/*
  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![-3, 0, 9, 0, -6, 0, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {37, 37, w^4 - 5*w^2 - w + 5}>;

primesArray := [
[3, 3, w],
[8, 2, w^3 - 3*w - 1],
[37, 37, w^4 - 5*w^2 - w + 5],
[37, 37, w^5 - 5*w^3 - w^2 + 4*w + 1],
[37, 37, -w^5 + w^4 + 5*w^3 - 4*w^2 - 5*w + 1],
[37, 37, w^5 + w^4 - 5*w^3 - 4*w^2 + 5*w + 1],
[37, 37, w^5 - 5*w^3 + w^2 + 4*w - 1],
[37, 37, -w^4 + 5*w^2 - w - 5],
[71, 71, w^5 - 5*w^3 - w^2 + 5*w + 4],
[71, 71, w^5 - 4*w^3 - w^2 + w + 2],
[71, 71, w^5 + w^4 - 5*w^3 - 5*w^2 + 4*w + 2],
[71, 71, w^5 - w^4 - 5*w^3 + 5*w^2 + 4*w - 2],
[71, 71, -w^5 + 4*w^3 - w^2 - w + 2],
[71, 71, -w^5 + 5*w^3 - w^2 - 5*w + 4],
[73, 73, w^5 - 5*w^3 - w^2 + 3*w + 2],
[73, 73, 2*w^5 - w^4 - 10*w^3 + 4*w^2 + 9*w - 2],
[73, 73, -w^5 + 4*w^3 - w^2 - 2*w + 2],
[73, 73, w^5 - 4*w^3 - w^2 + 2*w + 2],
[73, 73, -2*w^5 - w^4 + 10*w^3 + 4*w^2 - 9*w - 2],
[73, 73, w^5 - 5*w^3 + w^2 + 3*w - 2],
[107, 107, -2*w^5 - w^4 + 10*w^3 + 6*w^2 - 10*w - 5],
[107, 107, -w^5 + 6*w^3 - w^2 - 6*w + 2],
[107, 107, w^5 - w^4 - 4*w^3 + 6*w^2 + w - 7],
[107, 107, -w^5 - w^4 + 4*w^3 + 6*w^2 - w - 7],
[107, 107, w^5 - 6*w^3 - w^2 + 6*w + 2],
[107, 107, w^5 - 6*w^3 + 8*w + 2],
[109, 109, w^5 + w^4 - 4*w^3 - 5*w^2 + 2*w + 4],
[109, 109, w^3 + w^2 - 4*w - 2],
[109, 109, -w^5 - w^4 + 6*w^3 + 4*w^2 - 7*w - 2],
[109, 109, -w^5 + w^4 + 6*w^3 - 4*w^2 - 7*w + 2],
[109, 109, -w^3 + w^2 + 4*w - 2],
[109, 109, w^5 - w^4 - 4*w^3 + 5*w^2 + 2*w - 4],
[179, 179, w^5 - w^4 - 5*w^3 + 4*w^2 + 4*w - 4],
[179, 179, w^5 - w^4 - 5*w^3 + 5*w^2 + 5*w - 2],
[179, 179, w^2 - w - 4],
[179, 179, -w^2 - w + 4],
[179, 179, -w^5 - w^4 + 5*w^3 + 5*w^2 - 5*w - 2],
[179, 179, -w^5 - w^4 + 5*w^3 + 4*w^2 - 4*w - 4],
[181, 181, 2*w^5 - 10*w^3 - w^2 + 9*w + 2],
[181, 181, w^5 - w^4 - 5*w^3 + 4*w^2 + 6*w - 2],
[181, 181, w^5 - w^4 - 5*w^3 + 5*w^2 + 3*w - 4],
[181, 181, -w^5 - w^4 + 5*w^3 + 5*w^2 - 3*w - 4],
[181, 181, w^5 + w^4 - 5*w^3 - 4*w^2 + 6*w + 2],
[181, 181, -2*w^5 + 10*w^3 - w^2 - 9*w + 2],
[251, 251, -w^5 - w^4 + 4*w^3 + 3*w^2 + 1],
[251, 251, -w^5 - w^4 + 6*w^3 + 3*w^2 - 9*w + 1],
[251, 251, -w^5 - 2*w^4 + 6*w^3 + 10*w^2 - 8*w - 7],
[251, 251, w^5 - 2*w^4 - 6*w^3 + 10*w^2 + 8*w - 7],
[251, 251, 2*w^5 - 2*w^4 - 11*w^3 + 9*w^2 + 12*w - 7],
[251, 251, 2*w^5 + w^4 - 9*w^3 - 6*w^2 + 6*w + 5],
[289, 17, -2*w^4 + 9*w^2 - 7],
[289, 17, w^4 - 3*w^2 - 1],
[289, 17, w^4 - 6*w^2 + 5],
[359, 359, 2*w^5 - 9*w^3 + 5*w - 2],
[359, 359, 2*w^4 - 10*w^2 + w + 8],
[359, 359, -w^5 + w^4 + 5*w^3 - 3*w^2 - 3*w - 2],
[359, 359, 2*w^5 - 11*w^3 + 13*w + 2],
[359, 359, -2*w^5 + 2*w^4 + 10*w^3 - 9*w^2 - 9*w + 8],
[359, 359, -2*w^5 + 9*w^3 - 5*w - 2],
[361, 19, 3*w^4 - 12*w^2 + 4],
[361, 19, -3*w^4 + 14*w^2 - 7],
[361, 19, w^4 - 4*w^2 - 1],
[397, 397, -w^5 + w^4 + 4*w^3 - 5*w^2 - 2*w + 5],
[397, 397, w^5 - w^4 - 6*w^3 + 4*w^2 + 7*w - 1],
[397, 397, w^3 + w^2 - 4*w - 1],
[397, 397, -w^3 + w^2 + 4*w - 1],
[397, 397, -w^5 - w^4 + 6*w^3 + 4*w^2 - 7*w - 1],
[397, 397, -w^5 - w^4 + 4*w^3 + 5*w^2 - 2*w - 5],
[431, 431, -2*w^5 + 9*w^3 + w^2 - 7*w - 1],
[431, 431, -2*w^5 - w^4 + 9*w^3 + 4*w^2 - 6*w - 4],
[431, 431, w^5 - w^4 - 6*w^3 + 5*w^2 + 9*w - 2],
[431, 431, -w^5 - w^4 + 6*w^3 + 5*w^2 - 9*w - 2],
[431, 431, 2*w^5 - w^4 - 9*w^3 + 4*w^2 + 6*w - 4],
[431, 431, 2*w^5 - 9*w^3 + w^2 + 7*w - 1],
[433, 433, w^5 - w^4 - 7*w^3 + 4*w^2 + 10*w - 2],
[433, 433, -w^5 + w^4 + 3*w^3 - 5*w^2 + w + 4],
[433, 433, -2*w^3 - w^2 + 7*w + 2],
[433, 433, -2*w^3 + w^2 + 7*w - 2],
[433, 433, w^5 + w^4 - 3*w^3 - 5*w^2 - w + 4],
[433, 433, -w^5 - w^4 + 7*w^3 + 4*w^2 - 10*w - 2],
[467, 467, w^5 + w^4 - 5*w^3 - 3*w^2 + 6*w - 2],
[467, 467, -w^5 + 2*w^4 + 5*w^3 - 9*w^2 - 3*w + 8],
[467, 467, -w^5 + w^4 + 4*w^3 - 6*w^2 - w + 5],
[467, 467, 2*w^5 - w^4 - 10*w^3 + 6*w^2 + 9*w - 4],
[467, 467, -w^5 - 2*w^4 + 5*w^3 + 9*w^2 - 3*w - 8],
[467, 467, 2*w^5 + w^4 - 9*w^3 - 4*w^2 + 7*w + 2],
[503, 503, w^5 + w^4 - 5*w^3 - 4*w^2 + 4*w + 5],
[503, 503, w^4 - 2*w^3 - 5*w^2 + 6*w + 5],
[503, 503, -w^4 + 2*w^3 + 4*w^2 - 6*w - 1],
[503, 503, w^4 + 2*w^3 - 4*w^2 - 6*w + 1],
[503, 503, -w^4 - 2*w^3 + 5*w^2 + 6*w - 5],
[503, 503, w^5 + 2*w^4 - 6*w^3 - 8*w^2 + 7*w + 5],
[541, 541, -2*w^5 + 2*w^4 + 11*w^3 - 9*w^2 - 12*w + 5],
[541, 541, w^5 - w^4 - 4*w^3 + 6*w^2 + 3*w - 7],
[541, 541, -w^5 - w^4 + 4*w^3 + 3*w^2 - 1],
[541, 541, w^5 - w^4 - 4*w^3 + 3*w^2 - 1],
[541, 541, -w^5 - w^4 + 4*w^3 + 6*w^2 - 3*w - 7],
[541, 541, 2*w^5 + 2*w^4 - 11*w^3 - 9*w^2 + 12*w + 5],
[577, 577, w^3 + w^2 - 5*w - 1],
[577, 577, 2*w^5 + w^4 - 11*w^3 - 4*w^2 + 11*w + 1],
[577, 577, -2*w^5 + w^4 + 9*w^3 - 5*w^2 - 7*w + 5],
[577, 577, 2*w^5 + w^4 - 9*w^3 - 5*w^2 + 7*w + 5],
[577, 577, 2*w^5 - w^4 - 11*w^3 + 4*w^2 + 11*w - 1],
[577, 577, -w^3 + w^2 + 5*w - 1],
[613, 613, 2*w^5 - 10*w^3 - w^2 + 8*w + 1],
[613, 613, -2*w^5 + w^4 + 10*w^3 - 4*w^2 - 10*w + 1],
[613, 613, w^4 - 5*w^2 + 2*w + 5],
[613, 613, w^4 - 5*w^2 - 2*w + 5],
[613, 613, 2*w^5 + w^4 - 10*w^3 - 4*w^2 + 10*w + 1],
[613, 613, -2*w^5 + 10*w^3 - w^2 - 8*w + 1],
[647, 647, -w^5 + 2*w^4 + 5*w^3 - 10*w^2 - 6*w + 8],
[647, 647, 2*w^3 + w^2 - 5*w - 2],
[647, 647, -w^5 - w^4 + 7*w^3 + 5*w^2 - 11*w - 4],
[647, 647, w^5 - w^4 - 7*w^3 + 5*w^2 + 11*w - 4],
[647, 647, 2*w^3 - w^2 - 5*w + 2],
[647, 647, -w^5 - 2*w^4 + 5*w^3 + 10*w^2 - 6*w - 8],
[683, 683, -2*w^5 + 10*w^3 - 7*w - 1],
[683, 683, -3*w^5 + 15*w^3 - 14*w + 1],
[683, 683, -2*w^5 - w^4 + 11*w^3 + 6*w^2 - 12*w - 5],
[683, 683, 2*w^5 - w^4 - 11*w^3 + 6*w^2 + 12*w - 5],
[683, 683, 3*w^5 - 15*w^3 + 14*w + 1],
[683, 683, 2*w^5 - 10*w^3 + 7*w - 1],
[719, 719, 3*w^5 + w^4 - 15*w^3 - 5*w^2 + 14*w + 4],
[719, 719, -w^5 + 5*w^3 + w^2 - 7*w - 2],
[719, 719, -w^5 + 7*w^3 - 11*w + 1],
[719, 719, w^5 - 7*w^3 + 11*w + 1],
[719, 719, w^5 - 5*w^3 + w^2 + 7*w - 2],
[719, 719, -3*w^5 + w^4 + 15*w^3 - 5*w^2 - 14*w + 4],
[757, 757, w^5 - w^4 - 3*w^3 + 5*w^2 - 2*w - 5],
[757, 757, w^5 - w^4 - 6*w^3 + 4*w^2 + 9*w - 4],
[757, 757, 2*w^5 - 9*w^3 - w^2 + 6*w + 4],
[757, 757, 2*w^5 - 9*w^3 + w^2 + 6*w - 4],
[757, 757, -w^5 - w^4 + 6*w^3 + 4*w^2 - 9*w - 4],
[757, 757, w^5 + w^4 - 3*w^3 - 5*w^2 - 2*w + 5],
[827, 827, -w^5 - w^4 + 4*w^3 + 4*w^2 - w - 4],
[827, 827, 2*w^4 - 8*w^2 + w + 1],
[827, 827, w^5 + 3*w^4 - 5*w^3 - 12*w^2 + 4*w + 5],
[827, 827, -w^5 + 3*w^4 + 5*w^3 - 12*w^2 - 4*w + 5],
[827, 827, -2*w^4 + 8*w^2 + w - 1],
[827, 827, -w^5 + w^4 + 4*w^3 - 4*w^2 - w + 4],
[829, 829, -w^5 - 3*w^4 + 5*w^3 + 14*w^2 - 4*w - 11],
[829, 829, -w^5 - 2*w^4 + 5*w^3 + 11*w^2 - 5*w - 10],
[829, 829, -w^5 + 3*w^4 + 5*w^3 - 13*w^2 - 4*w + 8],
[829, 829, -w^5 - w^4 + 5*w^3 + 7*w^2 - 5*w - 7],
[829, 829, 2*w^4 - 7*w^2 + w + 1],
[829, 829, -w^5 + 3*w^4 + 5*w^3 - 14*w^2 - 4*w + 11],
[863, 863, w^5 - w^4 - 6*w^3 + 6*w^2 + 7*w - 4],
[863, 863, -w^5 - w^4 + 6*w^3 + 5*w^2 - 7*w - 1],
[863, 863, -w^4 + w^3 + 4*w^2 - 4*w - 5],
[863, 863, -w^4 - w^3 + 4*w^2 + 4*w - 5],
[863, 863, -w^5 + w^4 + 6*w^3 - 5*w^2 - 7*w + 1],
[863, 863, -w^5 - w^4 + 6*w^3 + 6*w^2 - 7*w - 4],
[937, 937, -w^5 + 4*w^4 + 5*w^3 - 16*w^2 - 4*w + 5],
[937, 937, w^5 + w^4 - 4*w^3 - 3*w^2 - w - 2],
[937, 937, w^5 - 3*w^4 - 4*w^3 + 12*w^2 + w - 7],
[937, 937, -2*w^5 + 10*w^3 - w^2 - 6*w + 2],
[937, 937, w^5 - w^4 - 4*w^3 + 3*w^2 - w + 2],
[937, 937, -w^5 - 4*w^4 + 5*w^3 + 16*w^2 - 4*w - 5],
[971, 971, w^4 - 2*w^3 - 4*w^2 + 6*w + 2],
[971, 971, 2*w^3 + w^2 - 6*w - 2],
[971, 971, w^4 - 2*w^3 - 5*w^2 + 6*w + 4],
[971, 971, w^4 + 2*w^3 - 5*w^2 - 6*w + 4],
[971, 971, 2*w^3 - w^2 - 6*w + 2],
[971, 971, w^4 + 2*w^3 - 4*w^2 - 6*w + 2]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^3 - 3*x^2 - x + 5;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, -e, 1, 2*e - 4, -2*e^2 + 6*e + 2, -5*e^2 + 4*e + 15, e^2 - 4*e - 3, 3*e^2 - 13, e^2 + 2*e - 11, -4*e^2 + 2*e + 16, -5*e^2 + 21, -14, -e^2 + 4*e + 5, -3*e^2 + 8*e + 3, 5*e^2 - 6*e - 18, 2*e^2 - 4*e - 9, -2*e^2 - 2*e + 5, -3*e^2 + 8, 3*e^2 - 10*e - 4, -5*e^2 + 6*e + 12, e^2 - 7*e + 3, 6*e^2 - 11*e - 16, 4*e^2 - 7*e - 2, -8*e^2 + 9*e + 18, e^2 - e - 7, 2*e^2 + 3*e - 10, 3*e^2 - 8*e + 1, -3*e^2 + 6*e - 3, -6*e + 10, -9*e^2 + 12*e + 17, -7*e^2 + 12*e + 7, 8*e^2 - 14*e - 14, 7*e^2 - 2*e - 31, -e^2 - 6*e + 13, 5*e^2 - 6*e - 1, e^2 - 10*e + 7, -7*e^2 + 12*e + 17, 5*e^2 - 6*e - 13, 4*e^2 + 4*e - 20, -6*e^2 + 12*e + 2, 8*e^2 - 12*e - 16, e^2 - 6*e - 11, 5*e^2 + 2*e - 23, 6*e^2 - 6*e - 28, 5*e^2 - 11*e + 7, -e^2 + 7*e + 7, -6*e^2 + 9*e + 4, 4*e^2 - e + 4, 3*e^2 - e - 25, 6*e^2 - 17*e - 18, 4*e^2 - 2*e - 31, -2*e - 15, -7*e^2 + 10*e + 16, 14*e^2 - 6*e - 54, -e^2 - 10*e + 7, 10*e^2 - 22*e - 6, 13*e^2 - 28*e - 21, -2*e^2 + 8*e - 4, -8*e^2 + 22*e + 8, 6*e - 11, 3*e^2 - 8*e - 8, 5*e^2 - 10*e - 20, -4*e^2 - 6*e + 8, -9*e^2 + 18*e + 9, 16*e^2 - 28*e - 26, -e^2 + 2*e - 23, 3*e^2 - 14*e + 17, -2*e^2 - 6*e + 14, 15*e^2 - 8*e - 63, 2*e^2 + 4*e, 9*e^2 - 10*e - 15, -8*e - 6, -17*e^2 + 30*e + 35, -12*e^2 + 10*e + 52, 8*e - 11, 17*e^2 - 6*e - 66, -8*e^2 + 12*e + 21, 18*e^2 - 14*e - 55, e^2 + 2*e + 6, -10*e^2 + 18*e + 25, 19*e^2 - 17*e - 57, -14*e^2 + 7*e + 58, -16*e^2 + 27*e + 32, e^2 - e + 25, 12*e^2 - 19*e - 22, e^2 + 9*e - 5, -4*e^2 + 8*e + 4, 7*e^2 - 6*e - 19, -e^2 + 12*e + 11, 6*e^2 - 14*e - 16, -2*e + 6, -4*e^2 + 10*e + 10, -5*e^2 - 8*e + 15, 8*e^2 - 14*e - 8, 10*e^2 - 22*e - 10, -11*e^2 + 12*e + 25, 4*e^2 - 16*e - 10, -3*e^2 - 2*e + 7, 14*e^2 - 20*e - 17, -12*e^2 + 26*e + 27, 10*e^2 - 4*e - 61, 2*e^2 - 12*e - 13, -5*e^2 + 4*e + 26, -8*e^2 + 20*e + 13, e^2 + 10*e - 21, -9*e^2 + 2*e + 29, 21*e^2 - 18*e - 57, -2*e^2 + 10*e - 14, 9*e^2 - 6*e - 45, 14*e^2 - 10*e - 50, 11*e^2 - 6*e - 29, -3*e^2 - 10*e + 25, 5*e^2 + 2*e - 23, -6*e + 12, -14*e^2 + 16*e + 12, 16*e^2 - 8*e - 62, 25*e^2 - 35*e - 47, -2*e^2 + 15*e + 16, 14*e^2 - 21*e - 28, -25*e^2 + 33*e + 51, 9*e^2 - 7*e - 31, 2*e^2 + 3*e - 16, -9*e^2 + 2*e + 39, -13*e^2 + 18*e - 1, -9*e^2 + 22*e - 5, 21*e^2 - 10*e - 79, -7*e^2 + 55, -4*e^2 + 8*e + 34, -5*e^2 + 2*e - 3, -11*e^2 + 12*e + 33, -2*e^2 + 2*e - 2, -2*e^2 - 10*e + 6, -7*e^2 + 22*e + 3, -11*e^2 + 18*e + 23, 13*e^2 - 34*e - 13, -17*e^2 + 20*e + 19, -7*e^2 + 2*e + 51, -23*e^2 + 34*e + 31, -3*e^2 - 6*e + 31, -5*e^2 + 18*e + 1, 15*e^2 - 26*e - 35, 13*e^2 - 24*e - 35, 22*e^2 - 28*e - 52, -17*e^2 + 2*e + 65, 7*e^2 + 3, 19*e^2 - 36*e - 37, -2*e^2 + 2*e - 14, -6*e^2 + 14*e - 18, 14*e^2 - 24*e - 16, -4*e^2 + 2*e + 24, 8*e^2 - 18*e - 32, 19*e^2 - 16*e - 59, -14*e^2 + 38*e + 22, 12*e^2 - 20*e - 22, -18*e^2 + 32*e + 32, 2*e^2 - 8*e + 16, -10*e^2 + 14*e + 18, 20*e^2 - 24*e - 22, -3*e^2 + 20*e + 17, -3*e^2 - 14*e + 47, 5*e^2 - 18*e - 29, 15*e^2 - 18*e - 67, -27*e^2 + 34*e + 55, -e^2 - 2*e + 25];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {37, 37, w^4 - 5*w^2 - w + 5}>] := -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;