/*
  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, 2/5*w^3 - 2/5*w^2 - 12/5*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 - 32*x^2 - 41*x + 63;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [-1, 1, e, 20/293*e^3 + 84/293*e^2 - 631/293*e - 1212/293, -18/293*e^3 - 17/293*e^2 + 480/293*e - 257/293, -2/293*e^3 - 67/293*e^2 - 142/293*e + 883/293, -10/293*e^3 - 42/293*e^2 + 169/293*e - 566/293, -8/293*e^3 + 25/293*e^2 + 604/293*e - 863/293, -10/293*e^3 - 42/293*e^2 + 169/293*e + 1192/293, 57/293*e^3 + 5/293*e^2 - 1813/293*e - 993/293, -25/293*e^3 - 105/293*e^2 + 569/293*e + 1515/293, -29/293*e^3 + 54/293*e^2 + 578/293*e - 1700/293, 23/293*e^3 + 38/293*e^2 - 418/293*e - 632/293, 2/293*e^3 + 67/293*e^2 + 142/293*e - 2348/293, -38/293*e^3 - 101/293*e^2 + 818/293*e + 76/293, -29/293*e^3 + 54/293*e^2 + 871/293*e - 1700/293, 43/293*e^3 + 122/293*e^2 - 1049/293*e - 1844/293, 29/293*e^3 - 54/293*e^2 - 1457/293*e + 1407/293, -95/293*e^3 - 106/293*e^2 + 2631/293*e + 483/293, 23/293*e^3 + 38/293*e^2 - 711/293*e - 3269/293, 65/293*e^3 - 20/293*e^2 - 1831/293*e + 163/293, 62/293*e^3 + 26/293*e^2 - 1751/293*e - 1003/293, -15/293*e^3 - 63/293*e^2 + 400/293*e + 2667/293, -87/293*e^3 - 131/293*e^2 + 2320/293*e + 2811/293, -6/293*e^3 + 92/293*e^2 - 426/293*e - 3797/293, 72/293*e^3 + 68/293*e^2 - 1920/293*e + 735/293, 80/293*e^3 + 336/293*e^2 - 2524/293*e - 3969/293, -62/293*e^3 - 26/293*e^2 + 1165/293*e - 3099/293, 88/293*e^3 + 18/293*e^2 - 2835/293*e - 3399/293, -88/293*e^3 - 18/293*e^2 + 2249/293*e + 1348/293, 28/293*e^3 - 234/293*e^2 - 1235/293*e + 3460/293, -4/293*e^3 - 134/293*e^2 + 888/293*e + 3817/293, 4/293*e^3 + 134/293*e^2 + 284/293*e - 4403/293, 73/293*e^3 - 45/293*e^2 - 1263/293*e + 2784/293, 37/293*e^3 - 79/293*e^2 - 889/293*e + 1098/293, 56/293*e^3 - 175/293*e^2 - 2470/293*e + 2818/293, -120/293*e^3 - 211/293*e^2 + 2614/293*e + 826/293, 101/293*e^3 + 14/293*e^2 - 3377/293*e - 1667/293, -75/293*e^3 - 315/293*e^2 + 2000/293*e + 3373/293, -33/293*e^3 - 80/293*e^2 + 1759/293*e + 3289/293, -109/293*e^3 + 11/293*e^2 + 3102/293*e + 511/293, -29/293*e^3 + 54/293*e^2 + 1457/293*e + 2109/293, -39/293*e^3 + 12/293*e^2 + 1333/293*e - 2559/293, 87/293*e^3 - 162/293*e^2 - 2027/293*e + 4807/293, -41/293*e^3 + 238/293*e^2 + 1777/293*e - 4313/293, 91/293*e^3 - 28/293*e^2 - 2036/293*e + 111/293, -61/293*e^3 - 432/293*e^2 + 1236/293*e + 5103/293, 116/293*e^3 + 77/293*e^2 - 4070/293*e - 2283/293, 30/293*e^3 - 167/293*e^2 - 800/293*e + 4335/293, 50/293*e^3 + 503/293*e^2 - 1431/293*e - 6839/293, 42/293*e^3 + 235/293*e^2 - 827/293*e - 5651/293, -92/293*e^3 + 141/293*e^2 + 2551/293*e - 5676/293, -4/293*e^3 + 159/293*e^2 + 9/293*e + 594/293, 80/293*e^3 + 43/293*e^2 - 3403/293*e + 426/293, 104/293*e^3 + 261/293*e^2 - 2285/293*e - 1380/293, 17/293*e^3 + 130/293*e^2 + 328/293*e - 620/293, -79/293*e^3 - 156/293*e^2 + 2888/293*e + 3088/293, -23/293*e^3 - 38/293*e^2 + 1297/293*e + 4734/293, -85/293*e^3 - 64/293*e^2 + 1583/293*e - 2760/293, 129/293*e^3 + 366/293*e^2 - 3147/293*e - 3774/293, 27/293*e^3 + 172/293*e^2 + 159/293*e - 5328/293, 21/293*e^3 - 29/293*e^2 - 267/293*e + 544/293, -87/293*e^3 - 131/293*e^2 + 2027/293*e + 2518/293, 156/293*e^3 - 48/293*e^2 - 5625/293*e - 2070/293, -60/293*e^3 + 41/293*e^2 + 1600/293*e - 759/293, 40/293*e^3 + 168/293*e^2 + 203/293*e - 2424/293, 38/293*e^3 + 101/293*e^2 - 818/293*e - 3885/293, -62/293*e^3 - 26/293*e^2 + 2923/293*e - 755/293, -148/293*e^3 - 270/293*e^2 + 4435/293*e + 4105/293, 4/293*e^3 - 159/293*e^2 + 577/293*e - 594/293, -44/293*e^3 - 595/293*e^2 + 1271/293*e + 8292/293, -160/293*e^3 - 379/293*e^2 + 4755/293*e + 5301/293, -204/293*e^3 - 95/293*e^2 + 6319/293*e + 3045/293, -82/293*e^3 - 696/293*e^2 + 2089/293*e + 8368/293, 86/293*e^3 + 244/293*e^2 - 2391/293*e - 758/293, 89/293*e^3 + 198/293*e^2 - 3350/293*e + 408/293, 21/293*e^3 + 264/293*e^2 + 26/293*e - 8832/293, 26/293*e^3 - 8/293*e^2 - 1377/293*e + 1120/293, -118/293*e^3 - 144/293*e^2 + 3635/293*e + 1408/293, 107/293*e^3 - 78/293*e^2 - 2658/293*e + 79/293, -39/293*e^3 + 12/293*e^2 + 454/293*e - 3731/293, 18/293*e^3 + 310/293*e^2 - 1066/293*e - 5603/293, -252/293*e^3 - 238/293*e^2 + 6134/293*e + 211/293, -116/293*e^3 - 77/293*e^2 + 2605/293*e - 3870/293, -88/293*e^3 + 275/293*e^2 + 2835/293*e - 996/293, -73/293*e^3 - 248/293*e^2 + 1849/293*e + 2490/293, 102/293*e^3 + 194/293*e^2 - 1548/293*e - 5478/293, 119/293*e^3 + 324/293*e^2 - 3271/293*e - 8442/293, 60/293*e^3 + 252/293*e^2 - 2186/293*e - 7152/293, 84/293*e^3 - 116/293*e^2 - 3998/293*e + 3348/293, -66/293*e^3 + 133/293*e^2 + 2053/293*e - 1919/293, 92/293*e^3 + 445/293*e^2 - 3137/293*e - 6923/293, -150/293*e^3 - 44/293*e^2 + 5465/293*e + 2058/293, -26/293*e^3 + 8/293*e^2 + 1377/293*e - 2292/293, 130/293*e^3 - 333/293*e^2 - 3662/293*e + 7065/293, -134/293*e^3 + 199/293*e^2 + 3378/293*e - 6471/293, -41/293*e^3 + 238/293*e^2 + 898/293*e - 6950/293, 183/293*e^3 + 124/293*e^2 - 6638/293*e - 1538/293, 106/293*e^3 + 328/293*e^2 - 3022/293*e - 5486/293, 59/293*e^3 + 365/293*e^2 - 499/293*e - 11252/293, -46/293*e^3 - 76/293*e^2 + 2594/293*e + 3022/293, -9/293*e^3 - 155/293*e^2 + 533/293*e + 7636/293, 10/293*e^3 - 251/293*e^2 - 169/293*e + 273/293, 72/293*e^3 + 361/293*e^2 - 455/293*e - 11571/293, -25/293*e^3 + 188/293*e^2 + 1155/293*e - 6689/293, -186/293*e^3 - 371/293*e^2 + 4960/293*e + 2716/293, 90/293*e^3 + 85/293*e^2 - 1228/293*e - 1352/293, -53/293*e^3 - 164/293*e^2 + 2683/293*e + 2743/293, -140/293*e^3 - 295/293*e^2 + 3831/293*e + 2038/293, 96/293*e^3 - 7/293*e^2 - 2853/293*e + 3910/293, -260/293*e^3 - 506/293*e^2 + 7617/293*e + 2278/293, 76/293*e^3 + 202/293*e^2 - 2515/293*e + 1606/293, -67/293*e^3 + 539/293*e^2 + 1982/293*e - 8363/293, 205/293*e^3 + 275/293*e^2 - 5662/293*e - 6563/293, 139/293*e^3 - 178/293*e^2 - 5367/293*e + 1480/293, -77/293*e^3 - 382/293*e^2 + 2737/293*e + 8944/293, -5/293*e^3 - 314/293*e^2 - 355/293*e + 2940/293, -13/293*e^3 + 590/293*e^2 + 249/293*e - 9936/293, -210/293*e^3 - 3/293*e^2 + 5600/293*e - 3389/293, 41/293*e^3 + 348/293*e^2 - 1484/293*e + 211/293, 32/293*e^3 + 193/293*e^2 - 658/293*e - 3287/293, 67/293*e^3 - 246/293*e^2 - 1396/293*e + 4261/293, 53/293*e^3 - 129/293*e^2 - 2976/293*e + 1359/293, 339/293*e^3 + 369/293*e^2 - 9626/293*e - 6831/293, -193/293*e^3 - 166/293*e^2 + 7393/293*e + 4488/293, -173/293*e^3 - 668/293*e^2 + 4125/293*e + 8550/293, -57/293*e^3 - 5/293*e^2 + 641/293*e - 5453/293, 89/293*e^3 - 95/293*e^2 - 3057/293*e - 6917/293, -4/293*e^3 + 159/293*e^2 - 870/293*e - 1164/293, -56/293*e^3 + 175/293*e^2 + 126/293*e - 9264/293, 132/293*e^3 + 613/293*e^2 - 3227/293*e - 2315/293, -230/293*e^3 - 87/293*e^2 + 6231/293*e - 419/293, 2*e^2 - e - 40, 80/293*e^3 - 250/293*e^2 - 3989/293*e + 2770/293, -77/293*e^3 - 89/293*e^2 + 1565/293*e - 432/293, 187/293*e^3 - 35/293*e^2 - 3717/293*e + 4314/293, -151/293*e^3 + 69/293*e^2 + 4222/293*e - 3507/293, -233/293*e^3 - 41/293*e^2 + 6604/293*e + 759/293, 12/293*e^3 + 109/293*e^2 - 320/293*e + 3492/293, -108/293*e^3 - 395/293*e^2 + 4052/293*e + 216/293, -73/293*e^3 - 248/293*e^2 + 3607/293*e + 4834/293, 91/293*e^3 - 28/293*e^2 - 2329/293*e - 5456/293, 283/293*e^3 - 42/293*e^2 - 8621/293*e - 566/293, 19/293*e^3 + 490/293*e^2 + 763/293*e - 8828/293, -108/293*e^3 - 102/293*e^2 + 4052/293*e - 6816/293, -68/293*e^3 + 66/293*e^2 + 3376/293*e + 1308/293, -154/293*e^3 - 471/293*e^2 + 5181/293*e + 7633/293, -66/293*e^3 + 133/293*e^2 + 881/293*e - 13346/293, -46/293*e^3 + 217/293*e^2 + 2887/293*e - 5768/293, 130/293*e^3 - 40/293*e^2 - 1611/293*e + 1791/293, 206/293*e^3 + 748/293*e^2 - 5591/293*e - 11253/293, -4/293*e^3 - 134/293*e^2 + 9/293*e - 2043/293, -36/293*e^3 - 34/293*e^2 + 1253/293*e - 807/293, -43/293*e^3 - 122/293*e^2 + 2221/293*e - 5481/293, -31/293*e^3 - 306/293*e^2 + 729/293*e - 1989/293, 167/293*e^3 + 467/293*e^2 - 3379/293*e - 9710/293, -9/293*e^3 - 155/293*e^2 + 1119/293*e + 604/293, 179/293*e^3 - 10/293*e^2 - 3699/293*e + 4623/293, 117/293*e^3 - 36/293*e^2 - 2827/293*e + 5919/293, -231/293*e^3 - 560/293*e^2 + 5281/293*e + 6322/293, -92/293*e^3 - 738/293*e^2 + 3137/293*e + 12783/293, -56/293*e^3 - 411/293*e^2 + 2470/293*e + 6558/293, -60/293*e^3 + 334/293*e^2 + 1893/293*e - 9549/293, -56/293*e^3 - 411/293*e^2 + 3642/293*e + 6558/293, -21/293*e^3 - 264/293*e^2 - 905/293*e - 251/293, -93/293*e^3 - 332/293*e^2 + 2187/293*e + 10441/293];
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 + 4/5*w^2 - 11/5*w - 7}>] := 1;
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;