/*
  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, 2, 11, -5, -2, 1];
F<w> := NumberField(g);
ZF := Integers(F);

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

primesArray := [
[13, 13, 3*w^5 - 2*w^4 - 17*w^3 + 10*w^2 + 16*w - 3],
[13, 13, w^5 - w^4 - 5*w^3 + 5*w^2 + 3*w - 3],
[13, 13, 2*w^5 - w^4 - 12*w^3 + 4*w^2 + 13*w],
[13, 13, -w^2 + 3],
[41, 41, -2*w^5 + w^4 + 11*w^3 - 4*w^2 - 10*w - 1],
[41, 41, -4*w^5 + 3*w^4 + 23*w^3 - 14*w^2 - 22*w + 4],
[41, 41, w^5 - 6*w^3 + w^2 + 7*w - 2],
[41, 41, -3*w^5 + w^4 + 18*w^3 - 4*w^2 - 20*w - 1],
[43, 43, 3*w^5 - 2*w^4 - 17*w^3 + 11*w^2 + 16*w - 6],
[43, 43, -2*w^5 + w^4 + 12*w^3 - 6*w^2 - 14*w + 5],
[49, 7, 2*w^5 - w^4 - 12*w^3 + 5*w^2 + 13*w - 4],
[64, 2, -2],
[71, 71, -2*w^5 + w^4 + 12*w^3 - 4*w^2 - 12*w - 1],
[71, 71, -3*w^5 + 2*w^4 + 17*w^3 - 9*w^2 - 15*w],
[83, 83, 4*w^5 - 2*w^4 - 24*w^3 + 9*w^2 + 26*w - 1],
[83, 83, -4*w^5 + 2*w^4 + 23*w^3 - 9*w^2 - 23*w],
[97, 97, -w^5 + 7*w^3 + w^2 - 11*w - 3],
[97, 97, -2*w^5 + w^4 + 11*w^3 - 6*w^2 - 9*w + 3],
[113, 113, -4*w^5 + 2*w^4 + 24*w^3 - 9*w^2 - 27*w + 2],
[113, 113, 3*w^5 - 2*w^4 - 18*w^3 + 10*w^2 + 20*w - 6],
[113, 113, -2*w^5 + w^4 + 12*w^3 - 5*w^2 - 12*w + 3],
[113, 113, 3*w^5 - w^4 - 18*w^3 + 4*w^2 + 20*w],
[127, 127, -w^5 + w^4 + 6*w^3 - 5*w^2 - 7*w],
[127, 127, -2*w^5 + w^4 + 11*w^3 - 4*w^2 - 10*w - 2],
[127, 127, -4*w^5 + 3*w^4 + 23*w^3 - 14*w^2 - 22*w + 5],
[127, 127, -8*w^5 + 5*w^4 + 46*w^3 - 23*w^2 - 45*w + 4],
[139, 139, w^3 + w^2 - 4*w - 1],
[139, 139, 5*w^5 - 2*w^4 - 29*w^3 + 9*w^2 + 29*w + 1],
[139, 139, -w^5 + w^4 + 5*w^3 - 6*w^2 - 2*w + 5],
[139, 139, w^5 - 6*w^3 + 5*w + 2],
[167, 167, w^4 - 4*w^2 + 1],
[167, 167, -3*w^5 + w^4 + 18*w^3 - 4*w^2 - 21*w - 1],
[169, 13, -3*w^5 + w^4 + 18*w^3 - 5*w^2 - 20*w],
[181, 181, -w^4 + w^3 + 5*w^2 - 3*w - 3],
[181, 181, -5*w^5 + 3*w^4 + 29*w^3 - 15*w^2 - 30*w + 6],
[181, 181, -w^5 + w^4 + 5*w^3 - 6*w^2 - 2*w + 4],
[181, 181, w^3 + w^2 - 4*w - 2],
[197, 197, w^5 - w^4 - 6*w^3 + 4*w^2 + 6*w + 2],
[197, 197, -4*w^5 + 2*w^4 + 24*w^3 - 9*w^2 - 26*w + 4],
[197, 197, -2*w^5 + 2*w^4 + 12*w^3 - 11*w^2 - 12*w + 5],
[197, 197, 2*w^5 - w^4 - 12*w^3 + 6*w^2 + 13*w - 7],
[197, 197, w^5 - 6*w^3 - w^2 + 7*w + 1],
[197, 197, -4*w^5 + 3*w^4 + 24*w^3 - 14*w^2 - 25*w + 2],
[211, 211, 2*w^5 - w^4 - 13*w^3 + 5*w^2 + 18*w - 1],
[211, 211, -2*w^5 + w^4 + 12*w^3 - 4*w^2 - 12*w - 4],
[223, 223, 4*w^5 - w^4 - 23*w^3 + 5*w^2 + 24*w - 1],
[223, 223, -3*w^5 + w^4 + 17*w^3 - 4*w^2 - 15*w - 4],
[223, 223, 3*w^5 - 3*w^4 - 16*w^3 + 15*w^2 + 11*w - 5],
[223, 223, 3*w^5 - 3*w^4 - 16*w^3 + 14*w^2 + 13*w - 5],
[239, 239, -w^5 + w^4 + 6*w^3 - 6*w^2 - 7*w + 4],
[239, 239, 2*w^5 - w^4 - 12*w^3 + 4*w^2 + 12*w + 2],
[251, 251, 3*w^5 - 3*w^4 - 16*w^3 + 14*w^2 + 13*w - 4],
[251, 251, 5*w^5 - 3*w^4 - 29*w^3 + 14*w^2 + 28*w - 1],
[251, 251, 6*w^5 - 4*w^4 - 35*w^3 + 19*w^2 + 37*w - 7],
[251, 251, -8*w^5 + 5*w^4 + 46*w^3 - 24*w^2 - 46*w + 8],
[281, 281, -6*w^5 + 4*w^4 + 35*w^3 - 18*w^2 - 35*w + 3],
[281, 281, 3*w^5 - 3*w^4 - 17*w^3 + 15*w^2 + 16*w - 10],
[281, 281, -3*w^5 + w^4 + 17*w^3 - 5*w^2 - 18*w + 3],
[281, 281, 5*w^5 - 3*w^4 - 30*w^3 + 14*w^2 + 33*w - 5],
[293, 293, -w^5 + w^4 + 6*w^3 - 5*w^2 - 8*w + 3],
[307, 307, 5*w^5 - 3*w^4 - 30*w^3 + 13*w^2 + 32*w],
[307, 307, -4*w^5 + 3*w^4 + 22*w^3 - 15*w^2 - 20*w + 5],
[307, 307, 3*w^5 - 2*w^4 - 17*w^3 + 8*w^2 + 15*w],
[307, 307, -2*w^5 + w^4 + 12*w^3 - 5*w^2 - 11*w - 1],
[307, 307, 5*w^5 - 4*w^4 - 28*w^3 + 20*w^2 + 26*w - 10],
[307, 307, -4*w^5 + 2*w^4 + 23*w^3 - 8*w^2 - 22*w - 3],
[337, 337, -w^5 + w^4 + 5*w^3 - 4*w^2 - 3*w + 3],
[337, 337, 5*w^5 - 3*w^4 - 29*w^3 + 14*w^2 + 29*w],
[337, 337, 5*w^5 - 3*w^4 - 28*w^3 + 15*w^2 + 27*w - 5],
[337, 337, -4*w^5 + 4*w^4 + 22*w^3 - 20*w^2 - 19*w + 10],
[349, 349, -8*w^5 + 5*w^4 + 46*w^3 - 25*w^2 - 45*w + 9],
[349, 349, -w^4 + 2*w^3 + 5*w^2 - 7*w - 3],
[419, 419, 5*w^5 - 3*w^4 - 29*w^3 + 13*w^2 + 28*w - 1],
[419, 419, -6*w^5 + 4*w^4 + 35*w^3 - 20*w^2 - 35*w + 10],
[421, 421, -6*w^5 + 3*w^4 + 34*w^3 - 15*w^2 - 32*w + 3],
[421, 421, -5*w^5 + 3*w^4 + 29*w^3 - 14*w^2 - 28*w],
[421, 421, w^5 - 5*w^3 - w^2 + 2*w + 3],
[421, 421, -6*w^5 + 3*w^4 + 35*w^3 - 14*w^2 - 37*w + 2],
[433, 433, -4*w^5 + 4*w^4 + 22*w^3 - 19*w^2 - 19*w + 6],
[433, 433, 7*w^5 - 4*w^4 - 40*w^3 + 19*w^2 + 40*w - 6],
[449, 449, 2*w^4 - w^3 - 10*w^2 + 4*w + 6],
[449, 449, -4*w^5 + 4*w^4 + 23*w^3 - 20*w^2 - 21*w + 10],
[461, 461, -6*w^5 + 3*w^4 + 36*w^3 - 15*w^2 - 40*w + 4],
[461, 461, -7*w^5 + 5*w^4 + 40*w^3 - 23*w^2 - 39*w + 6],
[461, 461, 4*w^5 - 2*w^4 - 22*w^3 + 9*w^2 + 19*w + 1],
[461, 461, 5*w^5 - 2*w^4 - 30*w^3 + 10*w^2 + 32*w - 1],
[463, 463, 4*w^5 - 2*w^4 - 23*w^3 + 10*w^2 + 23*w],
[463, 463, -6*w^5 + 4*w^4 + 35*w^3 - 20*w^2 - 34*w + 7],
[491, 491, 7*w^5 - 5*w^4 - 41*w^3 + 24*w^2 + 41*w - 8],
[491, 491, -2*w^5 + w^4 + 12*w^3 - 4*w^2 - 15*w - 2],
[491, 491, -w^5 + 6*w^3 + w^2 - 7*w],
[491, 491, w^5 - w^4 - 6*w^3 + 4*w^2 + 6*w + 3],
[491, 491, 5*w^5 - 3*w^4 - 29*w^3 + 16*w^2 + 30*w - 9],
[491, 491, -3*w^5 + 2*w^4 + 18*w^3 - 8*w^2 - 20*w - 2],
[503, 503, 3*w^5 - 2*w^4 - 18*w^3 + 10*w^2 + 20*w - 7],
[503, 503, 6*w^5 - 3*w^4 - 35*w^3 + 14*w^2 + 37*w - 3],
[547, 547, 2*w^5 - 12*w^3 - w^2 + 15*w + 3],
[547, 547, -w^5 + 2*w^4 + 6*w^3 - 9*w^2 - 6*w + 3],
[547, 547, 7*w^5 - 4*w^4 - 41*w^3 + 19*w^2 + 41*w - 3],
[547, 547, 2*w^5 - w^4 - 13*w^3 + 6*w^2 + 17*w - 3],
[547, 547, -4*w^5 + 3*w^4 + 23*w^3 - 13*w^2 - 21*w],
[547, 547, -7*w^5 + 4*w^4 + 41*w^3 - 20*w^2 - 43*w + 7],
[587, 587, 2*w^4 - w^3 - 10*w^2 + 4*w + 7],
[587, 587, 5*w^5 - 2*w^4 - 29*w^3 + 10*w^2 + 31*w - 2],
[601, 601, w^3 + 2*w^2 - 4*w - 4],
[601, 601, w^5 - 7*w^3 - 2*w^2 + 11*w + 5],
[617, 617, -7*w^5 + 4*w^4 + 41*w^3 - 20*w^2 - 43*w + 9],
[617, 617, -2*w^5 + 13*w^3 - 16*w],
[643, 643, -2*w^5 + 2*w^4 + 12*w^3 - 8*w^2 - 13*w],
[643, 643, -3*w^5 + w^4 + 18*w^3 - 3*w^2 - 21*w - 3],
[659, 659, 5*w^5 - 2*w^4 - 30*w^3 + 8*w^2 + 33*w + 2],
[659, 659, w^5 - 6*w^3 + 2*w^2 + 7*w - 4],
[673, 673, -2*w^5 + 2*w^4 + 10*w^3 - 11*w^2 - 5*w + 6],
[673, 673, 6*w^5 - 4*w^4 - 34*w^3 + 20*w^2 + 33*w - 9],
[673, 673, -3*w^5 + 2*w^4 + 17*w^3 - 9*w^2 - 16*w + 5],
[673, 673, 7*w^5 - 4*w^4 - 40*w^3 + 20*w^2 + 38*w - 6],
[673, 673, -3*w^5 + 2*w^4 + 16*w^3 - 11*w^2 - 12*w + 6],
[673, 673, w^5 + w^4 - 6*w^3 - 6*w^2 + 8*w + 5],
[701, 701, 5*w^5 - 4*w^4 - 28*w^3 + 20*w^2 + 26*w - 9],
[701, 701, 4*w^5 - 3*w^4 - 22*w^3 + 15*w^2 + 20*w - 6],
[727, 727, -2*w^5 + 2*w^4 + 11*w^3 - 8*w^2 - 9*w - 2],
[727, 727, -3*w^5 + 3*w^4 + 18*w^3 - 15*w^2 - 20*w + 9],
[729, 3, -3],
[743, 743, -3*w^5 + w^4 + 18*w^3 - 4*w^2 - 22*w],
[743, 743, -5*w^5 + 2*w^4 + 29*w^3 - 9*w^2 - 31*w - 1],
[743, 743, -5*w^5 + 3*w^4 + 29*w^3 - 14*w^2 - 31*w + 6],
[743, 743, -3*w^5 + 3*w^4 + 17*w^3 - 14*w^2 - 17*w + 3],
[757, 757, -6*w^5 + 3*w^4 + 35*w^3 - 13*w^2 - 37*w + 2],
[757, 757, 3*w^5 - 3*w^4 - 17*w^3 + 13*w^2 + 16*w - 1],
[757, 757, -8*w^5 + 5*w^4 + 47*w^3 - 25*w^2 - 48*w + 10],
[757, 757, 4*w^5 - 2*w^4 - 25*w^3 + 10*w^2 + 29*w - 4],
[797, 797, -6*w^5 + 4*w^4 + 35*w^3 - 19*w^2 - 34*w + 3],
[797, 797, -w^5 + w^4 + 7*w^3 - 6*w^2 - 10*w + 3],
[811, 811, 5*w^5 - 4*w^4 - 28*w^3 + 18*w^2 + 25*w - 4],
[811, 811, -4*w^5 + 3*w^4 + 22*w^3 - 15*w^2 - 17*w + 7],
[827, 827, w^5 - w^4 - 5*w^3 + 5*w^2 + w - 4],
[827, 827, -6*w^5 + 3*w^4 + 36*w^3 - 14*w^2 - 39*w + 3],
[827, 827, -6*w^5 + 3*w^4 + 36*w^3 - 14*w^2 - 39*w + 2],
[827, 827, -w^5 - w^4 + 7*w^3 + 6*w^2 - 11*w - 4],
[839, 839, -2*w^5 + 2*w^4 + 10*w^3 - 10*w^2 - 6*w + 7],
[839, 839, -6*w^5 + 4*w^4 + 34*w^3 - 20*w^2 - 32*w + 5],
[841, 29, -4*w^5 + 3*w^4 + 24*w^3 - 15*w^2 - 25*w + 7],
[841, 29, 5*w^5 - 3*w^4 - 30*w^3 + 15*w^2 + 32*w - 6],
[841, 29, 5*w^5 - 2*w^4 - 30*w^3 + 10*w^2 + 33*w - 2],
[853, 853, -5*w^5 + 3*w^4 + 29*w^3 - 16*w^2 - 29*w + 9],
[853, 853, -5*w^5 + 3*w^4 + 29*w^3 - 16*w^2 - 30*w + 7],
[881, 881, -3*w^5 + 3*w^4 + 16*w^3 - 14*w^2 - 13*w + 3],
[881, 881, 4*w^5 - 2*w^4 - 24*w^3 + 9*w^2 + 25*w + 2],
[881, 881, -8*w^5 + 5*w^4 + 46*w^3 - 24*w^2 - 46*w + 9],
[881, 881, 5*w^5 - 2*w^4 - 30*w^3 + 8*w^2 + 34*w + 2],
[883, 883, 3*w^5 - w^4 - 17*w^3 + 6*w^2 + 16*w - 5],
[883, 883, 7*w^5 - 5*w^4 - 40*w^3 + 24*w^2 + 37*w - 8],
[911, 911, w^5 - 8*w^3 + 14*w],
[911, 911, -3*w^5 + 2*w^4 + 17*w^3 - 9*w^2 - 17*w - 1],
[911, 911, -8*w^5 + 5*w^4 + 46*w^3 - 25*w^2 - 44*w + 9],
[911, 911, 2*w^5 - 2*w^4 - 12*w^3 + 9*w^2 + 11*w],
[937, 937, 5*w^5 - 2*w^4 - 30*w^3 + 9*w^2 + 32*w - 1],
[937, 937, 3*w^5 - 3*w^4 - 17*w^3 + 15*w^2 + 15*w - 4],
[967, 967, -w^5 + 7*w^3 - 9*w + 2],
[967, 967, 6*w^5 - 4*w^4 - 35*w^3 + 20*w^2 + 36*w - 11]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

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

heckeEigenvaluesArray := [-3, 2, -6, 1, -10, -5, -12, -2, -10, 5, -4, -9, 8, 8, 1, -4, -5, 5, -1, 13, -12, -16, 10, 5, 0, 5, -2, -7, -2, 8, -9, -9, -11, 3, -22, -13, 12, 21, 0, 12, -10, -24, 22, -23, 17, -12, -2, 3, 3, -20, 5, 3, 1, -24, 13, 7, -22, 2, 18, 0, -2, -12, -7, -22, -12, 23, -4, -9, -26, 4, 10, 35, -2, -22, 34, 34, 10, -5, -28, 37, 32, 17, -6, -10, -10, -6, 31, 16, -34, 6, -12, 33, -4, -44, -21, -16, -22, -32, -8, -43, 19, 19, 18, 13, 25, -30, 27, 12, -23, 12, 36, 16, 21, -11, -1, -26, 16, -46, -30, 0, -20, -10, 11, 9, 19, -8, 27, -30, 20, 10, 0, 12, -48, 9, 49, 28, -2, -52, -12, 16, -54, 8, -24, 26, -26, 19, -6, -55, 19, 0, 20, -20, 0, -53, 55, -18, 39, -16, -14, -24];
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;