w^2 - 46/5*w - 4],
[521, 521, -2/5*w^3 + 2/5*w^2 + 7/5*w - 7],
[521, 521, -3/5*w^3 + 3/5*w^2 + 23/5*w - 8],
[529, 23, -7/5*w^3 + 17/5*w^2 + 42/5*w - 15],
[529, 23, 2/5*w^3 + 13/5*w^2 - 27/5*w - 21],
[541, 541, 1/5*w^3 - 11/5*w^2 - 1/5*w + 10],
[541, 541, 3/5*w^3 - 8/5*w^2 - 13/5*w + 13],
[541, 541, -2*w^2 + w + 13],
[541, 541, 3/5*w^3 + 2/5*w^2 - 23/5*w + 2],
[569, 569, w^3 - 2*w^2 - 5*w + 7],
[569, 569, -1/5*w^3 + 1/5*w^2 + 1/5*w - 6],
[571, 571, w^2 + w - 9],
[571, 571, -2/5*w^3 + 12/5*w^2 + 7/5*w - 12],
[571, 571, 1/5*w^3 + 9/5*w^2 - 11/5*w - 11],
[571, 571, -2/5*w^3 + 7/5*w^2 + 17/5*w - 4],
[601, 601, -3/5*w^3 + 13/5*w^2 + 18/5*w - 11],
[601, 601, -4/5*w^3 + 4/5*w^2 + 34/5*w + 1],
[631, 631, 2/5*w^3 + 18/5*w^2 - 27/5*w - 30],
[631, 631, 3/5*w^3 - 23/5*w^2 - 3/5*w + 15],
[661, 661, -2*w^3 + 17*w + 12],
[661, 661, -9/5*w^3 + 4/5*w^2 + 74/5*w + 1],
[691, 691, -7/5*w^3 - 18/5*w^2 + 72/5*w + 35],
[691, 691, 6/5*w^3 - 31/5*w^2 - 6/5*w + 19],
[701, 701, -1/5*w^3 + 6/5*w^2 + 16/5*w - 6],
[701, 701, 2/5*w^3 - 7/5*w^2 - 22/5*w + 8],
[709, 709, -3/5*w^3 + 13/5*w^2 + 13/5*w - 18],
[709, 709, 1/5*w^3 + 14/5*w^2 - 11/5*w - 20],
[709, 709, -2/5*w^3 - 8/5*w^2 + 17/5*w + 5],
[709, 709, 1/5*w^3 + 4/5*w^2 - 21/5*w - 10],
[719, 719, -3/5*w^3 + 8/5*w^2 + 13/5*w - 12],
[719, 719, 3/5*w^3 + 2/5*w^2 - 23/5*w + 1],
[719, 719, -7/5*w^3 + 17/5*w^2 + 32/5*w - 14],
[719, 719, 7/5*w^3 + 3/5*w^2 - 52/5*w - 8],
[739, 739, 1/5*w^3 - 1/5*w^2 - 11/5*w - 5],
[739, 739, w - 6],
[761, 761, -4/5*w^3 + 9/5*w^2 + 19/5*w - 11],
[761, 761, 4/5*w^3 + 1/5*w^2 - 29/5*w],
[769, 769, 3/5*w^3 - 13/5*w^2 - 13/5*w + 8],
[769, 769, -2/5*w^3 - 8/5*w^2 + 17/5*w + 15],
[769, 769, 7/5*w^3 - 17/5*w^2 - 37/5*w + 18],
[769, 769, -6/5*w^3 - 4/5*w^2 + 41/5*w + 5],
[809, 809, -1/5*w^3 + 6/5*w^2 - 4/5*w - 14],
[809, 809, -2/5*w^3 - 3/5*w^2 + 22/5*w - 4],
[809, 809, -6/5*w^3 + 6/5*w^2 + 41/5*w - 5],
[809, 809, w^3 - w^2 - 5*w + 4],
[821, 821, -3/5*w^3 - 17/5*w^2 + 33/5*w + 30],
[821, 821, -4/5*w^3 + 24/5*w^2 + 9/5*w - 15],
[829, 829, 7/5*w^3 - 12/5*w^2 - 52/5*w + 10],
[829, 829, -1/5*w^3 + 11/5*w^2 - 4/5*w - 19],
[829, 829, -13/5*w^3 + 3/5*w^2 + 108/5*w + 10],
[829, 829, -w^3 + 4*w^2 + 5*w - 23],
[839, 839, 4/5*w^3 - 9/5*w^2 - 24/5*w + 4],
[839, 839, 3/5*w^3 + 2/5*w^2 - 18/5*w - 8],
[841, 29, -w^3 + w^2 + 6*w - 4],
[859, 859, 6/5*w^3 - 11/5*w^2 - 36/5*w + 9],
[859, 859, w^3 - 6*w - 3],
[881, 881, 4/5*w^3 - 4/5*w^2 - 9/5*w - 1],
[881, 881, 9/5*w^3 - 24/5*w^2 - 19/5*w + 14],
[881, 881, -w^3 + 6*w + 2],
[881, 881, 1/5*w^3 - 1/5*w^2 - 1/5*w - 6],
[911, 911, w^3 - w^2 - 5*w + 1],
[911, 911, 6/5*w^3 - 6/5*w^2 - 41/5*w + 2],
[919, 919, -7/5*w^3 + 12/5*w^2 + 42/5*w - 7],
[919, 919, -6/5*w^3 + 1/5*w^2 + 36/5*w + 5],
[929, 929, -2/5*w^3 - 8/5*w^2 + 17/5*w + 16],
[929, 929, -3/5*w^3 + 13/5*w^2 + 13/5*w - 7],
[961, 31, w^3 - w^2 - 6*w + 2],
[971, 971, 8/5*w^3 - 13/5*w^2 - 48/5*w + 6],
[971, 971, -2/5*w^3 - 3/5*w^2 + 22/5*w],
[971, 971, 7/5*w^3 - 7/5*w^2 - 57/5*w + 7],
[971, 971, -1/5*w^3 + 6/5*w^2 - 4/5*w - 10],
[991, 991, 2/5*w^3 + 8/5*w^2 - 22/5*w - 7],
[991, 991, -2/5*w^3 + 12/5*w^2 + 2/5*w - 15]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^4 - 2*x^3 - 10*x^2 + 15*x - 3;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, 0, -2/5*e^3 + 1/5*e^2 + 19/5*e + 6/5, 3/5*e^3 - 4/5*e^2 - 31/5*e + 36/5, -2/5*e^3 + 6/5*e^2 + 24/5*e - 34/5, 6/5*e^3 - 8/5*e^2 - 67/5*e + 52/5, -3/5*e^3 + 4/5*e^2 + 31/5*e - 1/5, 3/5*e^3 - 4/5*e^2 - 26/5*e + 26/5, -3/5*e^3 + 4/5*e^2 + 31/5*e - 26/5, 8/5*e^3 - 9/5*e^2 - 96/5*e + 66/5, -3/5*e^3 - 1/5*e^2 + 31/5*e + 24/5, -2/5*e^3 + 6/5*e^2 + 14/5*e - 34/5, 2/5*e^3 - 6/5*e^2 - 14/5*e + 29/5, -4/5*e^3 + 2/5*e^2 + 63/5*e - 13/5, -3/5*e^3 + 9/5*e^2 + 16/5*e - 46/5, 2*e^3 - 2*e^2 - 21*e + 14, -11/5*e^3 + 13/5*e^2 + 102/5*e - 47/5, 12/5*e^3 - 16/5*e^2 - 139/5*e + 99/5, e^3 - e^2 - 10*e, -1/5*e^3 - 2/5*e^2 + 32/5*e + 38/5, -4/5*e^3 + 2/5*e^2 + 48/5*e - 38/5, -e^2 + 5, -1/5*e^3 - 12/5*e^2 + 17/5*e + 78/5, e^3 - 2*e^2 - 11*e + 15, -11/5*e^3 + 13/5*e^2 + 112/5*e - 77/5, -2/5*e^3 - 9/5*e^2 + 19/5*e + 81/5, 4/5*e^3 + 3/5*e^2 - 43/5*e - 27/5, -16/5*e^3 + 18/5*e^2 + 172/5*e - 117/5, -9/5*e^3 + 17/5*e^2 + 93/5*e - 48/5, -3*e^3 + 4*e^2 + 29*e - 22, 4/5*e^3 - 7/5*e^2 - 58/5*e + 43/5, 13/5*e^3 - 19/5*e^2 - 141/5*e + 106/5, 2*e^3 - 2*e^2 - 19*e + 8, -8/5*e^3 + 9/5*e^2 + 101/5*e - 36/5, -7/5*e^3 + 16/5*e^2 + 54/5*e - 69/5, -4*e^3 + 7*e^2 + 38*e - 37, -9/5*e^3 + 12/5*e^2 + 98/5*e - 98/5, -11/5*e^3 + 23/5*e^2 + 112/5*e - 97/5, e^2 - e - 5, -8/5*e^3 + 14/5*e^2 + 66/5*e - 56/5, -13/5*e^3 + 24/5*e^2 + 116/5*e - 101/5, -e^3 + 15*e + 6, 18/5*e^3 - 24/5*e^2 - 191/5*e + 126/5, 11/5*e^3 - 13/5*e^2 - 102/5*e + 47/5, -1/5*e^3 + 8/5*e^2 + 17/5*e - 97/5, 18/5*e^3 - 29/5*e^2 - 191/5*e + 171/5, -19/5*e^3 + 22/5*e^2 + 183/5*e - 63/5, 13/5*e^3 - 4/5*e^2 - 136/5*e + 6/5, -24/5*e^3 + 32/5*e^2 + 248/5*e - 183/5, 13/5*e^3 - 24/5*e^2 - 141/5*e + 106/5, -14/5*e^3 + 32/5*e^2 + 123/5*e - 188/5, e^3 - 2*e^2 - 16*e + 9, -28/5*e^3 + 39/5*e^2 + 286/5*e - 216/5, -13/5*e^3 + 9/5*e^2 + 146/5*e - 81/5, -1/5*e^3 + 8/5*e^2 + 12/5*e - 12/5, -4/5*e^3 + 7/5*e^2 + 48/5*e - 83/5, 12/5*e^3 - 11/5*e^2 - 159/5*e + 104/5, -4*e^3 + 7*e^2 + 43*e - 24, 18/5*e^3 - 24/5*e^2 - 171/5*e + 51/5, -27/5*e^3 + 41/5*e^2 + 274/5*e - 204/5, 1/5*e^3 + 7/5*e^2 - 22/5*e - 138/5, 28/5*e^3 - 39/5*e^2 - 301/5*e + 166/5, -17/5*e^3 + 36/5*e^2 + 179/5*e - 169/5, e^3 + e^2 - 15*e - 18, -3/5*e^3 - 1/5*e^2 + 46/5*e - 51/5, -6/5*e^3 + 23/5*e^2 + 62/5*e - 87/5, -13/5*e^3 + 19/5*e^2 + 181/5*e - 141/5, -19/5*e^3 + 17/5*e^2 + 183/5*e - 58/5, e^3 - e^2 - 12*e - 1, 12/5*e^3 - 31/5*e^2 - 119/5*e + 159/5, -37/5*e^3 + 36/5*e^2 + 394/5*e - 279/5, 27/5*e^3 - 41/5*e^2 - 259/5*e + 204/5, 17/5*e^3 - 31/5*e^2 - 159/5*e + 144/5, -1/5*e^3 + 3/5*e^2 - 3/5*e + 53/5, -4/5*e^3 - 18/5*e^2 + 63/5*e + 52/5, 1/5*e^3 - 8/5*e^2 + 18/5*e + 42/5, 21/5*e^3 - 23/5*e^2 - 227/5*e + 222/5, 3/5*e^3 - 4/5*e^2 - 56/5*e + 146/5, 3*e^3 - e^2 - 31*e + 16, -1/5*e^3 + 8/5*e^2 - 23/5*e - 32/5, 24/5*e^3 - 32/5*e^2 - 263/5*e + 188/5, -3/5*e^3 - 6/5*e^2 + 36/5*e + 14/5, 4/5*e^3 + 8/5*e^2 - 78/5*e - 62/5, 4/5*e^3 + 3/5*e^2 - 33/5*e - 12/5, 2/5*e^3 + 4/5*e^2 - 49/5*e - 66/5, -29/5*e^3 + 32/5*e^2 + 273/5*e - 168/5, -22/5*e^3 + 16/5*e^2 + 259/5*e - 144/5, 17/5*e^3 - 21/5*e^2 - 139/5*e + 114/5, 6*e^3 - 7*e^2 - 65*e + 36, -27/5*e^3 + 41/5*e^2 + 254/5*e - 264/5, -3*e^3 + 4*e^2 + 34*e - 40, -16/5*e^3 + 28/5*e^2 + 157/5*e - 187/5, -6/5*e^3 + 18/5*e^2 + 62/5*e - 192/5, -32/5*e^3 + 26/5*e^2 + 324/5*e - 129/5, 18/5*e^3 - 29/5*e^2 - 141/5*e + 126/5, -3*e^3 + 39*e - 3, 32/5*e^3 - 36/5*e^2 - 364/5*e + 209/5, 26/5*e^3 - 43/5*e^2 - 297/5*e + 202/5, -26/5*e^3 + 33/5*e^2 + 287/5*e - 187/5, -24/5*e^3 + 32/5*e^2 + 228/5*e - 193/5, 2*e^3 - 3*e^2 - 21*e + 40, 28/5*e^3 - 24/5*e^2 - 286/5*e + 101/5, 6/5*e^3 - 13/5*e^2 - 67/5*e + 87/5, -e^3 + 4*e^2 + 17*e - 21, 14/5*e^3 - 7/5*e^2 - 153/5*e + 68/5, 2/5*e^3 + 14/5*e^2 - 39/5*e - 11/5, -14/5*e^3 - 3/5*e^2 + 183/5*e + 7/5, 3*e^3 - 6*e^2 - 36*e + 25, -e^3 + 3*e^2 + 12*e - 14, 14/5*e^3 + 3/5*e^2 - 153/5*e - 37/5, 34/5*e^3 - 57/5*e^2 - 358/5*e + 223/5, -16/5*e^3 + 3/5*e^2 + 202/5*e - 32/5, 31/5*e^3 - 33/5*e^2 - 327/5*e + 167/5, -13/5*e^3 - 11/5*e^2 + 171/5*e + 29/5, -3*e^3 + 7*e^2 + 25*e - 40, -2*e^3 + 8*e^2 + 17*e - 40, -16/5*e^3 + 13/5*e^2 + 202/5*e - 117/5, 18/5*e^3 - 4/5*e^2 - 201/5*e + 51/5, -e^3 + 3*e^2 + 10*e - 1, 8/5*e^3 - 9/5*e^2 - 81/5*e + 166/5, -e^3 - 2*e^2 + 10*e + 34, -18/5*e^3 + 39/5*e^2 + 171/5*e - 241/5, -23/5*e^3 + 34/5*e^2 + 276/5*e - 306/5, -6/5*e^3 - 12/5*e^2 + 87/5*e + 153/5, -2/5*e^3 + 1/5*e^2 + 14/5*e - 9/5, -16/5*e^3 + 33/5*e^2 + 102/5*e - 237/5, 2/5*e^3 - 11/5*e^2 + 11/5*e + 199/5, -16/5*e^3 + 13/5*e^2 + 107/5*e - 7/5, -3*e^3 - e^2 + 38*e - 9, -41/5*e^3 + 53/5*e^2 + 427/5*e - 282/5, -3/5*e^3 + 4/5*e^2 + 31/5*e - 16/5, 8*e^3 - 13*e^2 - 87*e + 65, 23/5*e^3 - 14/5*e^2 - 261/5*e + 86/5, 8/5*e^3 + 6/5*e^2 - 156/5*e - 119/5, 3*e^3 - 5*e^2 - 24*e + 33, -13/5*e^3 + 9/5*e^2 + 176/5*e + 24/5, -19/5*e^3 + 32/5*e^2 + 193/5*e - 3/5, -14/5*e^3 + 22/5*e^2 + 188/5*e - 183/5, -29/5*e^3 + 22/5*e^2 + 343/5*e - 213/5, -7*e^3 + 9*e^2 + 68*e - 57, -17/5*e^3 + 6/5*e^2 + 174/5*e - 79/5, -4/5*e^3 + 7/5*e^2 + 33/5*e - 53/5, -11/5*e^3 + 8/5*e^2 + 187/5*e - 97/5, -17/5*e^3 - 4/5*e^2 + 189/5*e + 1/5, -3*e^3 + 6*e^2 + 39*e - 54, 43/5*e^3 - 69/5*e^2 - 436/5*e + 306/5, -7/5*e^3 - 9/5*e^2 + 44/5*e + 71/5, -3/5*e^3 + 4/5*e^2 + 91/5*e - 11/5, -17/5*e^3 + 31/5*e^2 + 149/5*e - 49/5, 32/5*e^3 - 36/5*e^2 - 279/5*e + 114/5, -12/5*e^3 + 36/5*e^2 + 169/5*e - 249/5, 3*e^3 - 9*e^2 - 33*e + 54, -8/5*e^3 + 24/5*e^2 + 71/5*e - 111/5, -29/5*e^3 + 42/5*e^2 + 283/5*e - 273/5, -24/5*e^3 + 32/5*e^2 + 253/5*e - 228/5, 4/5*e^3 + 3/5*e^2 - 78/5*e - 17/5, -19/5*e^3 - 3/5*e^2 + 183/5*e + 77/5, 27/5*e^3 - 46/5*e^2 - 264/5*e + 204/5, -19/5*e^3 + 22/5*e^2 + 138/5*e - 123/5, 4/5*e^3 + 8/5*e^2 - 108/5*e - 77/5, -11/5*e^3 - 2/5*e^2 + 77/5*e + 93/5, -33/5*e^3 + 64/5*e^2 + 361/5*e - 336/5, 7/5*e^3 + 14/5*e^2 - 54/5*e - 96/5, -6*e^3 + 13*e^2 + 65*e - 75, 1/5*e^3 + 17/5*e^2 - 27/5*e - 43/5, 24/5*e^3 - 22/5*e^2 - 258/5*e + 173/5];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {5,5,1/5*w^3 - 6/5*w^2 - 1/5*w + 4}>] := 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;