/*
  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![29, 8, -12, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {16, 2, 2}>;

primesArray := [
[5, 5, -w + 2],
[5, 5, -3/5*w^3 - w^2 + 26/5*w + 42/5],
[9, 3, 2/5*w^3 - 19/5*w - 3/5],
[9, 3, -1/5*w^3 + 2/5*w + 4/5],
[11, 11, -2/5*w^3 - w^2 + 14/5*w + 28/5],
[11, 11, 1/5*w^3 - w^2 - 7/5*w + 36/5],
[16, 2, 2],
[29, 29, -w],
[29, 29, 1/5*w^3 - 12/5*w + 1/5],
[41, 41, -2/5*w^3 - w^2 + 14/5*w + 38/5],
[41, 41, -w^2 + 10],
[41, 41, 11/5*w^3 + 3*w^2 - 102/5*w - 149/5],
[41, 41, -1/5*w^3 + w^2 + 7/5*w - 26/5],
[49, 7, 1/5*w^3 + w^2 - 12/5*w - 19/5],
[49, 7, 1/5*w^3 + w^2 - 12/5*w - 44/5],
[59, 59, -4/5*w^3 - w^2 + 33/5*w + 36/5],
[59, 59, -2*w^3 - 3*w^2 + 17*w + 26],
[61, 61, -1/5*w^3 + w^2 + 12/5*w - 51/5],
[61, 61, 4/5*w^3 + w^2 - 28/5*w - 41/5],
[71, 71, 1/5*w^3 + w^2 - 7/5*w - 49/5],
[71, 71, -2*w^3 - 3*w^2 + 18*w + 28],
[89, 89, -3/5*w^3 + 16/5*w + 7/5],
[89, 89, 3/5*w^3 + w^2 - 26/5*w - 57/5],
[121, 11, -3/5*w^3 + 21/5*w + 2/5],
[131, 131, w^2 - w - 8],
[131, 131, 2/5*w^3 + w^2 - 19/5*w - 23/5],
[149, 149, 2/5*w^3 - w^2 - 14/5*w + 22/5],
[149, 149, 3/5*w^3 + w^2 - 21/5*w - 42/5],
[179, 179, -6/5*w^3 - w^2 + 57/5*w + 69/5],
[179, 179, -2/5*w^3 - 2*w^2 + 14/5*w + 73/5],
[179, 179, 2*w^2 - 11],
[179, 179, -3/5*w^3 + 2*w^2 + 16/5*w - 48/5],
[181, 181, -7/5*w^3 - 2*w^2 + 59/5*w + 78/5],
[181, 181, -2/5*w^3 + 24/5*w - 27/5],
[191, 191, -3/5*w^3 + 2*w^2 + 21/5*w - 78/5],
[191, 191, 2/5*w^3 - 2*w^2 - 9/5*w + 57/5],
[199, 199, -1/5*w^3 + 17/5*w - 16/5],
[199, 199, 1/5*w^3 - 17/5*w - 14/5],
[211, 211, 2/5*w^3 - w^2 + 1/5*w + 7/5],
[211, 211, -1/5*w^3 + w^2 - 3/5*w - 26/5],
[211, 211, 4/5*w^3 + w^2 - 38/5*w - 36/5],
[211, 211, -1/5*w^3 - 2*w^2 + 7/5*w + 44/5],
[239, 239, 3/5*w^3 + w^2 - 16/5*w - 37/5],
[239, 239, 2/5*w^3 + w^2 - 14/5*w - 53/5],
[251, 251, 2/5*w^3 - 24/5*w - 13/5],
[251, 251, 2*w - 3],
[251, 251, w^2 + w - 10],
[251, 251, -4/5*w^3 - w^2 + 28/5*w + 46/5],
[269, 269, 1/5*w^3 - 12/5*w - 24/5],
[269, 269, w - 5],
[271, 271, -3/5*w^3 + 31/5*w - 33/5],
[271, 271, -8/5*w^3 - 2*w^2 + 66/5*w + 77/5],
[281, 281, 4/5*w^3 - w^2 - 38/5*w + 39/5],
[281, 281, -14/5*w^3 - 3*w^2 + 128/5*w + 161/5],
[311, 311, 1/5*w^3 - 2*w^2 - 2/5*w + 51/5],
[311, 311, 1/5*w^3 - w^2 - 17/5*w + 21/5],
[331, 331, 1/5*w^3 + 3/5*w - 4/5],
[331, 331, 3/5*w^3 - 31/5*w - 2/5],
[359, 359, -w^3 + 6*w - 1],
[359, 359, -6/5*w^3 + 47/5*w - 6/5],
[359, 359, 3/5*w^3 + w^2 - 31/5*w - 27/5],
[359, 359, 3/5*w^3 + w^2 - 26/5*w - 22/5],
[361, 19, 4/5*w^3 - 28/5*w - 11/5],
[361, 19, 1/5*w^3 - 7/5*w - 24/5],
[379, 379, 9/5*w^3 + 3*w^2 - 78/5*w - 121/5],
[379, 379, -4/5*w^3 + 3*w^2 + 28/5*w - 99/5],
[389, 389, -3/5*w^3 - 2*w^2 + 26/5*w + 42/5],
[389, 389, -2*w^2 + w + 17],
[401, 401, 4/5*w^3 - w^2 - 23/5*w + 24/5],
[401, 401, 3/5*w^3 + w^2 - 26/5*w - 7/5],
[401, 401, -4/5*w^3 - w^2 + 38/5*w + 66/5],
[401, 401, 6/5*w^3 + w^2 - 52/5*w - 39/5],
[409, 409, -6/5*w^3 + 52/5*w + 19/5],
[409, 409, 1/5*w^3 + 2*w^2 - 7/5*w - 39/5],
[421, 421, -1/5*w^3 - w^2 + 17/5*w + 44/5],
[421, 421, 2/5*w^3 + w^2 - 24/5*w - 18/5],
[431, 431, -3/5*w^3 - w^2 + 36/5*w + 47/5],
[431, 431, -13/5*w^3 - 4*w^2 + 116/5*w + 182/5],
[431, 431, -2/5*w^3 + w^2 + 24/5*w - 47/5],
[431, 431, 2/5*w^3 - w^2 + 6/5*w - 8/5],
[439, 439, 6/5*w^3 + w^2 - 47/5*w - 49/5],
[439, 439, -2/5*w^3 + 2*w^2 + 9/5*w - 42/5],
[439, 439, 4/5*w^3 - w^2 - 23/5*w + 14/5],
[439, 439, -w^3 - 2*w^2 + 8*w + 17],
[461, 461, w^3 + w^2 - 10*w - 9],
[461, 461, -2*w^3 - 3*w^2 + 17*w + 27],
[461, 461, -1/5*w^3 + w^2 - 8/5*w - 16/5],
[461, 461, -3*w^3 - 4*w^2 + 26*w + 35],
[479, 479, 2*w - 1],
[479, 479, 2/5*w^3 - 24/5*w - 3/5],
[491, 491, w^3 - 2*w^2 - 6*w + 11],
[491, 491, w^3 + w^2 - 10*w - 15],
[499, 499, -1/5*w^3 + 2*w^2 + 2/5*w - 36/5],
[499, 499, 4/5*w^3 + 2*w^2 - 33/5*w - 91/5],
[509, 509, 8/5*w^3 + w^2 - 66/5*w - 42/5],
[509, 509, 1/5*w^3 - 2*w^2 + 8/5*w + 16/5],
[521, 521, w^3 - 9*w - 4],
[521, 521, 3/5*w^3 + w^2 - 16/5*w - 42/5],
[521, 521, 3/5*w^3 - w^2 - 26/5*w + 23/5],
[521, 521, -3/5*w^3 + 11/5*w + 22/5],
[529, 23, 3/5*w^3 + w^2 - 36/5*w - 62/5],
[529, 23, -4/5*w^3 + 2*w^2 + 23/5*w - 64/5],
[541, 541, -w^3 - w^2 + 9*w + 7],
[541, 541, 1/5*w^3 + 2*w^2 - 7/5*w - 64/5],
[541, 541, 3/5*w^3 + w^2 - 26/5*w - 12/5],
[571, 571, -w^3 - w^2 + 6*w + 9],
[571, 571, -w^3 + w^2 + 8*w - 4],
[619, 619, -w^3 - 2*w^2 + 8*w + 11],
[619, 619, -2/5*w^3 + 2*w^2 + 9/5*w - 72/5],
[619, 619, 2/5*w^3 - 2*w^2 - 4/5*w + 57/5],
[619, 619, -6/5*w^3 - 2*w^2 + 52/5*w + 69/5],
[631, 631, 1/5*w^3 + w^2 + 3/5*w - 24/5],
[631, 631, 2/5*w^3 - w^2 - 24/5*w + 42/5],
[659, 659, -13/5*w^3 - 4*w^2 + 111/5*w + 177/5],
[659, 659, -11/5*w^3 - 3*w^2 + 92/5*w + 119/5],
[659, 659, -8/5*w^3 - w^2 + 61/5*w + 27/5],
[659, 659, -8/5*w^3 + 71/5*w + 2/5],
[661, 661, -w^3 - w^2 + 7*w + 11],
[661, 661, 4/5*w^3 - w^2 - 28/5*w + 9/5],
[691, 691, 1/5*w^3 - w^2 - 17/5*w + 41/5],
[691, 691, 3/5*w^3 + 2*w^2 - 26/5*w - 67/5],
[701, 701, 12/5*w^3 + 4*w^2 - 99/5*w - 158/5],
[701, 701, -17/5*w^3 - 4*w^2 + 149/5*w + 193/5],
[709, 709, -1/5*w^3 + w^2 + 12/5*w - 6/5],
[709, 709, 1/5*w^3 + w^2 - 2/5*w - 59/5],
[719, 719, 2/5*w^3 + w^2 - 29/5*w - 8/5],
[719, 719, -w^3 + w^2 + 7*w - 4],
[719, 719, 6/5*w^3 + w^2 - 42/5*w - 44/5],
[719, 719, -2/5*w^3 - w^2 + 29/5*w + 53/5],
[739, 739, 14/5*w^3 + 4*w^2 - 123/5*w - 171/5],
[739, 739, -1/5*w^3 + 2*w^2 + 12/5*w - 56/5],
[739, 739, -6/5*w^3 + 3*w^2 + 37/5*w - 96/5],
[739, 739, 2/5*w^3 + 2*w^2 - 9/5*w - 73/5],
[751, 751, -9/5*w^3 - 2*w^2 + 73/5*w + 86/5],
[751, 751, -w^3 + 2*w^2 + 5*w - 8],
[761, 761, 1/5*w^3 + w^2 - 22/5*w + 16/5],
[761, 761, -16/5*w^3 - 4*w^2 + 147/5*w + 214/5],
[769, 769, 13/5*w^3 + 3*w^2 - 121/5*w - 172/5],
[769, 769, 3/5*w^3 - w^2 - 41/5*w + 78/5],
[769, 769, -12/5*w^3 - 4*w^2 + 99/5*w + 153/5],
[769, 769, -6/5*w^3 + 3*w^2 + 27/5*w - 61/5],
[809, 809, 1/5*w^3 + 3*w^2 - 22/5*w - 104/5],
[809, 809, w^3 + 3*w^2 - 10*w - 17],
[811, 811, -5*w + 11],
[811, 811, -16/5*w^3 - 5*w^2 + 137/5*w + 214/5],
[821, 821, -1/5*w^3 + w^2 + 17/5*w - 36/5],
[821, 821, w^2 + 2*w - 6],
[829, 829, 3*w^2 + w - 17],
[829, 829, 2/5*w^3 + 3*w^2 - 9/5*w - 108/5],
[841, 29, w^3 - 7*w - 3],
[859, 859, 6/5*w^3 + w^2 - 37/5*w - 34/5],
[859, 859, 2/5*w^3 + w^2 - 14/5*w - 68/5],
[881, 881, 1/5*w^3 + 2*w^2 - 17/5*w - 79/5],
[881, 881, 3/5*w^3 + 2*w^2 - 31/5*w - 47/5],
[911, 911, 4/5*w^3 - 3*w^2 - 28/5*w + 89/5],
[911, 911, 1/5*w^3 - w^2 - 7/5*w + 61/5],
[911, 911, w^3 - 6*w - 3],
[911, 911, -11/5*w^3 - 3*w^2 + 92/5*w + 139/5],
[919, 919, 7/5*w^3 + 2*w^2 - 54/5*w - 88/5],
[919, 919, 4/5*w^3 - 2*w^2 - 23/5*w + 39/5],
[929, 929, -7/5*w^3 + w^2 + 54/5*w - 17/5],
[929, 929, -1/5*w^3 + w^2 + 22/5*w - 71/5],
[941, 941, 9/5*w^3 + w^2 - 73/5*w - 31/5],
[941, 941, 1/5*w^3 - w^2 - 2/5*w + 66/5],
[941, 941, 2/5*w^3 + 2*w^2 - 19/5*w - 103/5],
[941, 941, -w^3 + w^2 + 5*w - 3],
[961, 31, w^3 - 7*w - 2],
[961, 31, 2/5*w^3 - 14/5*w - 33/5],
[971, 971, -2/5*w^3 - 2*w^2 + 24/5*w + 93/5],
[971, 971, 2/5*w^3 + 2*w^2 - 24/5*w - 33/5],
[991, 991, -w^3 + 2*w^2 + 7*w - 10],
[991, 991, -7/5*w^3 - 2*w^2 + 49/5*w + 78/5]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^6 + x^5 - 15*x^4 + 61*x^2 - 57*x + 13;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, -1/2*e^5 - e^4 + 13/2*e^3 + 11/2*e^2 - 26*e + 21/2, -1/2*e^4 - 3/2*e^3 + 4*e^2 + 17/2*e - 17/2, 1/2*e^4 + 3/2*e^3 - 4*e^2 - 17/2*e + 17/2, -1/4*e^5 + 17/4*e^3 - 3/4*e^2 - 35/2*e + 29/4, 3/4*e^5 + e^4 - 43/4*e^3 - 19/4*e^2 + 85/2*e - 79/4, 1, -3/4*e^5 - 3/2*e^4 + 37/4*e^3 + 31/4*e^2 - 35*e + 65/4, -1/4*e^5 - 1/2*e^4 + 11/4*e^3 + 9/4*e^2 - 8*e + 23/4, -1/4*e^5 + 1/2*e^4 + 27/4*e^3 - 11/4*e^2 - 30*e + 43/4, 7/4*e^5 + 3*e^4 - 91/4*e^3 - 67/4*e^2 + 165/2*e - 131/4, 1/4*e^5 + e^4 - 13/4*e^3 - 21/4*e^2 + 35/2*e - 69/4, 1/4*e^5 - 1/2*e^4 - 15/4*e^3 + 23/4*e^2 + 9*e - 47/4, 1/2*e^5 + e^4 - 17/2*e^3 - 13/2*e^2 + 41*e - 39/2, 1/2*e^5 + e^4 - 15/2*e^3 - 15/2*e^2 + 30*e - 17/2, -1/4*e^5 + 17/4*e^3 - 7/4*e^2 - 35/2*e + 65/4, -1/4*e^5 - e^4 + 5/4*e^3 + 25/4*e^2 - 1/2*e - 3/4, 5/4*e^5 + 2*e^4 - 65/4*e^3 - 37/4*e^2 + 119/2*e - 133/4, 3/4*e^5 + 2*e^4 - 31/4*e^3 - 43/4*e^2 + 53/2*e - 67/4, 3/4*e^5 + e^4 - 47/4*e^3 - 15/4*e^2 + 105/2*e - 135/4, 1/4*e^5 + e^4 - 5/4*e^3 - 29/4*e^2 - 5/2*e + 19/4, -5/4*e^5 - 5/2*e^4 + 63/4*e^3 + 57/4*e^2 - 57*e + 67/4, -5/4*e^5 - 5/2*e^4 + 67/4*e^3 + 53/4*e^2 - 68*e + 111/4, 1/2*e^5 + e^4 - 9/2*e^3 - 7/2*e^2 + 11*e - 35/2, -5/4*e^5 - 2*e^4 + 77/4*e^3 + 41/4*e^2 - 173/2*e + 161/4, 1/4*e^5 - 17/4*e^3 + 11/4*e^2 + 45/2*e - 77/4, 5/4*e^5 + 2*e^4 - 69/4*e^3 - 45/4*e^2 + 131/2*e - 77/4, 1/4*e^5 + e^4 - 13/4*e^3 - 25/4*e^2 + 33/2*e - 13/4, e^5 - 1/2*e^4 - 33/2*e^3 + 8*e^2 + 123/2*e - 67/2, 9/4*e^5 + 4*e^4 - 121/4*e^3 - 93/4*e^2 + 225/2*e - 181/4, 7/4*e^5 + 4*e^4 - 95/4*e^3 - 91/4*e^2 + 203/2*e - 203/4, -5/2*e^5 - 5/2*e^4 + 39*e^3 + 23/2*e^2 - 315/2*e + 70, 7/4*e^5 + 7/2*e^4 - 73/4*e^3 - 79/4*e^2 + 51*e - 25/4, -3/4*e^5 - 3/2*e^4 + 37/4*e^3 + 51/4*e^2 - 29*e - 35/4, -3*e^3 - 3*e^2 + 17*e - 5, 2*e^5 + 4*e^4 - 29*e^3 - 25*e^2 + 125*e - 47, 3/4*e^5 + 2*e^4 - 31/4*e^3 - 51/4*e^2 + 53/2*e + 33/4, -3/4*e^5 - 2*e^4 + 31/4*e^3 + 51/4*e^2 - 53/2*e + 47/4, -9/4*e^5 - 4*e^4 + 117/4*e^3 + 89/4*e^2 - 213/2*e + 177/4, 5/2*e^5 + 11/2*e^4 - 29*e^3 - 61/2*e^2 + 207/2*e - 34, e^5 + 3/2*e^4 - 27/2*e^3 - 5*e^2 + 101/2*e - 61/2, -5/4*e^5 - 3*e^4 + 65/4*e^3 + 65/4*e^2 - 137/2*e + 157/4, 1/2*e^5 + e^4 - 7/2*e^3 - 13/2*e^2 + 2*e + 31/2, -5/2*e^5 - 5*e^4 + 63/2*e^3 + 61/2*e^2 - 116*e + 69/2, -5/2*e^5 - 6*e^4 + 49/2*e^3 + 67/2*e^2 - 71*e + 31/2, -1/2*e^5 + 15/2*e^3 - 15/2*e^2 - 30*e + 81/2, -1/2*e^5 + e^4 + 25/2*e^3 - 13/2*e^2 - 52*e + 45/2, -2*e^4 - 2*e^3 + 16*e^2 - e - 12, -e^5 - 2*e^4 + 13*e^3 + 11*e^2 - 47*e + 9, -5/2*e^5 - 5*e^4 + 65/2*e^3 + 55/2*e^2 - 128*e + 81/2, -3/4*e^5 - 7/2*e^4 + 29/4*e^3 + 87/4*e^2 - 34*e + 73/4, -15/4*e^5 - 11/2*e^4 + 209/4*e^3 + 115/4*e^2 - 198*e + 333/4, 3/4*e^5 - e^4 - 83/4*e^3 + 29/4*e^2 + 207/2*e - 235/4, 1/4*e^5 + 3*e^4 - 1/4*e^3 - 105/4*e^2 + 5/2*e + 103/4, 17/4*e^5 + 19/2*e^4 - 207/4*e^3 - 221/4*e^2 + 197*e - 259/4, 3/4*e^5 + 1/2*e^4 - 53/4*e^3 + 1/4*e^2 + 53*e - 145/4, 3*e^5 + 5*e^4 - 36*e^3 - 26*e^2 + 116*e - 45, 1/2*e^5 + 2*e^4 - 9/2*e^3 - 15/2*e^2 + 24*e - 71/2, 1/2*e^5 + e^4 - 21/2*e^3 - 23/2*e^2 + 53*e - 23/2, -3/2*e^5 - 3*e^4 + 27/2*e^3 + 25/2*e^2 - 33*e + 17/2, -7/4*e^5 - 5/2*e^4 + 93/4*e^3 + 51/4*e^2 - 78*e + 141/4, -9/4*e^5 - 11/2*e^4 + 111/4*e^3 + 121/4*e^2 - 115*e + 223/4, 3/4*e^5 + 3/2*e^4 - 33/4*e^3 - 27/4*e^2 + 27*e - 101/4, -7/2*e^5 - 7*e^4 + 87/2*e^3 + 73/2*e^2 - 161*e + 133/2, 1/4*e^5 + 3/2*e^4 - 19/4*e^3 - 49/4*e^2 + 30*e - 83/4, 7/4*e^5 + 5/2*e^4 - 109/4*e^3 - 63/4*e^2 + 112*e - 213/4, 2*e^5 + 5*e^4 - 18*e^3 - 24*e^2 + 49*e - 25, 5/2*e^5 + 4*e^4 - 59/2*e^3 - 29/2*e^2 + 99*e - 117/2, -5/2*e^5 - 4*e^4 + 61/2*e^3 + 35/2*e^2 - 101*e + 85/2, 3/2*e^5 + 2*e^4 - 43/2*e^3 - 23/2*e^2 + 79*e - 43/2, -5/2*e^5 - 6*e^4 + 55/2*e^3 + 61/2*e^2 - 100*e + 83/2, 1/2*e^5 + 2*e^4 - 13/2*e^3 - 25/2*e^2 + 35*e - 21/2, 3/2*e^4 + 19/2*e^3 - 5*e^2 - 117/2*e + 45/2, e^5 + 1/2*e^4 - 21/2*e^3 + 6*e^2 + 49/2*e - 55/2, -1/2*e^5 - 2*e^4 + 9/2*e^3 + 25/2*e^2 - 19*e - 15/2, -1/2*e^5 + 17/2*e^3 - 3/2*e^2 - 31*e + 9/2, -11/2*e^5 - 8*e^4 + 155/2*e^3 + 79/2*e^2 - 303*e + 269/2, 5/2*e^5 + 13/2*e^4 - 27*e^3 - 77/2*e^2 + 173/2*e - 24, -1/2*e^5 - 4*e^4 + 1/2*e^3 + 53/2*e^2 + 3*e - 13/2, 11/2*e^5 + 19/2*e^4 - 75*e^3 - 95/2*e^2 + 599/2*e - 138, -15/4*e^5 - 13/2*e^4 + 193/4*e^3 + 123/4*e^2 - 186*e + 309/4, 11/2*e^5 + 19/2*e^4 - 72*e^3 - 97/2*e^2 + 545/2*e - 121, 1/4*e^5 - 1/2*e^4 - 35/4*e^3 + 7/4*e^2 + 53*e - 119/4, 3/2*e^5 + 9/2*e^4 - 15*e^3 - 49/2*e^2 + 99/2*e - 30, -2*e^5 - 5*e^4 + 19*e^3 + 33*e^2 - 49*e - 18, -5/4*e^5 + 101/4*e^3 - 15/4*e^2 - 251/2*e + 285/4, 2*e^5 + 5*e^4 - 20*e^3 - 34*e^2 + 56*e + 9, 19/4*e^5 + 7*e^4 - 267/4*e^3 - 123/4*e^2 + 545/2*e - 515/4, -13/4*e^5 - 15/2*e^4 + 155/4*e^3 + 169/4*e^2 - 142*e + 207/4, -17/4*e^5 - 15/2*e^4 + 227/4*e^3 + 153/4*e^2 - 219*e + 383/4, -1/2*e^5 - e^4 - 1/2*e^3 - 1/2*e^2 + 26*e - 11/2, -1/2*e^5 - e^4 + 1/2*e^3 - 3/2*e^2 + 15*e + 11/2, e^5 + 3*e^4 - 11*e^3 - 16*e^2 + 51*e - 39, -1/2*e^5 - 2*e^4 + 13/2*e^3 + 25/2*e^2 - 40*e + 5/2, -7/2*e^5 - 13/2*e^4 + 52*e^3 + 69/2*e^2 - 467/2*e + 115, -e^5 - 5/2*e^4 + 23/2*e^3 + 20*e^2 - 53/2*e - 19/2, -5/2*e^5 - 6*e^4 + 63/2*e^3 + 59/2*e^2 - 134*e + 153/2, -1/2*e^5 - e^4 + 9/2*e^3 + 15/2*e^2 + 3*e - 17/2, -7/2*e^5 - 7*e^4 + 95/2*e^3 + 73/2*e^2 - 203*e + 197/2, -9/2*e^5 - 8*e^4 + 111/2*e^3 + 87/2*e^2 - 188*e + 129/2, -1/4*e^5 - 5/2*e^4 - 9/4*e^3 + 69/4*e^2 + 11*e - 121/4, 5/4*e^5 + 9/2*e^4 - 47/4*e^3 - 117/4*e^2 + 46*e - 63/4, -15/4*e^5 - 7*e^4 + 175/4*e^3 + 155/4*e^2 - 277/2*e + 163/4, -3/2*e^5 - 3*e^4 + 43/2*e^3 + 37/2*e^2 - 89*e + 55/2, -7/4*e^5 - 4*e^4 + 91/4*e^3 + 67/4*e^2 - 203/2*e + 279/4, -9/2*e^5 - 8*e^4 + 121/2*e^3 + 93/2*e^2 - 221*e + 153/2, -11/2*e^5 - 12*e^4 + 147/2*e^3 + 135/2*e^2 - 307*e + 259/2, -9*e^5 - 20*e^4 + 114*e^3 + 114*e^2 - 450*e + 188, -13/2*e^5 - 11*e^4 + 179/2*e^3 + 117/2*e^2 - 339*e + 319/2, -17/4*e^5 - 11/2*e^4 + 231/4*e^3 + 89/4*e^2 - 217*e + 443/4, 5/4*e^5 - 1/2*e^4 - 103/4*e^3 + 15/4*e^2 + 116*e - 163/4, 3/2*e^5 + 5/2*e^4 - 10*e^3 - 15/2*e^2 - 11/2*e - 9, 3/2*e^5 + 7/2*e^4 - 13*e^3 - 19/2*e^2 + 87/2*e - 58, 21/4*e^5 + 11*e^4 - 261/4*e^3 - 233/4*e^2 + 511/2*e - 481/4, 5/4*e^5 + 2*e^4 - 57/4*e^3 - 33/4*e^2 + 69/2*e - 125/4, -3*e^5 - 4*e^4 + 53*e^3 + 23*e^2 - 246*e + 119, -2*e^5 - 6*e^4 + 26*e^3 + 46*e^2 - 102*e + 8, 5/4*e^5 + 4*e^4 - 45/4*e^3 - 101/4*e^2 + 63/2*e - 61/4, 9/4*e^5 + 3*e^4 - 133/4*e^3 - 49/4*e^2 + 273/2*e - 349/4, 3/2*e^5 + 2*e^4 - 45/2*e^3 - 25/2*e^2 + 88*e - 69/2, -1/2*e^5 + 11/2*e^3 - 5/2*e^2 - 10*e - 5/2, 7/2*e^5 + 6*e^4 - 93/2*e^3 - 69/2*e^2 + 161*e - 105/2, 6*e^5 + 13*e^4 - 79*e^3 - 72*e^2 + 328*e - 137, 7/2*e^5 + 9/2*e^4 - 45*e^3 - 31/2*e^2 + 321/2*e - 86, -e^5 + 1/2*e^4 + 47/2*e^3 - e^2 - 225/2*e + 77/2, -1/2*e^5 - 3*e^4 + 5/2*e^3 + 51/2*e^2 + e - 75/2, -7/4*e^5 - 3/2*e^4 + 85/4*e^3 + 27/4*e^2 - 59*e + 13/4, 17/4*e^5 + 13/2*e^4 - 231/4*e^3 - 153/4*e^2 + 212*e - 279/4, -3/2*e^5 - e^4 + 59/2*e^3 + 5/2*e^2 - 143*e + 147/2, -e^4 + 8*e^2 - 3*e + 5, -11/4*e^5 - 7/2*e^4 + 157/4*e^3 + 51/4*e^2 - 151*e + 285/4, -7/2*e^5 - 6*e^4 + 97/2*e^3 + 67/2*e^2 - 193*e + 159/2, -13/4*e^5 - 17/2*e^4 + 139/4*e^3 + 197/4*e^2 - 121*e + 99/4, -2*e^5 - 3/2*e^4 + 63/2*e^3 + 4*e^2 - 241/2*e + 181/2, -2*e^5 - 13/2*e^4 + 41/2*e^3 + 40*e^2 - 159/2*e + 99/2, -17/4*e^5 - 19/2*e^4 + 239/4*e^3 + 249/4*e^2 - 252*e + 347/4, -9/4*e^5 - 7/2*e^4 + 159/4*e^3 + 97/4*e^2 - 178*e + 315/4, 4*e^5 + 7*e^4 - 59*e^3 - 38*e^2 + 257*e - 108, e^5 + 7*e^4 - 7*e^3 - 55*e^2 + 32*e + 21, 4*e^5 + 3*e^4 - 71*e^3 - 13*e^2 + 309*e - 157, e^4 + e^3 - 12*e^2 - 15*e + 32, -3/2*e^5 - 2*e^4 + 53/2*e^3 + 19/2*e^2 - 124*e + 149/2, -5/2*e^5 - 6*e^4 + 61/2*e^3 + 79/2*e^2 - 111*e + 57/2, -1/4*e^5 - e^4 - 19/4*e^3 - 3/4*e^2 + 89/2*e + 17/4, -11/4*e^5 - 5*e^4 + 119/4*e^3 + 79/4*e^2 - 193/2*e + 251/4, 9/2*e^5 + 10*e^4 - 119/2*e^3 - 123/2*e^2 + 240*e - 179/2, 5/2*e^5 + 4*e^4 - 79/2*e^3 - 47/2*e^2 + 166*e - 163/2, 27/4*e^5 + 9*e^4 - 403/4*e^3 - 167/4*e^2 + 829/2*e - 771/4, 3/4*e^5 + 6*e^4 + 1/4*e^3 - 175/4*e^2 - 37/2*e + 169/4, 5/2*e^5 + 5*e^4 - 85/2*e^3 - 75/2*e^2 + 195*e - 135/2, 19/4*e^5 + 10*e^4 - 219/4*e^3 - 219/4*e^2 + 377/2*e - 303/4, 7/4*e^5 + 3*e^4 - 91/4*e^3 - 39/4*e^2 + 175/2*e - 295/4, -7/2*e^5 - 11/2*e^4 + 45*e^3 + 55/2*e^2 - 311/2*e + 54, -5/2*e^5 - 13/2*e^4 + 29*e^3 + 69/2*e^2 - 233/2*e + 48, 5*e^5 + 27/2*e^4 - 105/2*e^3 - 80*e^2 + 361/2*e - 101/2, 13/4*e^5 + 5/2*e^4 - 227/4*e^3 - 37/4*e^2 + 256*e - 555/4, -15/4*e^5 - 7/2*e^4 + 221/4*e^3 + 27/4*e^2 - 225*e + 445/4, 4*e^5 + 9/2*e^4 - 119/2*e^3 - 14*e^2 + 469/2*e - 275/2, 15/4*e^5 + 9/2*e^4 - 237/4*e^3 - 79/4*e^2 + 252*e - 481/4, 5/4*e^5 + 11/2*e^4 - 39/4*e^3 - 157/4*e^2 + 26*e + 93/4, 3/4*e^5 + 3/2*e^4 - 33/4*e^3 - 47/4*e^2 + 28*e + 115/4, -19/4*e^5 - 19/2*e^4 + 233/4*e^3 + 215/4*e^2 - 214*e + 357/4, 7/2*e^5 + 7/2*e^4 - 58*e^3 - 37/2*e^2 + 491/2*e - 116, -1/4*e^5 - 3/2*e^4 + 15/4*e^3 + 25/4*e^2 - 29*e + 123/4, -e^5 + 3/2*e^4 + 31/2*e^3 - 19*e^2 - 101/2*e + 63/2, -15/4*e^5 - 13/2*e^4 + 189/4*e^3 + 147/4*e^2 - 164*e + 201/4, 7*e^5 + 14*e^4 - 83*e^3 - 69*e^2 + 294*e - 136, 7/2*e^5 + 7*e^4 - 113/2*e^3 - 99/2*e^2 + 252*e - 195/2, 11/4*e^5 + 6*e^4 - 139/4*e^3 - 171/4*e^2 + 249/2*e - 87/4, -5/4*e^5 - 3*e^4 + 25/4*e^3 + 69/4*e^2 + 27/2*e - 171/4, -13/4*e^5 - 13/2*e^4 + 195/4*e^3 + 169/4*e^2 - 203*e + 387/4, -23/4*e^5 - 23/2*e^4 + 325/4*e^3 + 279/4*e^2 - 338*e + 597/4];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {16, 2, 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;