/*
  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, 3, 3, -4, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {67,67,2*w^2 - w - 4}>;

primesArray := [
[11, 11, w^4 + w^3 - 4*w^2 - 3*w + 2],
[23, 23, -w^4 + 3*w^2 + 1],
[23, 23, -w^4 + 3*w^2 + w - 2],
[23, 23, w^4 - w^3 - 3*w^2 + 3*w + 2],
[23, 23, -w^4 + w^3 + 4*w^2 - 3*w - 1],
[23, 23, -w^2 + w + 3],
[32, 2, 2],
[43, 43, -2*w^4 + w^3 + 6*w^2 - 2*w - 1],
[43, 43, -w^4 + 2*w^2 + w + 1],
[43, 43, w^3 + w^2 - 4*w - 2],
[43, 43, 2*w^4 - w^3 - 7*w^2 + 3*w + 3],
[43, 43, w^4 - w^3 - 4*w^2 + 4*w + 2],
[67, 67, 2*w^4 - 7*w^2 + 2],
[67, 67, w^4 - 2*w^3 - 3*w^2 + 6*w + 2],
[67, 67, 2*w^4 - 7*w^2 - w + 4],
[67, 67, w^4 - 2*w^3 - 4*w^2 + 6*w + 2],
[67, 67, -w^4 + w^3 + 5*w^2 - 3*w - 3],
[89, 89, w^3 + w^2 - 4*w - 1],
[89, 89, -2*w^4 + w^3 + 7*w^2 - 3*w - 2],
[89, 89, -w^4 + w^3 + 4*w^2 - 4*w - 3],
[89, 89, -w^4 + 2*w^2 + w + 2],
[89, 89, 2*w^4 - w^3 - 6*w^2 + 2*w + 2],
[109, 109, -w^3 + 2*w^2 + 3*w - 3],
[109, 109, w^4 - 4*w^2 - 2*w + 3],
[109, 109, -2*w^4 + 2*w^3 + 7*w^2 - 4*w - 3],
[109, 109, 2*w^3 - 5*w - 1],
[109, 109, -w^4 - w^3 + 5*w^2 + 2*w - 4],
[131, 131, w^4 - 3*w^3 - 2*w^2 + 7*w],
[131, 131, -w^4 + 2*w^3 + 5*w^2 - 7*w - 5],
[131, 131, 2*w^4 - 2*w^3 - 6*w^2 + 3*w + 2],
[131, 131, w^4 - 2*w^3 - 3*w^2 + 7*w],
[131, 131, 2*w^4 + w^3 - 8*w^2 - 3*w + 4],
[197, 197, 3*w^4 - w^3 - 10*w^2 + w + 5],
[197, 197, 2*w^4 - 7*w^2 + w + 1],
[197, 197, -3*w^4 + 2*w^3 + 10*w^2 - 5*w - 5],
[197, 197, -2*w^4 - w^3 + 9*w^2 + 2*w - 5],
[197, 197, -3*w^4 + 3*w^3 + 10*w^2 - 6*w - 5],
[199, 199, w^4 + 2*w^3 - 5*w^2 - 5*w + 3],
[199, 199, -2*w^4 + w^3 + 7*w^2 - 4*w - 2],
[199, 199, -w^4 + 2*w^3 + 4*w^2 - 7*w - 3],
[199, 199, 2*w^4 - 8*w^2 + w + 4],
[199, 199, 3*w^4 - w^3 - 10*w^2 + 2*w + 4],
[241, 241, w^4 - 2*w^3 - w^2 + 3*w - 3],
[241, 241, -w^4 + 3*w^3 + 2*w^2 - 9*w],
[241, 241, 3*w^4 - 11*w^2 - 2*w + 6],
[241, 241, -w^4 + 2*w^3 + 5*w^2 - 6*w - 4],
[241, 241, -2*w^4 + 2*w^3 + 9*w^2 - 5*w - 6],
[243, 3, -3],
[263, 263, -w^4 + 4*w^2 + w + 1],
[263, 263, 2*w^4 - w^3 - 8*w^2 + w + 2],
[263, 263, 3*w^4 - 3*w^3 - 9*w^2 + 7*w + 2],
[263, 263, -2*w^3 + w^2 + 4*w - 4],
[263, 263, -3*w^2 + w + 5],
[307, 307, 2*w^4 - w^3 - 6*w^2 + 3*w - 2],
[307, 307, w^4 - 2*w^2 - 2*w - 4],
[307, 307, -3*w^4 + 2*w^3 + 11*w^2 - 3*w - 6],
[307, 307, -2*w^4 + 2*w^3 + 7*w^2 - 6*w - 6],
[307, 307, -w^4 + 3*w^3 + 4*w^2 - 7*w - 3],
[331, 331, -w^4 - 2*w^3 + 3*w^2 + 7*w - 1],
[331, 331, -w^4 + 3*w^3 + 4*w^2 - 8*w - 4],
[331, 331, w^4 - w^2 - 4],
[331, 331, 3*w^4 - 2*w^3 - 9*w^2 + 5*w + 2],
[331, 331, -3*w^4 + 2*w^3 + 11*w^2 - 6*w - 5],
[353, 353, -2*w^4 + w^3 + 9*w^2 - 3*w - 5],
[353, 353, -2*w^4 - w^3 + 7*w^2 + 4*w - 4],
[353, 353, -w^4 + 3*w^3 + 3*w^2 - 9*w - 3],
[353, 353, -3*w^4 + 11*w^2 + w - 7],
[353, 353, w^4 - w^3 - 6*w^2 + 3*w + 4],
[373, 373, w^3 + 2*w^2 - 5*w - 3],
[373, 373, 2*w^4 - 3*w^3 - 6*w^2 + 6*w],
[373, 373, -w^4 + w^3 + 3*w^2 - 3*w + 3],
[373, 373, -2*w^4 + 2*w^3 + 8*w^2 - 7*w - 5],
[373, 373, -w^4 + w^3 + 4*w^2 - 3*w - 6],
[397, 397, -w^4 - w^3 + 6*w^2 + 3*w - 5],
[397, 397, 2*w^4 - 2*w^3 - 7*w^2 + 3*w + 3],
[397, 397, -w^4 + 5*w^2 + 2*w - 5],
[397, 397, w^4 + 2*w^3 - 5*w^2 - 5*w + 4],
[397, 397, -w^4 + 3*w^3 + 3*w^2 - 7*w - 2],
[419, 419, 3*w^3 - w^2 - 8*w],
[419, 419, 2*w^4 - 3*w^3 - 6*w^2 + 6*w + 1],
[419, 419, w^4 - 2*w^3 - 5*w^2 + 7*w + 4],
[419, 419, -2*w^4 - w^3 + 9*w^2 + 3*w - 7],
[419, 419, -w^4 + 3*w^3 + w^2 - 8*w + 1],
[439, 439, -w^4 + w^3 + 5*w^2 - 5*w - 5],
[439, 439, 2*w^4 - 9*w^2 - 2*w + 8],
[439, 439, -3*w^4 + 2*w^3 + 11*w^2 - 5*w - 3],
[439, 439, 2*w^4 - 2*w^3 - 8*w^2 + 3*w + 4],
[439, 439, -w^4 + 3*w^3 + 3*w^2 - 6*w - 3],
[461, 461, -w^4 + 5*w^2 + 2*w - 6],
[461, 461, -w^4 + 3*w^2 - w + 3],
[461, 461, -w^4 - 2*w^3 + 5*w^2 + 5*w - 5],
[461, 461, -2*w^4 + 7*w^2 + w - 6],
[461, 461, -w^4 + 3*w^3 + 3*w^2 - 7*w - 3],
[463, 463, 2*w^4 - 2*w^3 - 8*w^2 + 5*w + 3],
[463, 463, w^4 + w^3 - 3*w^2 - 2*w - 2],
[463, 463, 2*w^4 - 7*w^2 - 2*w + 5],
[463, 463, w^4 - 2*w^3 - 2*w^2 + 6*w + 1],
[463, 463, -w^3 + 2*w^2 + w - 5],
[571, 571, w^4 - w^3 - 3*w^2 + 2*w - 3],
[571, 571, 3*w^4 - 2*w^3 - 10*w^2 + 6*w + 2],
[571, 571, -2*w^4 + 2*w^3 + 8*w^2 - 7*w - 6],
[571, 571, 2*w^4 - 5*w^2 - 2*w - 2],
[571, 571, 3*w^4 - w^3 - 9*w^2 + 2*w + 3],
[593, 593, 3*w^4 - w^3 - 9*w^2 + w + 3],
[593, 593, -w^4 + 2*w^3 + 5*w^2 - 7*w - 3],
[593, 593, -2*w^4 + 2*w^3 + 7*w^2 - 7*w - 4],
[593, 593, w^4 - w^2 - 2*w - 4],
[593, 593, 3*w^4 - w^3 - 10*w^2 + 3*w + 2],
[617, 617, -w^4 - w^3 + 6*w^2 + 4*w - 5],
[617, 617, 3*w^3 - w^2 - 7*w],
[617, 617, 2*w^4 - w^3 - 8*w^2 + 6],
[617, 617, 2*w^4 - 3*w^3 - 7*w^2 + 6*w + 3],
[617, 617, -w^4 - 2*w^3 + 6*w^2 + 5*w - 6],
[659, 659, -w^4 + w^3 + 4*w^2 - 5*w - 4],
[659, 659, 2*w^3 + w^2 - 7*w],
[659, 659, -3*w^4 + 2*w^3 + 9*w^2 - 4*w - 4],
[659, 659, 3*w^4 - w^3 - 11*w^2 + 3*w + 3],
[659, 659, w^4 - w^2 - w - 5],
[661, 661, -5*w^4 + 2*w^3 + 18*w^2 - 4*w - 9],
[661, 661, -w^4 + 3*w^3 + 6*w^2 - 9*w - 8],
[661, 661, w^4 + 2*w^3 - 2*w^2 - 7*w - 2],
[661, 661, -w^4 + w^3 + 5*w^2 - 2*w - 9],
[661, 661, -3*w^4 + 2*w^3 + 12*w^2 - 2*w - 8],
[683, 683, -w^4 + w^3 + 4*w^2 - 5*w - 3],
[683, 683, 2*w^3 + w^2 - 7*w - 1],
[683, 683, w^4 - w^2 - w - 4],
[683, 683, 3*w^4 - 2*w^3 - 9*w^2 + 4*w + 3],
[683, 683, 3*w^4 - w^3 - 11*w^2 + 3*w + 4],
[727, 727, -5*w^4 + 3*w^3 + 19*w^2 - 7*w - 9],
[727, 727, -2*w^4 - 2*w^3 + 7*w^2 + 9*w - 4],
[727, 727, 3*w^4 + 2*w^3 - 13*w^2 - 6*w + 6],
[727, 727, w^4 - 2*w^2 + 3*w - 4],
[727, 727, -2*w^4 + 5*w^3 + 6*w^2 - 14*w - 4],
[769, 769, -2*w^4 - w^3 + 9*w^2 + 2*w - 7],
[769, 769, -w^4 + 4*w^2 + 3*w - 4],
[769, 769, -w^3 + 3*w^2 + 3*w - 4],
[769, 769, 3*w^4 - 3*w^3 - 10*w^2 + 6*w + 3],
[769, 769, 3*w^3 - 8*w - 2],
[857, 857, w^4 - 5*w^3 - 2*w^2 + 14*w - 1],
[857, 857, 4*w^4 - w^3 - 13*w^2 + 4],
[857, 857, -2*w^4 + 4*w^3 + 6*w^2 - 10*w - 5],
[857, 857, 4*w^4 + w^3 - 14*w^2 - 5*w + 6],
[857, 857, 4*w^4 + w^3 - 16*w^2 - 4*w + 8],
[859, 859, 3*w^4 - w^3 - 11*w^2 + 5*w + 5],
[859, 859, -2*w^4 + 5*w^3 + 6*w^2 - 13*w - 1],
[859, 859, -2*w^4 - 2*w^3 + 5*w^2 + 7*w + 1],
[859, 859, -w^3 + w^2 + 6*w + 1],
[859, 859, 4*w^4 - w^3 - 13*w^2 + 3*w + 4],
[881, 881, 2*w^4 + w^3 - 11*w^2 - 3*w + 11],
[881, 881, w^4 - 6*w^2 - 3*w + 5],
[881, 881, -2*w^4 + 5*w^3 + 6*w^2 - 12*w - 1],
[881, 881, -2*w^4 - 3*w^3 + 9*w^2 + 8*w - 4],
[881, 881, 3*w^4 + w^3 - 12*w^2 - 3*w + 7],
[947, 947, 2*w^4 + 3*w^3 - 7*w^2 - 8*w],
[947, 947, 2*w^4 - 3*w^3 - 6*w^2 + 4*w],
[947, 947, 2*w^4 + 2*w^3 - 7*w^2 - 8*w + 4],
[947, 947, -w^4 + 2*w^3 - 7*w + 3],
[947, 947, -4*w^4 + 5*w^3 + 14*w^2 - 12*w - 5],
[967, 967, 2*w^4 - w^3 - 9*w^2 + 3*w + 4],
[967, 967, -2*w^4 - w^3 + 7*w^2 + 4*w - 5],
[967, 967, w^4 - 2*w^2 + w - 4],
[967, 967, w^4 - w^3 - 2*w^2 - 3],
[967, 967, -2*w^4 + 3*w^3 + 6*w^2 - 8*w - 4],
[991, 991, -w^4 + 4*w^3 + 2*w^2 - 12*w - 1],
[991, 991, w^4 + 3*w^3 - 3*w^2 - 11*w - 1],
[991, 991, 2*w^3 - 2*w^2 - 5*w + 8],
[991, 991, -2*w^4 + 3*w^3 + 4*w^2 - 5*w + 3],
[991, 991, 4*w^4 - 4*w^3 - 11*w^2 + 9*w]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^2 - 24;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, -1/2*e - 5, 4, -e + 1, 1/2*e + 3, e + 2, -1/2*e - 1, 9, 4, e - 3, 4, -2*e - 2, e - 4, 1, -2*e - 3, -2, -2*e + 2, -e - 8, -5/2*e - 5, e + 8, e - 12, e - 12, e + 8, -3*e - 4, 0, -e + 7, 1/2*e - 1, 1/2*e - 9, -3*e - 2, -3/2*e - 5, e, 2, -3/2*e + 11, 18, -4*e - 4, -e + 10, -7, 7/2*e - 7, -7/2*e - 3, -e - 18, -5, -3*e - 4, e + 10, 3*e - 4, 3/2*e + 9, e - 5, e - 10, -2*e - 2, 3/2*e - 9, 3*e + 8, e - 13, e + 12, -3*e + 5, 6*e + 1, -1/2*e - 21, 4*e + 10, -2, 18, -3*e + 8, -e - 16, 7/2*e + 5, -5*e + 2, 7, 4*e + 16, -5*e - 11, 3*e - 12, e - 18, e + 2, 5*e + 4, e + 2, -2*e + 3, -6, -e - 9, 5*e + 8, e + 16, -4*e - 4, 3*e - 13, 3*e + 2, -7*e - 6, 4*e - 18, -e + 22, -10, -3/2*e - 27, 3*e + 14, 3/2*e + 27, -3*e - 14, 3*e - 6, -2*e + 14, 11/2*e + 11, -4*e + 20, -e - 26, -2*e + 26, -2*e + 6, 2*e + 20, e + 2, 5*e + 4, 3*e + 18, -16, -3*e - 2, 8*e - 4, 1/2*e + 11, 4*e - 26, -3*e - 32, 6*e + 2, 5*e + 9, 2*e - 30, -6, -7*e + 8, e - 4, e - 24, -7/2*e + 15, 4*e + 5, -2*e - 3, 11/2*e + 19, -3*e - 4, -e - 18, -2*e - 36, -7/2*e - 3, -2*e + 26, 3*e + 36, 3*e - 4, -6*e + 4, 2*e + 8, 5/2*e + 9, e + 17, 3*e + 3, -5*e + 14, 3/2*e + 1, -5*e + 3, 2*e - 16, 3*e + 22, -6*e - 10, -6*e - 10, 5*e + 20, -6*e - 8, -2*e - 6, -10, 6*e + 23, 7/2*e + 41, -42, 7/2*e + 21, -7*e - 18, -1/2*e + 29, 2*e - 14, 40, -2*e + 19, -2*e + 14, 3*e - 26, 3/2*e - 21, 7, -4*e + 10, 3*e - 24, -7/2*e - 11, -11*e, -6*e + 10, 9*e, 4*e - 25, 4*e - 20, 3/2*e - 35, 9/2*e + 39, 3*e + 17, 2*e - 36, 6*e + 16, -8, 5*e + 7, 6*e, -9/2*e + 11, 3*e - 4];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {67,67,2*w^2 - w - 4}>] := -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;