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

NN := ideal<ZF | {13, 13, -w^2 + 3}>;

primesArray := [
[8, 2, -2*w^5 - w^4 + 13*w^3 + w^2 - 16*w - 1],
[8, 2, -3*w^5 - w^4 + 19*w^3 - 2*w^2 - 20*w + 1],
[13, 13, -w^2 + 3],
[29, 29, -w^5 - w^4 + 5*w^3 + 2*w^2 - 3*w - 1],
[41, 41, -2*w^5 + 13*w^3 - 4*w^2 - 12*w],
[41, 41, 4*w^5 + 2*w^4 - 25*w^3 - 2*w^2 + 26*w + 5],
[43, 43, -w^2 - w + 2],
[43, 43, 2*w^5 + w^4 - 13*w^3 - 2*w^2 + 14*w + 5],
[43, 43, -w^5 + 7*w^3 - w^2 - 8*w - 3],
[43, 43, 2*w^5 + w^4 - 13*w^3 - 2*w^2 + 15*w + 3],
[49, 7, -w^5 - w^4 + 6*w^3 + 3*w^2 - 7*w],
[71, 71, -5*w^5 - 2*w^4 + 32*w^3 - w^2 - 35*w - 2],
[71, 71, 2*w^5 + w^4 - 13*w^3 - 2*w^2 + 14*w + 6],
[71, 71, -3*w^5 - 2*w^4 + 19*w^3 + 5*w^2 - 22*w - 7],
[71, 71, 4*w^5 + 2*w^4 - 25*w^3 - 2*w^2 + 26*w + 6],
[83, 83, -2*w^5 - w^4 + 12*w^3 - 12*w + 1],
[83, 83, -2*w^5 - w^4 + 13*w^3 - 16*w + 1],
[83, 83, -4*w^5 - 2*w^4 + 26*w^3 + 3*w^2 - 29*w - 6],
[83, 83, 2*w^5 - 13*w^3 + 5*w^2 + 13*w - 1],
[97, 97, -3*w^5 + 20*w^3 - 8*w^2 - 22*w + 5],
[97, 97, -4*w^5 - w^4 + 27*w^3 - w^2 - 31*w - 4],
[113, 113, 4*w^5 + w^4 - 26*w^3 + 4*w^2 + 27*w - 1],
[113, 113, 6*w^5 + 3*w^4 - 39*w^3 - 3*w^2 + 46*w + 4],
[113, 113, w^5 - 8*w^3 + w^2 + 12*w],
[113, 113, -6*w^5 - 2*w^4 + 39*w^3 - w^2 - 42*w - 8],
[139, 139, -3*w^5 - w^4 + 20*w^3 - 22*w - 4],
[139, 139, -w^5 + 7*w^3 - 3*w^2 - 9*w + 3],
[167, 167, -3*w^5 - w^4 + 19*w^3 - 2*w^2 - 19*w + 3],
[167, 167, -2*w^5 - 2*w^4 + 13*w^3 + 6*w^2 - 18*w - 3],
[167, 167, -3*w^5 - w^4 + 19*w^3 - 2*w^2 - 21*w + 1],
[167, 167, 4*w^5 + 2*w^4 - 26*w^3 - 3*w^2 + 30*w + 5],
[169, 13, -3*w^5 - 2*w^4 + 19*w^3 + 4*w^2 - 22*w - 5],
[169, 13, w^4 + w^3 - 5*w^2 - w + 2],
[181, 181, 4*w^5 + 2*w^4 - 26*w^3 - 3*w^2 + 29*w + 8],
[181, 181, 3*w^5 + 2*w^4 - 19*w^3 - 4*w^2 + 20*w + 4],
[197, 197, w^5 + 2*w^4 - 6*w^3 - 9*w^2 + 8*w + 7],
[197, 197, -5*w^5 - w^4 + 34*w^3 - 4*w^2 - 39*w - 2],
[197, 197, -6*w^5 - w^4 + 40*w^3 - 7*w^2 - 44*w - 1],
[197, 197, 8*w^5 + 3*w^4 - 51*w^3 + w^2 + 53*w + 7],
[211, 211, 4*w^5 + 2*w^4 - 25*w^3 - w^2 + 26*w + 4],
[211, 211, -7*w^5 - 3*w^4 + 45*w^3 - 49*w - 3],
[211, 211, 4*w^5 + 2*w^4 - 27*w^3 - 3*w^2 + 34*w + 4],
[211, 211, -w^5 - w^4 + 7*w^3 + 4*w^2 - 11*w - 5],
[223, 223, 2*w^5 + w^4 - 11*w^3 + 7*w + 1],
[223, 223, -7*w^5 - 2*w^4 + 45*w^3 - 5*w^2 - 46*w - 3],
[223, 223, -w^5 - 2*w^4 + 5*w^3 + 8*w^2 - 5*w - 4],
[223, 223, -4*w^5 - w^4 + 27*w^3 - w^2 - 31*w - 5],
[239, 239, 2*w^5 + w^4 - 13*w^3 - w^2 + 17*w - 1],
[239, 239, -7*w^5 - 3*w^4 + 44*w^3 - 46*w - 6],
[239, 239, -w^5 + w^4 + 7*w^3 - 9*w^2 - 7*w + 7],
[239, 239, 4*w^5 + w^4 - 26*w^3 + 4*w^2 + 29*w - 3],
[251, 251, -w^3 - 2*w^2 + 4*w + 3],
[251, 251, 4*w^5 - 26*w^3 + 10*w^2 + 25*w - 4],
[251, 251, w^3 + 2*w^2 - 4*w - 4],
[251, 251, 3*w^5 + 2*w^4 - 17*w^3 - 2*w^2 + 15*w + 1],
[281, 281, 3*w^5 + w^4 - 19*w^3 + 3*w^2 + 19*w - 1],
[281, 281, 4*w^5 + 2*w^4 - 26*w^3 - 4*w^2 + 29*w + 8],
[281, 281, 4*w^5 + 2*w^4 - 24*w^3 - w^2 + 23*w + 4],
[281, 281, -3*w^5 - w^4 + 21*w^3 + w^2 - 27*w - 6],
[293, 293, -4*w^5 - 2*w^4 + 25*w^3 + 3*w^2 - 25*w - 6],
[293, 293, 4*w^5 + w^4 - 26*w^3 + 2*w^2 + 26*w + 6],
[307, 307, -3*w^5 - w^4 + 19*w^3 - 2*w^2 - 18*w + 2],
[307, 307, 3*w^5 + w^4 - 19*w^3 + w^2 + 17*w + 3],
[349, 349, -5*w^5 - 2*w^4 + 32*w^3 + w^2 - 34*w - 5],
[349, 349, -6*w^5 - 3*w^4 + 39*w^3 + 4*w^2 - 46*w - 7],
[349, 349, -5*w^5 - 2*w^4 + 32*w^3 - 33*w - 6],
[349, 349, 3*w^5 - 20*w^3 + 6*w^2 + 20*w - 3],
[379, 379, -3*w^5 - 2*w^4 + 19*w^3 + 5*w^2 - 23*w - 5],
[379, 379, 7*w^5 + 2*w^4 - 45*w^3 + 4*w^2 + 47*w + 4],
[379, 379, -w^5 + w^4 + 8*w^3 - 6*w^2 - 10*w],
[379, 379, -3*w^5 - 2*w^4 + 18*w^3 + 4*w^2 - 18*w - 7],
[379, 379, -2*w^5 + 14*w^3 - 2*w^2 - 15*w - 4],
[379, 379, 2*w^5 + 2*w^4 - 12*w^3 - 5*w^2 + 15*w],
[419, 419, -w^4 + 6*w^2 - 2*w - 1],
[419, 419, 5*w^5 + w^4 - 32*w^3 + 6*w^2 + 33*w + 1],
[421, 421, 5*w^5 + 2*w^4 - 33*w^3 - 2*w^2 + 37*w + 9],
[421, 421, 2*w^2 + w - 4],
[433, 433, 3*w^5 + 2*w^4 - 19*w^3 - 4*w^2 + 21*w + 7],
[433, 433, -7*w^5 - 3*w^4 + 45*w^3 + w^2 - 51*w - 6],
[433, 433, -4*w^5 - 2*w^4 + 26*w^3 + 2*w^2 - 31*w - 6],
[433, 433, -4*w^5 + 27*w^3 - 8*w^2 - 28*w],
[449, 449, 5*w^5 + w^4 - 33*w^3 + 5*w^2 + 36*w - 1],
[449, 449, -2*w^5 - w^4 + 12*w^3 + w^2 - 12*w - 5],
[461, 461, -8*w^5 - 3*w^4 + 51*w^3 - 2*w^2 - 55*w - 3],
[461, 461, 7*w^5 + 3*w^4 - 44*w^3 + w^2 + 46*w + 3],
[463, 463, 3*w^5 - 21*w^3 + 5*w^2 + 23*w + 1],
[463, 463, -4*w^5 - w^4 + 26*w^3 - 2*w^2 - 26*w - 5],
[463, 463, -6*w^5 - 3*w^4 + 38*w^3 + 3*w^2 - 40*w - 8],
[463, 463, 8*w^5 + 4*w^4 - 51*w^3 - 4*w^2 + 56*w + 9],
[491, 491, -3*w^5 + 19*w^3 - 7*w^2 - 16*w + 3],
[491, 491, 2*w^5 - 14*w^3 + 3*w^2 + 16*w + 4],
[491, 491, w^5 + w^4 - 7*w^3 - 5*w^2 + 11*w + 4],
[491, 491, -4*w^5 - 2*w^4 + 25*w^3 + 3*w^2 - 26*w - 8],
[503, 503, 2*w^5 + 2*w^4 - 11*w^3 - 5*w^2 + 11*w + 3],
[503, 503, -3*w^5 + 19*w^3 - 8*w^2 - 16*w + 5],
[503, 503, 7*w^5 + 2*w^4 - 45*w^3 + 4*w^2 + 46*w + 4],
[503, 503, -w^5 + 8*w^3 - 13*w - 2],
[547, 547, -5*w^5 - 3*w^4 + 32*w^3 + 6*w^2 - 37*w - 6],
[547, 547, -3*w^5 - 2*w^4 + 20*w^3 + 6*w^2 - 25*w - 6],
[587, 587, -2*w^5 - w^4 + 14*w^3 + w^2 - 20*w - 1],
[587, 587, -2*w^5 + 13*w^3 - 4*w^2 - 11*w + 2],
[587, 587, 7*w^5 + 3*w^4 - 45*w^3 - w^2 + 48*w + 7],
[587, 587, 3*w^5 + 2*w^4 - 18*w^3 - 4*w^2 + 17*w + 8],
[601, 601, 4*w^5 + 2*w^4 - 25*w^3 - 2*w^2 + 24*w + 5],
[601, 601, -2*w^5 - w^4 + 14*w^3 + 3*w^2 - 20*w - 5],
[601, 601, 3*w^5 + w^4 - 20*w^3 - w^2 + 22*w + 7],
[601, 601, 3*w^5 + w^4 - 20*w^3 - w^2 + 22*w + 5],
[617, 617, -4*w^5 - w^4 + 26*w^3 - 3*w^2 - 26*w - 4],
[617, 617, -4*w^5 - w^4 + 26*w^3 - 2*w^2 - 26*w - 4],
[617, 617, -5*w^5 - 2*w^4 + 33*w^3 + w^2 - 37*w - 7],
[617, 617, 2*w^5 + w^4 - 13*w^3 - w^2 + 16*w - 2],
[617, 617, -4*w^5 - 3*w^4 + 24*w^3 + 6*w^2 - 26*w - 4],
[617, 617, w^5 + w^4 - 6*w^3 - 3*w^2 + 6*w - 1],
[643, 643, -3*w^5 + 20*w^3 - 7*w^2 - 22*w + 1],
[643, 643, -3*w^5 - 2*w^4 + 20*w^3 + 5*w^2 - 25*w - 6],
[643, 643, 2*w^5 + w^4 - 12*w^3 + 13*w - 2],
[643, 643, -7*w^5 - 3*w^4 + 45*w^3 - 50*w - 4],
[673, 673, -6*w^5 - 2*w^4 + 39*w^3 - 3*w^2 - 42*w],
[673, 673, -3*w^5 - w^4 + 21*w^3 - 27*w - 1],
[701, 701, 2*w^3 + 2*w^2 - 7*w - 2],
[701, 701, 4*w^5 + 2*w^4 - 27*w^3 - 4*w^2 + 34*w + 6],
[727, 727, 7*w^5 + 3*w^4 - 46*w^3 - 2*w^2 + 53*w + 7],
[727, 727, 3*w^5 + w^4 - 19*w^3 + 19*w + 5],
[727, 727, -2*w^5 - w^4 + 11*w^3 - 2*w^2 - 8*w + 3],
[727, 727, -4*w^5 - 2*w^4 + 26*w^3 + w^2 - 30*w - 1],
[729, 3, -3],
[743, 743, 3*w^5 + 2*w^4 - 18*w^3 - 2*w^2 + 18*w],
[743, 743, 3*w^5 + 2*w^4 - 20*w^3 - 6*w^2 + 26*w + 7],
[757, 757, 3*w^5 + w^4 - 19*w^3 + 2*w^2 + 22*w],
[757, 757, 4*w^5 + 2*w^4 - 25*w^3 - w^2 + 28*w],
[757, 757, -w^5 + w^4 + 7*w^3 - 8*w^2 - 7*w + 3],
[757, 757, -3*w^5 + 20*w^3 - 7*w^2 - 23*w + 4],
[769, 769, 5*w^5 + 2*w^4 - 33*w^3 - 2*w^2 + 37*w + 6],
[769, 769, 8*w^5 + 3*w^4 - 51*w^3 + 3*w^2 + 54*w + 2],
[797, 797, 7*w^5 + 3*w^4 - 44*w^3 + 47*w + 3],
[797, 797, -2*w^5 + 13*w^3 - 5*w^2 - 15*w + 3],
[797, 797, -4*w^5 - w^4 + 26*w^3 - 4*w^2 - 30*w + 1],
[797, 797, 4*w^5 - 26*w^3 + 9*w^2 + 26*w - 1],
[811, 811, 6*w^5 + w^4 - 39*w^3 + 9*w^2 + 41*w - 2],
[811, 811, 4*w^5 + w^4 - 26*w^3 + 4*w^2 + 26*w],
[811, 811, -2*w^5 + 15*w^3 - 3*w^2 - 21*w],
[811, 811, 7*w^5 + 3*w^4 - 44*w^3 + w^2 + 45*w + 3],
[827, 827, w^5 - 5*w^3 + 4*w^2 - w - 2],
[827, 827, 3*w^5 + 2*w^4 - 19*w^3 - 4*w^2 + 24*w + 5],
[827, 827, -4*w^5 - 2*w^4 + 24*w^3 - 21*w - 1],
[827, 827, -3*w^5 - w^4 + 20*w^3 - 21*w - 4],
[827, 827, w^5 + 2*w^4 - 5*w^3 - 8*w^2 + 4*w + 7],
[827, 827, -5*w^5 - 3*w^4 + 31*w^3 + 4*w^2 - 35*w - 5],
[839, 839, -2*w^5 - 2*w^4 + 13*w^3 + 7*w^2 - 18*w - 8],
[839, 839, 5*w^5 + 3*w^4 - 30*w^3 - 4*w^2 + 29*w + 5],
[841, 29, -4*w^5 - 3*w^4 + 25*w^3 + 7*w^2 - 29*w - 7],
[841, 29, -5*w^5 - 2*w^4 + 33*w^3 - 38*w - 2],
[853, 853, 3*w^5 - 20*w^3 + 5*w^2 + 20*w + 1],
[853, 853, -6*w^5 - 3*w^4 + 38*w^3 + 4*w^2 - 41*w - 10],
[881, 881, -8*w^5 - 4*w^4 + 51*w^3 + 4*w^2 - 57*w - 8],
[881, 881, -7*w^5 - 3*w^4 + 44*w^3 - 47*w - 6],
[881, 881, w^5 + w^4 - 7*w^3 - 5*w^2 + 10*w + 3],
[881, 881, -7*w^5 - 2*w^4 + 45*w^3 - 6*w^2 - 48*w],
[883, 883, -4*w^5 + 26*w^3 - 9*w^2 - 24*w + 2],
[883, 883, 5*w^5 + 3*w^4 - 30*w^3 - 4*w^2 + 29*w + 6],
[911, 911, -3*w^5 - 2*w^4 + 18*w^3 + 3*w^2 - 18*w - 6],
[911, 911, 4*w^5 + 3*w^4 - 26*w^3 - 8*w^2 + 32*w + 7],
[911, 911, 7*w^5 + 3*w^4 - 45*w^3 - w^2 + 49*w + 8],
[911, 911, -8*w^5 - 4*w^4 + 51*w^3 + 3*w^2 - 57*w - 5],
[937, 937, -2*w^5 + w^4 + 14*w^3 - 9*w^2 - 14*w + 3],
[937, 937, 5*w^5 + 3*w^4 - 31*w^3 - 5*w^2 + 34*w + 7],
[953, 953, -8*w^5 - 3*w^4 + 52*w^3 - 2*w^2 - 58*w - 2],
[953, 953, -4*w^5 - 2*w^4 + 27*w^3 + 3*w^2 - 33*w - 4],
[967, 967, w^4 - 5*w^2 + 2*w - 1],
[967, 967, 7*w^5 + 3*w^4 - 45*w^3 - w^2 + 50*w + 3]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

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

heckeEigenvaluesArray := [1, 5, -1, 2, -2, 6, 4, -4, 4, 4, -2, 8, -8, 0, -16, 12, -12, -12, -12, -6, 18, -6, -2, -6, 14, -4, 12, 0, -8, 8, 0, -14, -14, -14, 10, -6, 2, 18, 10, -4, 4, 20, 20, 24, 16, -8, -8, 8, 0, 0, 0, -20, -4, 28, -12, 14, -2, -26, -10, 22, -18, 20, 12, 2, 30, -2, -14, 20, -4, -4, 20, -20, 4, 12, -4, -6, -22, 2, -18, -14, -2, 30, 6, -14, -30, 16, 24, -8, -8, 36, -12, -20, -20, 16, 0, -40, -40, -20, 20, 12, 28, 36, -36, 10, 22, 26, -18, -38, -22, 26, -18, -30, 6, 12, 28, 12, -44, -26, -34, -2, -2, 16, 16, 0, 48, -26, 40, -24, 22, 22, -42, -26, 34, 50, -30, -2, 54, -14, -28, 20, 28, -36, 36, -4, -4, -28, 12, -28, 24, 24, 26, -22, -22, 10, -14, -42, -42, 50, -52, -20, 8, -40, -24, -8, -26, 22, -6, 18, -8, -56];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {13,13,-w^2+3}>] := 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;