/*
  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![-1, 1, 5, -3, -2, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {49, 7, -w^4 + 2*w^3 + 4*w^2 - 6*w - 2}>;

primesArray := [
[3, 3, w^4 - 2*w^3 - 3*w^2 + 4*w + 1],
[13, 13, w^3 - 2*w^2 - 2*w + 2],
[23, 23, 2*w^4 - 3*w^3 - 6*w^2 + 5*w + 1],
[25, 5, -w^2 + 2*w + 2],
[29, 29, -w^4 + w^3 + 4*w^2 - 3*w - 1],
[31, 31, w^4 - w^3 - 5*w^2 + 2*w + 4],
[32, 2, 2],
[37, 37, w^4 - 2*w^3 - 2*w^2 + 4*w + 1],
[47, 47, w^4 - w^3 - 5*w^2 + 2*w + 3],
[47, 47, 2*w^4 - 3*w^3 - 6*w^2 + 6*w + 2],
[49, 7, -w^4 + 2*w^3 + 4*w^2 - 6*w - 2],
[53, 53, -2*w^4 + 3*w^3 + 7*w^2 - 7*w - 2],
[59, 59, -2*w^4 + 3*w^3 + 6*w^2 - 5*w - 2],
[67, 67, -w^4 + 3*w^3 + 2*w^2 - 7*w],
[67, 67, -w^4 + 3*w^3 + w^2 - 7*w + 1],
[71, 71, w^4 - 2*w^3 - 4*w^2 + 3*w + 3],
[71, 71, -w^2 + 5],
[79, 79, 2*w^4 - 3*w^3 - 5*w^2 + 3*w + 1],
[81, 3, -w^4 + 3*w^3 + 3*w^2 - 8*w - 2],
[83, 83, -3*w^4 + 3*w^3 + 10*w^2 - 2*w - 2],
[89, 89, -w^4 + 2*w^3 + 3*w^2 - 6*w - 1],
[101, 101, w^4 - w^3 - 4*w^2 + 3],
[103, 103, w^4 - w^3 - 5*w^2 + 3*w + 1],
[103, 103, 2*w^4 - 3*w^3 - 8*w^2 + 8*w + 3],
[103, 103, -w^4 + 2*w^3 + 2*w^2 - 5*w + 4],
[109, 109, -2*w^4 + 4*w^3 + 7*w^2 - 10*w - 4],
[121, 11, w^4 - 3*w^3 - 2*w^2 + 7*w + 1],
[125, 5, -2*w^4 + 2*w^3 + 7*w^2 - 3*w + 1],
[127, 127, -2*w^3 + 2*w^2 + 7*w - 4],
[149, 149, 2*w^4 - 5*w^3 - 4*w^2 + 11*w + 1],
[157, 157, -2*w^4 + 2*w^3 + 9*w^2 - 5*w - 6],
[163, 163, 2*w^4 - 2*w^3 - 6*w^2 + w - 1],
[163, 163, 3*w^4 - 5*w^3 - 9*w^2 + 10*w + 4],
[163, 163, -3*w^4 + 5*w^3 + 10*w^2 - 12*w - 3],
[167, 167, w^4 - 2*w^3 - 2*w^2 + 4*w - 4],
[173, 173, 2*w^4 - 3*w^3 - 6*w^2 + 7*w],
[173, 173, 2*w^4 - 5*w^3 - 4*w^2 + 10*w],
[179, 179, -w^4 + 3*w^3 + 2*w^2 - 6*w - 2],
[181, 181, -w^4 + w^3 + 2*w^2 + w + 2],
[181, 181, -2*w^4 + 3*w^3 + 7*w^2 - 4*w - 4],
[193, 193, 2*w^4 - 4*w^3 - 5*w^2 + 8*w + 3],
[197, 197, 2*w^4 - 2*w^3 - 9*w^2 + 3*w + 6],
[199, 199, w^4 - 2*w^3 - w^2 + 3*w - 4],
[211, 211, -w^4 + 6*w^2 + w - 3],
[223, 223, 3*w^4 - 5*w^3 - 9*w^2 + 11*w + 1],
[223, 223, -w^3 + w^2 + w - 3],
[229, 229, 2*w^2 - w - 4],
[241, 241, -w^4 + 6*w^2 - 6],
[241, 241, -w^4 + 3*w^3 + 3*w^2 - 8*w],
[257, 257, -2*w^3 + 2*w^2 + 5*w],
[269, 269, 2*w^3 - w^2 - 5*w],
[269, 269, 3*w^4 - 6*w^3 - 7*w^2 + 12*w],
[277, 277, -w^4 + 3*w^3 + w^2 - 7*w],
[277, 277, -w^3 - w^2 + 5*w + 4],
[283, 283, -3*w^4 + 5*w^3 + 8*w^2 - 8*w - 1],
[283, 283, 3*w^4 - 4*w^3 - 9*w^2 + 7*w + 1],
[283, 283, 3*w^4 - 5*w^3 - 9*w^2 + 8*w + 4],
[293, 293, 2*w^4 - 4*w^3 - 7*w^2 + 11*w + 2],
[307, 307, -w^3 + 3*w^2 - 5],
[311, 311, w^2 - 2*w - 4],
[317, 317, w^4 - w^3 - 2*w^2 + w - 4],
[317, 317, -w^3 + 6*w],
[331, 331, -w^4 + w^3 + 2*w^2 - 2*w + 3],
[343, 7, -2*w^3 + w^2 + 7*w - 1],
[349, 349, 3*w^4 - 5*w^3 - 8*w^2 + 10*w],
[349, 349, w^4 + w^3 - 6*w^2 - 4*w + 4],
[353, 353, w^3 - 3*w - 4],
[359, 359, 2*w^4 - 2*w^3 - 8*w^2 + w + 4],
[367, 367, -5*w^4 + 8*w^3 + 17*w^2 - 18*w - 8],
[367, 367, -w^4 + 4*w^3 - 9*w + 1],
[373, 373, 3*w^4 - 6*w^3 - 8*w^2 + 12*w + 1],
[379, 379, 4*w^4 - 7*w^3 - 12*w^2 + 15*w + 6],
[379, 379, -w^4 - w^3 + 6*w^2 + 5*w],
[379, 379, -3*w^4 + 4*w^3 + 9*w^2 - 6*w - 1],
[421, 421, -3*w^4 + 4*w^3 + 11*w^2 - 7*w - 3],
[433, 433, 2*w^4 - 5*w^3 - 5*w^2 + 10*w],
[439, 439, -4*w^4 + 7*w^3 + 11*w^2 - 15*w - 1],
[457, 457, -2*w^4 + 4*w^3 + 6*w^2 - 7*w - 2],
[461, 461, w^3 - 2*w - 3],
[463, 463, -3*w^4 + 5*w^3 + 10*w^2 - 11*w - 2],
[463, 463, -2*w^4 + 2*w^3 + 8*w^2 - 2*w - 7],
[467, 467, w^4 - 2*w^3 - 4*w^2 + 4*w + 6],
[487, 487, w^2 - 3*w - 3],
[487, 487, w^2 + w - 5],
[491, 491, -3*w^4 + 6*w^3 + 8*w^2 - 12*w],
[491, 491, -w^4 + 3*w^3 + 3*w^2 - 9*w - 1],
[503, 503, -w^4 + 7*w^2 - 7],
[503, 503, -w^4 + 2*w^3 + 3*w^2 - 5*w + 3],
[509, 509, -2*w^4 + 4*w^3 + 7*w^2 - 7*w - 3],
[521, 521, -w^4 + w^3 + 6*w^2 - 2*w - 5],
[541, 541, -w^4 + 2*w^3 + 3*w^2 - 4*w - 5],
[541, 541, -3*w^4 + 5*w^3 + 10*w^2 - 10*w - 2],
[541, 541, -w^4 + 3*w^2 + 5*w - 1],
[547, 547, -3*w^4 + 5*w^3 + 10*w^2 - 9*w - 4],
[557, 557, 5*w^4 - 5*w^3 - 18*w^2 + 6*w + 3],
[557, 557, w^4 - 2*w^2 - 2*w - 2],
[563, 563, -2*w^4 + 4*w^3 + 5*w^2 - 6*w],
[563, 563, 3*w^4 - 4*w^3 - 10*w^2 + 8*w + 1],
[587, 587, -w^4 + 5*w^2 + 2*w - 6],
[599, 599, 3*w^4 - 4*w^3 - 11*w^2 + 9*w + 3],
[599, 599, 3*w^3 - 3*w^2 - 9*w + 1],
[601, 601, -w^4 + 3*w^3 + w^2 - 8*w],
[607, 607, 4*w^4 - 8*w^3 - 12*w^2 + 17*w + 6],
[607, 607, w^4 + 2*w^3 - 7*w^2 - 6*w + 3],
[613, 613, -3*w^4 + 4*w^3 + 9*w^2 - 7*w - 2],
[617, 617, 2*w^3 - 4*w^2 - 5*w + 2],
[619, 619, -w^4 + w^3 + 2*w^2 + 3*w + 2],
[641, 641, -2*w^4 + 4*w^3 + 6*w^2 - 11*w - 3],
[641, 641, -2*w^4 + 5*w^3 + 5*w^2 - 11*w - 2],
[643, 643, -3*w^4 + 6*w^3 + 7*w^2 - 10*w + 3],
[653, 653, -4*w^4 + 6*w^3 + 15*w^2 - 15*w - 6],
[659, 659, 2*w^4 - w^3 - 9*w^2 + w + 5],
[659, 659, -w^4 + w^3 + w^2 + 2*w + 2],
[659, 659, -2*w^4 + 4*w^3 + 5*w^2 - 9*w - 3],
[673, 673, 2*w^4 + w^3 - 10*w^2 - 7*w + 3],
[677, 677, -3*w^4 + w^3 + 14*w^2 + 3*w - 5],
[677, 677, -3*w^4 + 4*w^3 + 8*w^2 - 4*w - 1],
[677, 677, -2*w^4 + 2*w^3 + 6*w^2 + w + 1],
[683, 683, -4*w^4 + 7*w^3 + 12*w^2 - 15*w],
[691, 691, -3*w^4 + 4*w^3 + 9*w^2 - 4*w + 1],
[691, 691, 3*w^4 - 5*w^3 - 8*w^2 + 9*w + 1],
[709, 709, -w^4 + 4*w^3 - 8*w - 1],
[733, 733, 4*w^4 - 6*w^3 - 13*w^2 + 10*w + 2],
[743, 743, 2*w^4 - 5*w^3 - 4*w^2 + 14*w - 3],
[751, 751, -w^4 - 2*w^3 + 7*w^2 + 6*w - 4],
[757, 757, 3*w^3 - 2*w^2 - 9*w + 1],
[761, 761, 4*w^4 - 5*w^3 - 14*w^2 + 9*w + 3],
[761, 761, 5*w^4 - 8*w^3 - 16*w^2 + 15*w + 8],
[761, 761, w^4 - 2*w^3 - 2*w^2 + 2*w - 3],
[769, 769, 3*w^3 - 4*w^2 - 8*w + 7],
[769, 769, 3*w^4 - 5*w^3 - 11*w^2 + 12*w + 4],
[769, 769, -2*w^3 + 3*w^2 + 6*w - 1],
[773, 773, 3*w^4 - 3*w^3 - 12*w^2 + 7*w + 8],
[773, 773, -2*w^4 + 6*w^3 + 3*w^2 - 14*w + 2],
[773, 773, -2*w^4 + w^3 + 9*w^2 - w - 6],
[787, 787, w^4 - 4*w^2 - 2*w + 6],
[809, 809, -2*w^4 + 2*w^3 + 8*w^2 - w - 6],
[809, 809, -2*w^4 + 3*w^3 + 9*w^2 - 5*w - 8],
[821, 821, -2*w^4 + 9*w^2 + 3*w + 1],
[821, 821, w^4 - 4*w^2 - 3*w + 4],
[829, 829, -w^4 + 3*w^3 - 4*w + 4],
[853, 853, -3*w^4 + 5*w^3 + 8*w^2 - 9*w - 2],
[859, 859, 2*w^4 - 2*w^3 - 6*w^2 - 3],
[863, 863, 5*w^4 - 9*w^3 - 14*w^2 + 17*w + 3],
[863, 863, 2*w^4 - 4*w^3 - 7*w^2 + 12*w + 6],
[863, 863, 2*w^4 - 2*w^3 - 11*w^2 + 6*w + 9],
[877, 877, -2*w^4 + 3*w^3 + 8*w^2 - 7*w - 2],
[877, 877, 3*w^4 - 6*w^3 - 6*w^2 + 13*w - 1],
[877, 877, w^4 + w^3 - 4*w^2 - 8*w],
[881, 881, -w^4 - w^3 + 7*w^2 + 2*w - 4],
[883, 883, 4*w^4 - 4*w^3 - 16*w^2 + 7*w + 8],
[887, 887, -2*w^4 + 3*w^3 + 10*w^2 - 8*w - 9],
[929, 929, -w^3 + 2*w^2 - 4],
[929, 929, -3*w^4 + 6*w^3 + 9*w^2 - 14*w],
[929, 929, -4*w^4 + 7*w^3 + 11*w^2 - 14*w - 4],
[937, 937, 2*w^4 - 2*w^3 - 5*w^2 - w - 2],
[941, 941, -3*w^4 + 3*w^3 + 13*w^2 - 8*w - 8],
[941, 941, w^4 - w^3 - 3*w^2 + 3*w - 3],
[947, 947, -3*w^4 + 6*w^3 + 10*w^2 - 16*w - 3],
[961, 31, -w^4 + 5*w^3 - 14*w],
[961, 31, -w^4 - w^3 + 8*w^2 + 2*w - 4],
[971, 971, -2*w^4 + 3*w^3 + 4*w^2 - 3*w + 2],
[983, 983, 4*w^4 - 7*w^3 - 11*w^2 + 16*w],
[983, 983, -3*w^4 + 2*w^3 + 12*w^2],
[997, 997, 3*w^4 - 4*w^3 - 8*w^2 + 5*w - 2],
[997, 997, w^4 - 3*w^3 - w^2 + 3*w - 2],
[997, 997, -4*w^4 + 5*w^3 + 13*w^2 - 9*w - 2]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x;
K := Rationals(); e := 1;

heckeEigenvaluesArray := [0, -6, 0, -2, -2, 8, -3, -6, 8, -4, -1, -6, -12, 8, -12, -12, 0, 8, 2, 4, 14, -10, 0, 8, -8, -2, 2, -14, -8, -6, -18, 8, -20, -8, 12, -14, 14, -4, 22, -2, -2, -22, -24, -20, 24, -20, -10, 2, 14, 6, 10, -30, 2, -22, 12, 0, -28, 26, -32, 0, 26, -14, 20, -24, -2, -10, -18, 32, 12, -12, 10, -4, 12, -16, -2, -2, -32, 14, -2, -8, -16, 16, 24, -16, -28, 16, 16, 16, -30, 6, -6, 26, -30, -12, 42, 18, -20, 12, 12, 32, -24, 14, -8, 12, 38, -18, -40, 18, 2, 36, -42, -12, 12, -12, -6, 18, -14, 30, -8, 28, 4, 22, 34, -8, -32, -26, -10, 38, -6, -14, -46, -50, -6, 22, 46, -36, -6, 6, -6, 22, 2, -42, 24, 40, -24, -20, -38, 42, 2, 42, 56, -8, -50, -54, 42, 22, 54, 38, -16, 10, 2, -4, 24, 16, -26, 46, -38];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {49, 7, -w^4 + 2*w^3 + 4*w^2 - 6*w - 2}>] := 1;

// EXAMPLE:
// pp := Factorization(2*ZF)[1][1];
// heckeEigenvalues[pp];

/* EXTRA CODE: recompute eigenform (warning, may take a few minutes or longer!):
M := HilbertCuspForms(F, NN);
S := NewSubspace(M);
// SetVerbose("ModFrmHil", 1);
newspaces := NewformDecomposition(S);
newforms := [Eigenform(U) : U in newspaces];
ppind := 0;
while #newforms gt 1 do
  pp := primes[ppind];
  newforms := [f : f in newforms | HeckeEigenvalue(f,pp) eq heckeEigenvalues[pp]];
end while;
f := newforms[1];
// [HeckeEigenvalue(f,pp) : pp in primes] eq heckeEigenvaluesArray;
*/