/*
  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![9, 3, -7, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {41, 41, -1/3*w^3 + 4/3*w^2 + 7/3*w - 6}>;

primesArray := [
[5, 5, -1/3*w^3 - 2/3*w^2 + 7/3*w + 4],
[5, 5, 1/3*w^3 - 4/3*w^2 - 1/3*w + 3],
[9, 3, -w],
[9, 3, 1/3*w^3 - 1/3*w^2 - 7/3*w + 1],
[16, 2, 2],
[19, 19, 2/3*w^3 - 2/3*w^2 - 11/3*w],
[19, 19, -1/3*w^3 + 1/3*w^2 + 1/3*w + 1],
[31, 31, 2/3*w^3 + 1/3*w^2 - 11/3*w - 2],
[31, 31, 2/3*w^3 - 5/3*w^2 - 5/3*w + 5],
[41, 41, -1/3*w^3 + 4/3*w^2 + 7/3*w - 6],
[41, 41, -1/3*w^3 + 4/3*w^2 + 7/3*w - 3],
[61, 61, 1/3*w^3 - 4/3*w^2 + 2/3*w + 4],
[61, 61, 2/3*w^3 + 1/3*w^2 - 14/3*w - 2],
[71, 71, -1/3*w^3 + 4/3*w^2 + 1/3*w - 7],
[71, 71, 1/3*w^3 + 2/3*w^2 - 7/3*w],
[89, 89, -1/3*w^3 + 1/3*w^2 + 10/3*w - 3],
[89, 89, 1/3*w^3 - 1/3*w^2 - 10/3*w - 1],
[101, 101, 1/3*w^3 - 1/3*w^2 - 7/3*w - 3],
[101, 101, -2/3*w^3 + 5/3*w^2 + 11/3*w - 4],
[101, 101, 2/3*w^3 - 5/3*w^2 - 8/3*w + 3],
[101, 101, w - 4],
[109, 109, 2/3*w^3 + 1/3*w^2 - 8/3*w - 4],
[109, 109, -w^3 + 2*w^2 + 4*w - 4],
[121, 11, w^3 - w^2 - 4*w + 1],
[121, 11, -w^3 + w^2 + 4*w - 2],
[131, 131, 4/3*w^3 - 1/3*w^2 - 22/3*w - 1],
[131, 131, -w^3 + 7*w + 2],
[139, 139, 2/3*w^3 - 2/3*w^2 - 14/3*w + 1],
[139, 139, 2*w - 1],
[139, 139, 1/3*w^3 + 5/3*w^2 - 7/3*w - 10],
[139, 139, -2/3*w^3 + 8/3*w^2 + 5/3*w - 5],
[149, 149, w^3 + w^2 - 7*w - 11],
[149, 149, -w^3 - w^2 + 5*w + 4],
[151, 151, -2/3*w^3 - 4/3*w^2 + 14/3*w + 5],
[151, 151, 2/3*w^3 - 8/3*w^2 - 2/3*w + 9],
[169, 13, -2/3*w^3 - 1/3*w^2 + 11/3*w],
[169, 13, -2/3*w^3 + 5/3*w^2 + 5/3*w - 7],
[179, 179, 4/3*w^3 - 7/3*w^2 - 13/3*w + 6],
[179, 179, -4/3*w^3 + 1/3*w^2 + 19/3*w + 1],
[181, 181, 5/3*w^3 - 8/3*w^2 - 20/3*w + 5],
[181, 181, -1/3*w^3 + 1/3*w^2 + 10/3*w - 1],
[181, 181, -4/3*w^3 + 1/3*w^2 + 16/3*w + 3],
[191, 191, w^2 - 3*w - 1],
[191, 191, -1/3*w^3 + 4/3*w^2 + 10/3*w - 3],
[199, 199, -w^3 + 3*w^2 + w - 7],
[199, 199, -4/3*w^3 + 1/3*w^2 + 19/3*w - 1],
[211, 211, -1/3*w^3 - 5/3*w^2 + 4/3*w + 8],
[211, 211, w^3 - 3*w^2 - 4*w + 8],
[229, 229, -4/3*w^3 + 4/3*w^2 + 13/3*w - 4],
[229, 229, -5/3*w^3 + 5/3*w^2 + 23/3*w - 5],
[229, 229, 1/3*w^3 - 7/3*w^2 + 2/3*w + 10],
[229, 229, -2/3*w^3 + 2/3*w^2 + 2/3*w - 3],
[239, 239, 4/3*w^3 - 1/3*w^2 - 19/3*w - 2],
[239, 239, -2/3*w^3 + 2/3*w^2 + 11/3*w - 6],
[251, 251, 2*w^2 - w - 8],
[251, 251, -1/3*w^3 - 2/3*w^2 + 7/3*w - 2],
[251, 251, 1/3*w^3 - 4/3*w^2 - 1/3*w + 9],
[251, 251, -1/3*w^3 + 7/3*w^2 + 1/3*w - 7],
[269, 269, -5/3*w^3 + 8/3*w^2 + 17/3*w - 8],
[269, 269, -5/3*w^3 + 11/3*w^2 + 17/3*w - 12],
[271, 271, 1/3*w^3 + 2/3*w^2 - 13/3*w - 3],
[271, 271, 1/3*w^3 + 2/3*w^2 - 13/3*w - 2],
[311, 311, -1/3*w^3 + 7/3*w^2 - 5/3*w - 8],
[311, 311, -w^3 + 5*w - 1],
[331, 331, -2/3*w^3 - 1/3*w^2 + 8/3*w + 6],
[331, 331, -w^3 + 2*w^2 + 4*w - 2],
[349, 349, -w^3 + 2*w^2 + w + 1],
[349, 349, 1/3*w^3 - 1/3*w^2 - 7/3*w - 4],
[349, 349, -4/3*w^3 + 13/3*w^2 + 10/3*w - 11],
[349, 349, w - 5],
[361, 19, -4/3*w^3 + 4/3*w^2 + 16/3*w - 3],
[379, 379, 7/3*w^3 - 1/3*w^2 - 43/3*w - 9],
[379, 379, 2*w^3 - 3*w^2 - 9*w + 8],
[401, 401, 1/3*w^3 + 2/3*w^2 - 7/3*w - 8],
[401, 401, -1/3*w^3 + 10/3*w^2 - 2/3*w - 15],
[401, 401, -4/3*w^3 + 4/3*w^2 + 13/3*w - 3],
[401, 401, -1/3*w^3 + 4/3*w^2 - 5/3*w + 3],
[409, 409, -5/3*w^3 + 8/3*w^2 + 20/3*w - 7],
[409, 409, 4/3*w^3 - 1/3*w^2 - 16/3*w - 1],
[419, 419, 4/3*w^3 - 16/3*w^2 - 1/3*w + 10],
[419, 419, 5/3*w^3 + 7/3*w^2 - 35/3*w - 17],
[421, 421, -2*w^3 + 4*w^2 + 7*w - 10],
[421, 421, -5/3*w^3 + 8/3*w^2 + 29/3*w - 7],
[421, 421, 1/3*w^3 + 2/3*w^2 + 5/3*w - 4],
[421, 421, -5/3*w^3 - 1/3*w^2 + 23/3*w + 5],
[431, 431, -2/3*w^3 + 5/3*w^2 + 11/3*w - 1],
[431, 431, -w^2 - w + 8],
[439, 439, -2/3*w^3 + 11/3*w^2 - 1/3*w - 8],
[439, 439, -2/3*w^3 + 11/3*w^2 + 8/3*w - 15],
[479, 479, w^3 + 2*w^2 - 6*w - 10],
[479, 479, -5/3*w^3 + 5/3*w^2 + 14/3*w],
[479, 479, -5/3*w^3 + 5/3*w^2 + 23/3*w - 3],
[479, 479, -4/3*w^3 + 4/3*w^2 + 13/3*w - 2],
[521, 521, -w^3 + 2*w^2 + 5*w + 1],
[521, 521, 1/3*w^3 + 2/3*w^2 - 1/3*w - 10],
[541, 541, 1/3*w^3 + 5/3*w^2 + 2/3*w - 7],
[541, 541, 5/3*w^3 - 5/3*w^2 - 26/3*w],
[569, 569, -w - 5],
[569, 569, 1/3*w^3 - 1/3*w^2 - 7/3*w + 6],
[571, 571, w^3 - 4*w - 8],
[571, 571, -1/3*w^3 + 7/3*w^2 + 7/3*w - 4],
[599, 599, 5/3*w^3 - 5/3*w^2 - 14/3*w + 4],
[599, 599, -4/3*w^3 + 13/3*w^2 + 13/3*w - 15],
[599, 599, 2*w^3 - 3*w^2 - 8*w + 5],
[599, 599, 7/3*w^3 - 7/3*w^2 - 34/3*w + 6],
[601, 601, -4/3*w^3 + 13/3*w^2 + 13/3*w - 13],
[601, 601, -w^3 + 8*w + 5],
[619, 619, -5/3*w^3 + 11/3*w^2 + 17/3*w - 13],
[619, 619, 4/3*w^3 + 2/3*w^2 - 19/3*w - 2],
[631, 631, w^3 - w^2 - 7*w + 1],
[631, 631, 3*w - 2],
[641, 641, -2/3*w^3 + 5/3*w^2 + 2/3*w - 7],
[641, 641, -1/3*w^3 + 7/3*w^2 - 5/3*w - 9],
[661, 661, -5/3*w^3 + 11/3*w^2 + 20/3*w - 9],
[661, 661, w^3 + w^2 - 4*w - 7],
[691, 691, -5/3*w^3 + 8/3*w^2 + 17/3*w - 6],
[691, 691, 2*w^2 - 3*w - 10],
[691, 691, 5/3*w^3 - 2/3*w^2 - 23/3*w - 1],
[691, 691, 1/3*w^3 + 5/3*w^2 - 13/3*w - 3],
[701, 701, 2/3*w^3 + 4/3*w^2 - 20/3*w - 11],
[701, 701, -2/3*w^3 - 1/3*w^2 + 17/3*w + 9],
[709, 709, -2/3*w^3 - 7/3*w^2 + 17/3*w + 11],
[709, 709, -1/3*w^3 + 10/3*w^2 - 2/3*w - 8],
[709, 709, -2/3*w^3 + 11/3*w^2 - 1/3*w - 10],
[709, 709, 3*w^2 - 2*w - 14],
[719, 719, 7/3*w^3 + 8/3*w^2 - 46/3*w - 21],
[719, 719, -2*w^3 + 7*w^2 + 2*w - 13],
[719, 719, 2/3*w^3 + 7/3*w^2 - 14/3*w - 9],
[719, 719, -w^3 + 4*w^2 + 2*w - 13],
[761, 761, 1/3*w^3 + 5/3*w^2 - 1/3*w - 8],
[761, 761, 5/3*w^3 - 2/3*w^2 - 14/3*w + 1],
[761, 761, -8/3*w^3 + 11/3*w^2 + 38/3*w - 11],
[761, 761, -4/3*w^3 + 10/3*w^2 + 19/3*w - 9],
[809, 809, -2/3*w^3 + 11/3*w^2 + 5/3*w - 15],
[809, 809, 3*w^2 - w - 8],
[811, 811, 2/3*w^3 + 7/3*w^2 - 14/3*w - 11],
[811, 811, 1/3*w^3 + 8/3*w^2 - 13/3*w - 10],
[811, 811, 1/3*w^3 - 10/3*w^2 + 5/3*w + 11],
[811, 811, -w^3 + 4*w^2 + 2*w - 11],
[821, 821, 2*w^3 - w^2 - 12*w - 5],
[821, 821, 1/3*w^3 + 5/3*w^2 - 7/3*w - 14],
[829, 829, -2/3*w^3 + 11/3*w^2 + 5/3*w - 10],
[829, 829, 3*w^2 - w - 13],
[839, 839, 4/3*w^3 - 1/3*w^2 - 25/3*w + 1],
[839, 839, -2/3*w^3 + 5/3*w^2 - 1/3*w - 6],
[841, 29, 5/3*w^3 - 5/3*w^2 - 20/3*w + 1],
[841, 29, -5/3*w^3 + 5/3*w^2 + 20/3*w - 4],
[859, 859, -1/3*w^3 - 8/3*w^2 + 13/3*w + 11],
[859, 859, -1/3*w^3 + 10/3*w^2 - 5/3*w - 10],
[859, 859, 2/3*w^3 - 8/3*w^2 + 1/3*w + 9],
[859, 859, -2/3*w^3 + 5/3*w^2 + 2/3*w - 8],
[881, 881, -4/3*w^3 + 16/3*w^2 + 4/3*w - 11],
[881, 881, 1/3*w^3 - 13/3*w^2 - 4/3*w + 18],
[911, 911, -1/3*w^3 + 10/3*w^2 + 1/3*w - 14],
[911, 911, -1/3*w^3 + 10/3*w^2 + 1/3*w - 9],
[929, 929, 4/3*w^3 - 7/3*w^2 - 22/3*w + 4],
[929, 929, 1/3*w^3 + 2/3*w^2 + 2/3*w - 6],
[961, 31, 5/3*w^3 - 5/3*w^2 - 20/3*w + 2],
[971, 971, -2/3*w^3 + 11/3*w^2 + 2/3*w - 13],
[971, 971, 1/3*w^3 + 8/3*w^2 - 10/3*w - 9],
[991, 991, -4/3*w^3 + 10/3*w^2 + 16/3*w - 17],
[991, 991, -2*w^3 + 4*w^2 + 11*w - 11]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^7 - 4*x^6 - 17*x^5 + 80*x^4 + 29*x^3 - 386*x^2 + 396*x - 72;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, 1/6*e^6 - 5/12*e^5 - 43/12*e^4 + 97/12*e^3 + 113/6*e^2 - 115/3*e + 6, 1/6*e^6 - 5/12*e^5 - 10/3*e^4 + 47/6*e^3 + 193/12*e^2 - 215/6*e + 6, -1/3*e^6 + 5/6*e^5 + 20/3*e^4 - 47/3*e^3 - 187/6*e^2 + 215/3*e - 18, -1/12*e^6 + 1/3*e^5 + 23/12*e^4 - 20/3*e^3 - 125/12*e^2 + 95/3*e - 8, -1/12*e^6 + 1/3*e^5 + 17/12*e^4 - 17/3*e^3 - 65/12*e^2 + 133/6*e - 4, 1/4*e^6 - 1/2*e^5 - 5*e^4 + 39/4*e^3 + 23*e^2 - 46*e + 14, -1/6*e^6 + 5/12*e^5 + 10/3*e^4 - 25/3*e^3 - 187/12*e^2 + 118/3*e - 7, -1/4*e^4 + 1/4*e^3 + 15/4*e^2 - 3/2*e - 6, -1, -1/3*e^6 + 7/12*e^5 + 43/6*e^4 - 35/3*e^3 - 455/12*e^2 + 167/3*e + 1, -1/2*e^6 + e^5 + 41/4*e^4 - 83/4*e^3 - 195/4*e^2 + 215/2*e - 30, -1/4*e^5 + 1/4*e^4 + 11/4*e^3 - 3/2*e^2 - 6, 1/2*e^6 - 3/2*e^5 - 43/4*e^4 + 113/4*e^3 + 223/4*e^2 - 251/2*e + 24, -5/12*e^6 + 11/12*e^5 + 53/6*e^4 - 211/12*e^3 - 553/12*e^2 + 244/3*e - 3, 5/12*e^6 - 11/12*e^5 - 25/3*e^4 + 217/12*e^3 + 463/12*e^2 - 262/3*e + 23, -5/6*e^6 + 11/6*e^5 + 103/6*e^4 - 107/3*e^3 - 254/3*e^2 + 503/3*e - 24, -7/12*e^6 + 4/3*e^5 + 38/3*e^4 - 317/12*e^3 - 397/6*e^2 + 389/3*e - 24, 1/12*e^6 - 1/12*e^5 - 5/3*e^4 + 35/12*e^3 + 83/12*e^2 - 133/6*e + 12, -1/4*e^5 - 1/2*e^4 + 9/2*e^3 + 19/4*e^2 - 33/2*e + 12, -1/4*e^6 + 3/4*e^5 + 11/2*e^4 - 63/4*e^3 - 113/4*e^2 + 79*e - 21, 1/4*e^6 - 1/4*e^5 - 5*e^4 + 23/4*e^3 + 95/4*e^2 - 65/2*e - 6, 1/2*e^6 - 5/4*e^5 - 39/4*e^4 + 93/4*e^3 + 87/2*e^2 - 104*e + 34, -1/12*e^6 + 1/12*e^5 + 23/12*e^4 - 19/6*e^3 - 26/3*e^2 + 77/3*e - 16, 1/4*e^6 - 1/2*e^5 - 19/4*e^4 + 23/2*e^3 + 77/4*e^2 - 133/2*e + 22, -7/12*e^6 + 11/6*e^5 + 73/6*e^4 - 419/12*e^3 - 355/6*e^2 + 470/3*e - 48, -1/2*e^6 + e^5 + 41/4*e^4 - 77/4*e^3 - 209/4*e^2 + 91*e - 3, 5/12*e^6 - 7/6*e^5 - 53/6*e^4 + 301/12*e^3 + 245/6*e^2 - 403/3*e + 52, 1/3*e^6 - 7/12*e^5 - 23/3*e^4 + 41/3*e^3 + 503/12*e^2 - 469/6*e + 10, 3/4*e^6 - 2*e^5 - 15*e^4 + 153/4*e^3 + 139/2*e^2 - 179*e + 54, 1/2*e^6 - 5/4*e^5 - 21/2*e^4 + 51/2*e^3 + 205/4*e^2 - 126*e + 39, 11/12*e^6 - 23/12*e^5 - 55/3*e^4 + 439/12*e^3 + 1027/12*e^2 - 517/3*e + 41, 1/12*e^6 - 1/12*e^5 - 23/12*e^4 + 13/6*e^3 + 32/3*e^2 - 38/3*e + 4, -3/4*e^6 + 7/4*e^5 + 16*e^4 - 135/4*e^3 - 331/4*e^2 + 158*e - 15, 1/4*e^6 - e^5 - 19/4*e^4 + 39/2*e^3 + 71/4*e^2 - 181/2*e + 44, -13/12*e^6 + 7/3*e^5 + 269/12*e^4 - 140/3*e^3 - 1301/12*e^2 + 1369/6*e - 52, 5/4*e^6 - 11/4*e^5 - 105/4*e^4 + 56*e^3 + 128*e^2 - 282*e + 78, -3/4*e^6 + 2*e^5 + 65/4*e^4 - 79/2*e^3 - 337/4*e^2 + 379/2*e - 42, -7/6*e^6 + 29/12*e^5 + 295/12*e^4 - 589/12*e^3 - 749/6*e^2 + 742/3*e - 40, 7/12*e^6 - 11/6*e^5 - 137/12*e^4 + 104/3*e^3 + 617/12*e^2 - 479/3*e + 49, 1/12*e^6 - 1/12*e^5 - 17/12*e^4 + 5/3*e^3 + 37/6*e^2 - 32/3*e - 4, -e^6 + 2*e^5 + 41/2*e^4 - 79/2*e^3 - 199/2*e^2 + 190*e - 30, -1/6*e^6 + 2/3*e^5 + 13/3*e^4 - 71/6*e^3 - 88/3*e^2 + 133/3*e + 10, -1/12*e^6 + 1/3*e^5 + 13/6*e^4 - 101/12*e^3 - 35/3*e^2 + 155/3*e - 22, -5/12*e^6 + 11/12*e^5 + 97/12*e^4 - 58/3*e^3 - 103/3*e^2 + 307/3*e - 40, 7/6*e^6 - 35/12*e^5 - 73/3*e^4 + 347/6*e^3 + 1435/12*e^2 - 1697/6*e + 74, e^6 - 11/4*e^5 - 85/4*e^4 + 217/4*e^3 + 213/2*e^2 - 259*e + 60, 1/4*e^6 - 3/4*e^5 - 21/4*e^4 + 14*e^3 + 25*e^2 - 66*e + 36, -1/12*e^6 + 7/12*e^5 + 29/12*e^4 - 41/3*e^3 - 41/3*e^2 + 230/3*e - 34, -5/4*e^6 + 5/2*e^5 + 105/4*e^4 - 52*e^3 - 523/4*e^2 + 537/2*e - 56, -5/12*e^6 + 17/12*e^5 + 97/12*e^4 - 79/3*e^3 - 203/6*e^2 + 349/3*e - 52, -1/2*e^5 + 1/2*e^4 + 15/2*e^3 - 8*e^2 - 20*e + 32, 2/3*e^6 - 17/12*e^5 - 83/6*e^4 + 167/6*e^3 + 847/12*e^2 - 803/6*e + 18, 1/2*e^6 - 7/4*e^5 - 43/4*e^4 + 133/4*e^3 + 54*e^2 - 148*e + 36, 3/4*e^6 - 2*e^5 - 29/2*e^4 + 143/4*e^3 + 66*e^2 - 151*e + 30, -5/12*e^6 + 2/3*e^5 + 53/6*e^4 - 181/12*e^3 - 127/3*e^2 + 259/3*e - 40, -7/12*e^6 + 13/12*e^5 + 143/12*e^4 - 62/3*e^3 - 176/3*e^2 + 275/3*e - 6, -7/12*e^6 + 4/3*e^5 + 73/6*e^4 - 275/12*e^3 - 191/3*e^2 + 281/3*e + 4, 5/12*e^6 - 5/12*e^5 - 25/3*e^4 + 115/12*e^3 + 475/12*e^2 - 347/6*e - 4, -1/2*e^5 - 1/4*e^4 + 45/4*e^3 - 3/4*e^2 - 117/2*e + 36, 2/3*e^6 - 7/6*e^5 - 181/12*e^4 + 289/12*e^3 + 1021/12*e^2 - 725/6*e - 8, 1/4*e^6 - 1/2*e^5 - 9/2*e^4 + 37/4*e^3 + 33/2*e^2 - 43*e + 12, 7/12*e^6 - 13/12*e^5 - 73/6*e^4 + 263/12*e^3 + 689/12*e^2 - 685/6*e + 48, 1/6*e^6 - 7/6*e^5 - 10/3*e^4 + 131/6*e^3 + 95/6*e^2 - 286/3*e + 24, 1/4*e^6 - 1/2*e^5 - 23/4*e^4 + 21/2*e^3 + 141/4*e^2 - 99/2*e - 12, -5/3*e^6 + 11/3*e^5 + 103/3*e^4 - 217/3*e^3 - 502/3*e^2 + 1072/3*e - 80, -1/4*e^6 + 23/4*e^4 - 5/2*e^3 - 127/4*e^2 + 57/2*e + 12, 7/4*e^4 - 7/4*e^3 - 81/4*e^2 + 33/2*e + 14, 1/4*e^6 - 1/4*e^5 - 23/4*e^4 + 17/2*e^3 + 29*e^2 - 59*e + 36, 7/4*e^6 - 4*e^5 - 35*e^4 + 301/4*e^3 + 331/2*e^2 - 346*e + 62, -e^6 + 11/4*e^5 + 85/4*e^4 - 217/4*e^3 - 213/2*e^2 + 255*e - 64, -1/2*e^6 + 3/2*e^5 + 21/2*e^4 - 29*e^3 - 50*e^2 + 134*e - 44, -5/4*e^6 + 13/4*e^5 + 107/4*e^4 - 129/2*e^3 - 136*e^2 + 308*e - 66, 2/3*e^6 - 5/3*e^5 - 46/3*e^4 + 106/3*e^3 + 247/3*e^2 - 547/3*e + 48, 1/4*e^6 - 21/4*e^4 + 105/4*e^2 - 7/2*e - 6, 1/4*e^6 - 5/4*e^5 - 17/4*e^4 + 22*e^3 + 29/2*e^2 - 90*e + 30, -1/3*e^6 + 1/3*e^5 + 37/6*e^4 - 43/6*e^3 - 145/6*e^2 + 125/3*e - 22, -2/3*e^6 + 7/6*e^5 + 163/12*e^4 - 283/12*e^3 - 787/12*e^2 + 719/6*e - 32, 3/2*e^6 - 4*e^5 - 127/4*e^4 + 317/4*e^3 + 633/4*e^2 - 757/2*e + 110, 1/12*e^6 - 1/3*e^5 - 5/3*e^4 + 47/12*e^3 + 55/6*e^2 + 4/3*e - 12, 13/12*e^6 - 7/3*e^5 - 62/3*e^4 + 527/12*e^3 + 541/6*e^2 - 608/3*e + 60, 1/4*e^6 - 3/4*e^5 - 9/2*e^4 + 61/4*e^3 + 59/4*e^2 - 157/2*e + 50, 7/12*e^6 - 4/3*e^5 - 155/12*e^4 + 86/3*e^3 + 791/12*e^2 - 919/6*e + 32, 7/6*e^6 - 8/3*e^5 - 70/3*e^4 + 299/6*e^3 + 325/3*e^2 - 694/3*e + 70, -7/12*e^6 + 19/12*e^5 + 35/3*e^4 - 365/12*e^3 - 641/12*e^2 + 883/6*e - 52, 2/3*e^6 - 11/12*e^5 - 40/3*e^4 + 58/3*e^3 + 739/12*e^2 - 617/6*e + 28, 5/6*e^6 - 11/6*e^5 - 97/6*e^4 + 107/3*e^3 + 212/3*e^2 - 518/3*e + 64, -1/3*e^6 + 5/6*e^5 + 23/3*e^4 - 47/3*e^3 - 241/6*e^2 + 197/3*e - 20, 1/4*e^6 - 23/4*e^4 + 2*e^3 + 137/4*e^2 - 20*e - 25, 5/6*e^6 - 7/3*e^5 - 50/3*e^4 + 271/6*e^3 + 230/3*e^2 - 626/3*e + 48, -3/2*e^6 + 4*e^5 + 31*e^4 - 149/2*e^3 - 154*e^2 + 330*e - 60, -1/12*e^6 + 1/12*e^5 + 7/6*e^4 - 41/12*e^3 - 29/12*e^2 + 187/6*e - 10, -5/3*e^6 + 14/3*e^5 + 203/6*e^4 - 539/6*e^3 - 953/6*e^2 + 1243/3*e - 120, 5/3*e^6 - 41/12*e^5 - 409/12*e^4 + 793/12*e^3 + 499/3*e^2 - 946/3*e + 52, 3/4*e^6 - 7/4*e^5 - 31/2*e^4 + 137/4*e^3 + 309/4*e^2 - 161*e + 9, -9/4*e^6 + 11/2*e^5 + 187/4*e^4 - 109*e^3 - 911/4*e^2 + 525*e - 129, -1/2*e^6 + e^5 + 10*e^4 - 35/2*e^3 - 52*e^2 + 72*e + 10, -11/12*e^6 + 29/12*e^5 + 247/12*e^4 - 154/3*e^3 - 325/3*e^2 + 793/3*e - 68, -7/6*e^6 + 19/6*e^5 + 143/6*e^4 - 193/3*e^3 - 334/3*e^2 + 952/3*e - 106, -5/4*e^6 + 3*e^5 + 99/4*e^4 - 113/2*e^3 - 451/4*e^2 + 511/2*e - 66, 3/4*e^6 - 7/4*e^5 - 61/4*e^4 + 65/2*e^3 + 77*e^2 - 148*e + 16, -11/12*e^6 + 13/6*e^5 + 119/6*e^4 - 535/12*e^3 - 623/6*e^2 + 676/3*e - 48, 7/6*e^6 - 13/6*e^5 - 76/3*e^4 + 281/6*e^3 + 791/6*e^2 - 757/3*e + 42, -1/12*e^6 + 5/6*e^5 + 11/12*e^4 - 85/6*e^3 + 7/12*e^2 + 319/6*e - 12, 1/3*e^6 - 5/6*e^5 - 49/6*e^4 + 103/6*e^3 + 149/3*e^2 - 248/3*e - 4, -2*e^6 + 11/2*e^5 + 40*e^4 - 104*e^3 - 373/2*e^2 + 471*e - 130, -1/6*e^6 + 1/6*e^5 + 11/6*e^4 - 7/3*e^3 + 8/3*e^2 + 37/3*e - 14, 1/4*e^6 - 11/2*e^4 - 3/4*e^3 + 32*e^2 + 5*e - 36, 1/4*e^6 - 1/2*e^5 - 4*e^4 + 27/4*e^3 + 13*e^2 - 15*e, 5/2*e^6 - 23/4*e^5 - 107/2*e^4 + 113*e^3 + 1117/4*e^2 - 1085/2*e + 70, 2*e^6 - 19/4*e^5 - 167/4*e^4 + 375/4*e^3 + 207*e^2 - 452*e + 114, -1/4*e^6 + 1/2*e^5 + 7*e^4 - 43/4*e^3 - 47*e^2 + 55*e + 6, 2/3*e^6 - 23/12*e^5 - 40/3*e^4 + 103/3*e^3 + 775/12*e^2 - 893/6*e + 28, -3/4*e^6 + 2*e^5 + 17*e^4 - 161/4*e^3 - 191/2*e^2 + 193*e, -3/2*e^6 + 7/2*e^5 + 61/2*e^4 - 68*e^3 - 148*e^2 + 329*e - 64, -1/2*e^6 + 5/4*e^5 + 19/2*e^4 - 20*e^3 - 175/4*e^2 + 141/2*e - 16, e^6 - 13/4*e^5 - 87/4*e^4 + 239/4*e^3 + 231/2*e^2 - 252*e + 36, -1/4*e^5 - 7/4*e^4 + 27/4*e^3 + 27/2*e^2 - 38*e + 52, -e^6 + 7/4*e^5 + 85/4*e^4 - 145/4*e^3 - 215/2*e^2 + 189*e - 34, 2/3*e^6 - 23/12*e^5 - 157/12*e^4 + 445/12*e^3 + 347/6*e^2 - 520/3*e + 78, -5/6*e^6 + 31/12*e^5 + 209/12*e^4 - 563/12*e^3 - 523/6*e^2 + 602/3*e - 54, -3/4*e^5 - 1/4*e^4 + 53/4*e^3 + 7/2*e^2 - 54*e + 2, -5/4*e^6 + 11/4*e^5 + 97/4*e^4 - 51*e^3 - 106*e^2 + 230*e - 70, 1/4*e^6 - e^5 - 3*e^4 + 59/4*e^3 - 3/2*e^2 - 43*e + 48, 11/12*e^6 - 8/3*e^5 - 55/3*e^4 + 601/12*e^3 + 521/6*e^2 - 691/3*e + 40, 13/6*e^6 - 53/12*e^5 - 547/12*e^4 + 1093/12*e^3 + 1355/6*e^2 - 1399/3*e + 106, -17/12*e^6 + 41/12*e^5 + 85/3*e^4 - 799/12*e^3 - 1579/12*e^2 + 1901/6*e - 100, 2/3*e^6 - 23/12*e^5 - 46/3*e^4 + 227/6*e^3 + 1057/12*e^2 - 529/3*e + 15, 4/3*e^6 - 31/12*e^5 - 347/12*e^4 + 623/12*e^3 + 470/3*e^2 - 770/3*e - 2, 7/6*e^6 - 47/12*e^5 - 143/6*e^4 + 229/3*e^3 + 1309/12*e^2 - 2105/6*e + 130, 17/12*e^6 - 35/12*e^5 - 88/3*e^4 + 667/12*e^3 + 1789/12*e^2 - 1529/6*e + 14, 2/3*e^6 - 5/3*e^5 - 181/12*e^4 + 409/12*e^3 + 1003/12*e^2 - 995/6*e + 28, 29/12*e^6 - 31/6*e^5 - 583/12*e^4 + 599/6*e^3 + 2737/12*e^2 - 2849/6*e + 120, 1/4*e^5 - 5/4*e^4 - 19/4*e^3 + 37/2*e^2 + 19*e - 24, -3/4*e^6 + 13/4*e^5 + 15*e^4 - 245/4*e^3 - 259/4*e^2 + 545/2*e - 108, 5/6*e^6 - 25/12*e^5 - 215/12*e^4 + 485/12*e^3 + 553/6*e^2 - 560/3*e + 28, -1/4*e^6 + 1/4*e^5 + 23/4*e^4 - 21/2*e^3 - 29*e^2 + 85*e - 12, 1/2*e^6 - 7/4*e^5 - 43/4*e^4 + 129/4*e^3 + 52*e^2 - 137*e + 60, -13/6*e^6 + 59/12*e^5 + 541/12*e^4 - 1135/12*e^3 - 1367/6*e^2 + 1336/3*e - 62, 11/3*e^6 - 49/6*e^5 - 913/12*e^4 + 1903/12*e^3 + 4543/12*e^2 - 2254/3*e + 115, -13/6*e^6 + 71/12*e^5 + 535/12*e^4 - 1357/12*e^3 - 643/3*e^2 + 1540/3*e - 144, -8/3*e^6 + 37/6*e^5 + 329/6*e^4 - 719/6*e^3 - 793/3*e^2 + 1726/3*e - 140, -e^6 + 11/4*e^5 + 77/4*e^4 - 201/4*e^3 - 171/2*e^2 + 219*e - 60, 11/6*e^6 - 55/12*e^5 - 229/6*e^4 + 535/6*e^3 + 2255/12*e^2 - 1244/3*e + 111, 29/12*e^6 - 17/3*e^5 - 299/6*e^4 + 1321/12*e^3 + 724/3*e^2 - 1579/3*e + 132, -35/12*e^6 + 20/3*e^5 + 353/6*e^4 - 1579/12*e^3 - 829/3*e^2 + 1912/3*e - 160, -5/2*e^6 + 19/4*e^5 + 205/4*e^4 - 375/4*e^3 - 505/2*e^2 + 462*e - 74, 7/4*e^6 - 7/2*e^5 - 149/4*e^4 + 145/2*e^3 + 747/4*e^2 - 747/2*e + 84, -1/12*e^6 - 5/12*e^5 + 23/12*e^4 + 29/6*e^3 - 97/6*e^2 + 11/3*e + 50, -1/4*e^5 + 5/4*e^4 + 3/4*e^3 - 31/2*e^2 + 15*e + 6, 1/2*e^6 + 1/2*e^5 - 12*e^4 - 9/2*e^3 + 147/2*e^2 - 15*e - 40, 1/12*e^6 - 1/3*e^5 - 41/12*e^4 + 49/6*e^3 + 335/12*e^2 - 146/3*e - 3, 11/12*e^6 - 8/3*e^5 - 58/3*e^4 + 601/12*e^3 + 563/6*e^2 - 646/3*e + 74, 1/4*e^6 - 5/4*e^5 - 23/4*e^4 + 55/2*e^3 + 24*e^2 - 148*e + 84, -e^6 + 9/4*e^5 + 41/2*e^4 - 41*e^3 - 417/4*e^2 + 180*e - 3, 19/12*e^6 - 43/12*e^5 - 193/6*e^4 + 851/12*e^3 + 1799/12*e^2 - 2119/6*e + 120, 7/2*e^6 - 35/4*e^5 - 291/4*e^4 + 685/4*e^3 + 354*e^2 - 814*e + 216, 2*e^6 - 4*e^5 - 169/4*e^4 + 321/4*e^3 + 871/4*e^2 - 813/2*e + 36, -4*e^6 + 35/4*e^5 + 83*e^4 - 351/2*e^3 - 1621/4*e^2 + 869*e - 201, -19/6*e^6 + 101/12*e^5 + 389/6*e^4 - 469/3*e^3 - 3871/12*e^2 + 4133/6*e - 94, -1/4*e^6 + 3/2*e^5 + 5*e^4 - 83/4*e^3 - 27*e^2 + 41*e, 1/6*e^6 - 11/12*e^5 - 67/12*e^4 + 253/12*e^3 + 106/3*e^2 - 334/3*e + 70];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {41, 41, -1/3*w^3 + 4/3*w^2 + 7/3*w - 6}>] := 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;