/*
  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 - 28*x^2 + 91*x - 69;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [1, -1, e, -20/37*e^3 - 4/37*e^2 + 529/37*e - 708/37, -2/37*e^3 + 7/37*e^2 + 64/37*e - 241/37, -32/37*e^3 + 1/37*e^2 + 876/37*e - 1303/37, 10/37*e^3 + 2/37*e^2 - 283/37*e + 428/37, 22/37*e^3 - 3/37*e^2 - 630/37*e + 875/37, 10/37*e^3 + 2/37*e^2 - 283/37*e + 206/37, 15/37*e^3 + 3/37*e^2 - 443/37*e + 531/37, 33/37*e^3 - 23/37*e^2 - 945/37*e + 1479/37, -13/37*e^3 - 10/37*e^2 + 342/37*e - 364/37, -1/37*e^3 + 22/37*e^2 + 106/37*e - 472/37, 18/37*e^3 + 11/37*e^2 - 502/37*e + 356/37, -22/37*e^3 + 3/37*e^2 + 630/37*e - 1060/37, -13/37*e^3 - 10/37*e^2 + 379/37*e - 364/37, -21/37*e^3 + 18/37*e^2 + 635/37*e - 1180/37, -15/37*e^3 + 34/37*e^2 + 443/37*e - 1197/37, 61/37*e^3 - 10/37*e^2 - 1693/37*e + 2559/37, 25/37*e^3 - 32/37*e^2 - 763/37*e + 1403/37, 7/37*e^3 - 6/37*e^2 - 187/37*e + 11/37, 18/37*e^3 - 26/37*e^2 - 539/37*e + 985/37, -3/37*e^3 + 29/37*e^2 + 96/37*e - 417/37, 5/37*e^3 + 1/37*e^2 - 160/37*e + 399/37, 26/37*e^3 + 20/37*e^2 - 610/37*e + 395/37, 4*e + 3, 8/37*e^3 - 28/37*e^2 - 256/37*e + 927/37, 52/37*e^3 - 34/37*e^2 - 1479/37*e + 2085/37, 22/37*e^3 + 34/37*e^2 - 519/37*e - 87/37, 72/37*e^3 - 30/37*e^2 - 2119/37*e + 3200/37, -12/37*e^3 + 42/37*e^2 + 421/37*e - 928/37, -44/37*e^3 + 6/37*e^2 + 1260/37*e - 2083/37, -36/37*e^3 - 22/37*e^2 + 856/37*e - 823/37, 13/37*e^3 - 27/37*e^2 - 453/37*e + 882/37, -23/37*e^3 - 49/37*e^2 + 477/37*e + 96/37, 40/37*e^3 + 45/37*e^2 - 1058/37*e + 602/37, 104/37*e^3 - 31/37*e^2 - 2958/37*e + 4466/37, -57/37*e^3 - 4/37*e^2 + 1491/37*e - 1855/37, -41/37*e^3 + 51/37*e^2 + 1238/37*e - 2221/37, 69/37*e^3 - 38/37*e^2 - 1949/37*e + 3005/37, 25/37*e^3 + 5/37*e^2 - 800/37*e + 737/37, -31/37*e^3 + 16/37*e^2 + 881/37*e - 1719/37, -61/37*e^3 + 10/37*e^2 + 1693/37*e - 2115/37, -89/37*e^3 + 34/37*e^2 + 2589/37*e - 4009/37, 23/37*e^3 - 62/37*e^2 - 847/37*e + 2087/37, 71/37*e^3 - 8/37*e^2 - 2124/37*e + 2691/37, -1/37*e^3 - 52/37*e^2 - 116/37*e + 675/37, 30/37*e^3 - 31/37*e^2 - 960/37*e + 1839/37, 84/37*e^3 - 35/37*e^2 - 2318/37*e + 3573/37, -42/37*e^3 - 1/37*e^2 + 1233/37*e - 1879/37, -50/37*e^3 + 27/37*e^2 + 1637/37*e - 2251/37, -44/37*e^3 - 31/37*e^2 + 1223/37*e - 936/37, 100/37*e^3 - 17/37*e^2 - 2719/37*e + 4206/37, -16/37*e^3 + 19/37*e^2 + 549/37*e - 522/37, -48/37*e^3 + 57/37*e^2 + 1499/37*e - 3120/37, -51/37*e^3 + 12/37*e^2 + 1336/37*e - 2020/37, -67/37*e^3 - 6/37*e^2 + 1848/37*e - 2320/37, -39/37*e^3 - 30/37*e^2 + 1137/37*e - 870/37, -77/37*e^3 - 8/37*e^2 + 2131/37*e - 3192/37, -47/37*e^3 - 2/37*e^2 + 1245/37*e - 1242/37, 35/37*e^3 + 44/37*e^2 - 861/37*e + 240/37, 17/37*e^3 + 33/37*e^2 - 433/37*e + 254/37, -11/37*e^3 - 17/37*e^2 + 241/37*e - 160/37, -100/37*e^3 - 20/37*e^2 + 2719/37*e - 3540/37, -18/37*e^3 - 11/37*e^2 + 502/37*e - 837/37, 144/37*e^3 - 60/37*e^2 - 4053/37*e + 6030/37, -20/37*e^3 + 33/37*e^2 + 640/37*e - 1263/37, -32/37*e^3 + 38/37*e^2 + 839/37*e - 1747/37, -98/37*e^3 + 10/37*e^2 + 2655/37*e - 4039/37, 64/37*e^3 - 39/37*e^2 - 2085/37*e + 2976/37, -24/37*e^3 - 27/37*e^2 + 509/37*e - 894/37, -96/37*e^3 + 77/37*e^2 + 2739/37*e - 4575/37, -20/37*e^3 + 33/37*e^2 + 455/37*e - 1263/37, -6/37*e^3 - 16/37*e^2 + 229/37*e + 350/37, 82/37*e^3 - 28/37*e^2 - 2587/37*e + 3332/37, -71/37*e^3 + 8/37*e^2 + 2050/37*e - 3468/37, 61/37*e^3 - 10/37*e^2 - 1582/37*e + 2892/37, -42/37*e^3 + 36/37*e^2 + 1159/37*e - 2212/37, 54/37*e^3 - 4/37*e^2 - 1469/37*e + 2252/37, 29/37*e^3 + 28/37*e^2 - 706/37*e + 257/37, 23/37*e^3 - 62/37*e^2 - 810/37*e + 1643/37, 12/37*e^3 + 106/37*e^2 - 162/37*e - 1477/37, 2/37*e^3 + 30/37*e^2 + 158/37*e - 499/37, -92/37*e^3 + 63/37*e^2 + 2759/37*e - 4056/37, 16/37*e^3 + 55/37*e^2 - 327/37*e - 810/37, -9/37*e^3 - 24/37*e^2 + 325/37*e - 474/37, -82/37*e^3 - 46/37*e^2 + 2180/37*e - 2814/37, 23/37*e^3 + 12/37*e^2 - 699/37*e + 570/37, 156/37*e^3 - 28/37*e^2 - 4326/37*e + 6588/37, 20/37*e^3 + 4/37*e^2 - 418/37*e + 264/37, 8/37*e^3 + 9/37*e^2 + 3/37*e + 113/37, -54/37*e^3 + 41/37*e^2 + 1617/37*e - 2659/37, -54/37*e^3 + 4/37*e^2 + 1469/37*e - 2400/37, -130/37*e^3 + 48/37*e^2 + 3605/37*e - 5490/37, -46/37*e^3 + 87/37*e^2 + 1546/37*e - 3471/37, 90/37*e^3 - 93/37*e^2 - 2806/37*e + 5073/37, 147/37*e^3 - 52/37*e^2 - 4038/37*e + 6410/37, -29/37*e^3 + 46/37*e^2 + 1002/37*e - 2218/37, -94/37*e^3 - 4/37*e^2 + 2490/37*e - 3298/37, -1/37*e^3 - 15/37*e^2 - 79/37*e + 860/37, -6/37*e^3 - 16/37*e^2 + 266/37*e - 94/37, -109/37*e^3 - 7/37*e^2 + 3081/37*e - 4828/37, -132/37*e^3 - 19/37*e^2 + 3669/37*e - 5583/37, 30/37*e^3 - 31/37*e^2 - 997/37*e + 1173/37, -77/37*e^3 - 8/37*e^2 + 1983/37*e - 2785/37, -70/37*e^3 - 51/37*e^2 + 1796/37*e - 1960/37, 6/37*e^3 + 53/37*e^2 - 192/37*e - 868/37, -65/37*e^3 + 24/37*e^2 + 1895/37*e - 3337/37, 44/37*e^3 - 43/37*e^2 - 1297/37*e + 2786/37, e^2 - e - 22, 24/37*e^3 - 10/37*e^2 - 583/37*e + 1634/37, -40/37*e^3 + 66/37*e^2 + 1021/37*e - 3118/37, 25/37*e^3 - 69/37*e^2 - 726/37*e + 1625/37, -103/37*e^3 + 83/37*e^2 + 3222/37*e - 5215/37, -13/37*e^3 - 10/37*e^2 + 157/37*e + 80/37, 171/37*e^3 - 62/37*e^2 - 4843/37*e + 7304/37, -77/37*e^3 + 66/37*e^2 + 2501/37*e - 4080/37, 75/37*e^3 - 22/37*e^2 - 2215/37*e + 2988/37, -32/37*e^3 + 1/37*e^2 + 950/37*e - 1303/37, 23/37*e^3 - 62/37*e^2 - 884/37*e + 1865/37, -50/37*e^3 + 101/37*e^2 + 1600/37*e - 3805/37, -11/37*e^3 + 20/37*e^2 + 500/37*e - 49/37, 71/37*e^3 - 119/37*e^2 - 2050/37*e + 4245/37, 137/37*e^3 - 17/37*e^2 - 3792/37*e + 5427/37, 83/37*e^3 + 24/37*e^2 - 2471/37*e + 2694/37, -177/37*e^3 + 46/37*e^2 + 4813/37*e - 7176/37, 81/37*e^3 - 43/37*e^2 - 2333/37*e + 2675/37, 47/37*e^3 + 39/37*e^2 - 1171/37*e + 539/37, 56/37*e^3 + 63/37*e^2 - 1274/37*e + 384/37, 44/37*e^3 + 31/37*e^2 - 1038/37*e + 1380/37, -50/37*e^3 + 101/37*e^2 + 1563/37*e - 3583/37, -72/37*e^3 - 7/37*e^2 + 2193/37*e - 1831/37, 180/37*e^3 - 38/37*e^2 - 5057/37*e + 7334/37, -60/37*e^3 + 62/37*e^2 + 2031/37*e - 3604/37, -e^3 - 3*e^2 + 19*e + 18, 27/37*e^3 + 35/37*e^2 - 679/37*e + 312/37, -77/37*e^3 + 103/37*e^2 + 2316/37*e - 4413/37, -59/37*e^3 + 77/37*e^2 + 1814/37*e - 3687/37, -52/37*e^3 - 3/37*e^2 + 1220/37*e - 2640/37, -12/37*e^3 + 5/37*e^2 + 384/37*e + 108/37, -149/37*e^3 + 22/37*e^2 + 4287/37*e - 6244/37, 147/37*e^3 - 126/37*e^2 - 4297/37*e + 7298/37, 19/37*e^3 - 48/37*e^2 - 645/37*e + 680/37, -97/37*e^3 - 12/37*e^2 + 2475/37*e - 3382/37, -132/37*e^3 + 18/37*e^2 + 3632/37*e - 5028/37, -92/37*e^3 + 26/37*e^2 + 2500/37*e - 4944/37, -46/37*e^3 + 13/37*e^2 + 1065/37*e - 1843/37, -154/37*e^3 + 21/37*e^2 + 4299/37*e - 6532/37, 26/37*e^3 + 57/37*e^2 - 499/37*e - 1270/37, -46/37*e^3 - 24/37*e^2 + 1509/37*e - 1473/37, -90/37*e^3 - 92/37*e^2 + 2177/37*e - 1965/37, -24/37*e^3 + 10/37*e^2 + 657/37*e - 1227/37, 24/37*e^3 - 10/37*e^2 - 731/37*e + 561/37, 41/37*e^3 - 14/37*e^2 - 1275/37*e + 963/37, -67/37*e^3 - 6/37*e^2 + 1737/37*e - 3393/37, -41/37*e^3 - 23/37*e^2 + 979/37*e - 1444/37, -97/37*e^3 - 49/37*e^2 + 2697/37*e - 3382/37, 13/37*e^3 - 64/37*e^2 - 527/37*e + 2103/37, -29/37*e^3 - 102/37*e^2 + 521/37*e + 927/37, 81/37*e^3 + 68/37*e^2 - 2259/37*e + 1898/37, -80/37*e^3 + 58/37*e^2 + 2449/37*e - 4497/37, -104/37*e^3 - 43/37*e^2 + 2514/37*e - 3282/37, 32/37*e^3 - 38/37*e^2 - 1283/37*e + 2043/37, -24/37*e^3 - 27/37*e^2 + 398/37*e - 450/37, 43/37*e^3 + 16/37*e^2 - 1265/37*e + 2129/37, 131/37*e^3 + 4/37*e^2 - 3637/37*e + 4445/37];
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;