/*
  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, -8, 8, 6, -6, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {41,41,-w^4 - 2*w^3 + 4*w^2 + 6*w - 3}>;

primesArray := [
[7, 7, -w^5 + 5*w^3 - 5*w - 1],
[27, 3, -2*w^5 + 10*w^3 - w^2 - 10*w + 2],
[41, 41, -w^5 + 6*w^3 - w^2 - 7*w + 2],
[41, 41, w^4 - w^3 - 4*w^2 + 3*w + 1],
[41, 41, -2*w^5 + 12*w^3 - 2*w^2 - 17*w + 5],
[41, 41, w^5 - 5*w^3 + 2*w^2 + 5*w - 5],
[41, 41, -w^4 - 2*w^3 + 4*w^2 + 6*w - 3],
[41, 41, -2*w^5 + 10*w^3 - w^2 - 10*w + 3],
[43, 43, -w^5 + w^4 + 6*w^3 - 5*w^2 - 9*w + 4],
[43, 43, -w^4 - w^3 + 4*w^2 + 4*w - 3],
[43, 43, -w^4 + 3*w^2 + 1],
[43, 43, -w^3 + w^2 + 4*w - 2],
[43, 43, -w^5 + w^4 + 6*w^3 - 4*w^2 - 8*w + 3],
[43, 43, -w^2 - w + 3],
[64, 2, -2],
[83, 83, -w^5 + 6*w^3 - w^2 - 10*w + 4],
[83, 83, -2*w^5 + w^4 + 12*w^3 - 6*w^2 - 17*w + 6],
[83, 83, w^5 - 6*w^3 + 2*w^2 + 8*w - 3],
[83, 83, w^5 - 4*w^3 + w - 1],
[83, 83, -2*w^5 + 11*w^3 - w^2 - 13*w + 1],
[83, 83, w^5 - 5*w^3 + 2*w^2 + 6*w - 6],
[125, 5, w^5 + w^4 - 5*w^3 - 4*w^2 + 6*w + 2],
[125, 5, -w^5 - w^4 + 5*w^3 + 4*w^2 - 6*w - 1],
[127, 127, w^5 - w^4 - 5*w^3 + 5*w^2 + 6*w - 4],
[127, 127, -w^5 + 7*w^3 - 2*w^2 - 12*w + 5],
[127, 127, -2*w^5 + w^4 + 11*w^3 - 5*w^2 - 13*w + 5],
[127, 127, -2*w^3 + w^2 + 7*w - 2],
[127, 127, -w^5 + 6*w^3 - 7*w],
[127, 127, -w^4 - 2*w^3 + 4*w^2 + 7*w - 3],
[167, 167, -w^5 - w^4 + 4*w^3 + 4*w^2 - 1],
[167, 167, -2*w^4 + 7*w^2 + w - 3],
[167, 167, w^5 - w^4 - 4*w^3 + 3*w^2 + w + 2],
[167, 167, w^5 + w^4 - 4*w^3 - 4*w^2 + 3*w - 1],
[167, 167, 2*w^5 - 11*w^3 + w^2 + 12*w - 3],
[167, 167, 3*w^5 + w^4 - 16*w^3 - w^2 + 19*w - 6],
[169, 13, w^5 - 4*w^3 + w^2 + 2*w - 3],
[169, 13, -2*w^5 + 11*w^3 - 2*w^2 - 13*w + 4],
[169, 13, -w^5 + 7*w^3 - w^2 - 11*w + 4],
[211, 211, 3*w^5 - w^4 - 17*w^3 + 7*w^2 + 23*w - 9],
[211, 211, -3*w^3 + 10*w - 1],
[211, 211, w^5 + w^4 - 7*w^3 - 4*w^2 + 12*w - 1],
[211, 211, 3*w^5 - w^4 - 16*w^3 + 6*w^2 + 18*w - 5],
[211, 211, -2*w^5 + 12*w^3 - 3*w^2 - 16*w + 6],
[211, 211, w^3 - 2*w^2 - 4*w + 3],
[251, 251, 3*w^5 - 18*w^3 + 3*w^2 + 26*w - 8],
[251, 251, 2*w^5 - 2*w^4 - 13*w^3 + 9*w^2 + 20*w - 7],
[251, 251, w^5 + 2*w^4 - 5*w^3 - 8*w^2 + 6*w + 2],
[251, 251, 2*w^5 + w^4 - 10*w^3 - 4*w^2 + 11*w],
[251, 251, 2*w^4 - w^3 - 8*w^2 + 3*w + 3],
[251, 251, w^4 + 3*w^3 - 5*w^2 - 9*w + 3],
[293, 293, -2*w^5 - 2*w^4 + 10*w^3 + 7*w^2 - 11*w],
[293, 293, -3*w^5 - w^4 + 16*w^3 + 2*w^2 - 18*w + 2],
[293, 293, 2*w^5 - 13*w^3 + w^2 + 20*w - 5],
[293, 293, w^5 - 2*w^4 - 7*w^3 + 8*w^2 + 12*w - 5],
[293, 293, -2*w^4 - w^3 + 9*w^2 + 2*w - 7],
[293, 293, -w^5 - w^4 + 4*w^3 + 5*w^2 - 3*w - 3],
[337, 337, -w^5 + w^4 + 5*w^3 - 5*w^2 - 6*w + 3],
[337, 337, -w^5 + 7*w^3 - 2*w^2 - 12*w + 4],
[337, 337, -w^5 + 6*w^3 - 7*w - 1],
[337, 337, 2*w^5 - w^4 - 11*w^3 + 5*w^2 + 13*w - 6],
[337, 337, -w^4 - 2*w^3 + 4*w^2 + 7*w - 4],
[337, 337, w^5 + w^4 - 5*w^3 - 2*w^2 + 5*w - 4],
[379, 379, 2*w^5 - 2*w^4 - 11*w^3 + 8*w^2 + 13*w - 4],
[379, 379, w^5 - 2*w^4 - 5*w^3 + 9*w^2 + 5*w - 4],
[379, 379, 3*w^5 - 16*w^3 + w^2 + 18*w - 5],
[379, 379, w^5 - 2*w^4 - 6*w^3 + 9*w^2 + 10*w - 7],
[379, 379, -2*w^5 - 2*w^4 + 11*w^3 + 6*w^2 - 15*w + 1],
[379, 379, w^5 - 7*w^3 + 3*w^2 + 13*w - 8],
[419, 419, 2*w^5 - 10*w^3 + 3*w^2 + 11*w - 9],
[419, 419, -3*w^5 + w^4 + 18*w^3 - 7*w^2 - 25*w + 8],
[419, 419, -w^4 + 2*w^3 + 3*w^2 - 6*w + 3],
[419, 419, -2*w^5 + 9*w^3 - w^2 - 6*w + 4],
[419, 419, -w^5 + w^4 + 8*w^3 - 4*w^2 - 14*w + 5],
[419, 419, 2*w^5 + w^4 - 11*w^3 - 2*w^2 + 12*w - 1],
[421, 421, -w^3 - 2*w^2 + 4*w + 3],
[421, 421, -3*w^5 + w^4 + 17*w^3 - 6*w^2 - 23*w + 9],
[421, 421, 3*w^5 + w^4 - 17*w^3 - w^2 + 23*w - 7],
[421, 421, 3*w^5 - 15*w^3 + w^2 + 16*w - 2],
[421, 421, -2*w^4 - 3*w^3 + 9*w^2 + 9*w - 6],
[421, 421, w^5 - w^4 - 6*w^3 + 5*w^2 + 6*w - 5],
[461, 461, 2*w^5 - w^4 - 13*w^3 + 7*w^2 + 20*w - 8],
[461, 461, -2*w^5 + w^4 + 10*w^3 - 5*w^2 - 9*w + 6],
[461, 461, 2*w^4 + w^3 - 9*w^2 - 3*w + 4],
[461, 461, w^5 - w^4 - 8*w^3 + 4*w^2 + 15*w - 5],
[461, 461, 2*w^5 + w^4 - 10*w^3 - w^2 + 11*w - 7],
[461, 461, w^5 - w^4 - 4*w^3 + 4*w^2 + 3*w],
[463, 463, w^5 - 2*w^4 - 7*w^3 + 8*w^2 + 10*w - 6],
[463, 463, w^5 - w^4 - 6*w^3 + 5*w^2 + 9*w - 8],
[463, 463, w^5 - w^4 - 6*w^3 + 4*w^2 + 8*w + 1],
[463, 463, -3*w^5 - w^4 + 15*w^3 + 3*w^2 - 15*w],
[463, 463, -w^4 + 3*w^2 - 3],
[463, 463, -w^4 - w^3 + 4*w^2 + 4*w + 1],
[503, 503, w^5 - 7*w^3 + 12*w - 3],
[503, 503, -3*w^5 - w^4 + 17*w^3 + w^2 - 23*w + 6],
[503, 503, 2*w^5 - 10*w^3 + 3*w^2 + 10*w - 7],
[503, 503, -w^5 + w^4 + 8*w^3 - 5*w^2 - 15*w + 8],
[503, 503, -3*w^5 + 15*w^3 - w^2 - 16*w + 3],
[503, 503, -2*w^5 + w^4 + 10*w^3 - 5*w^2 - 10*w + 7],
[547, 547, 2*w^5 + w^4 - 11*w^3 - 4*w^2 + 14*w - 1],
[547, 547, 3*w^5 - w^4 - 17*w^3 + 7*w^2 + 21*w - 9],
[547, 547, -3*w^5 + w^4 + 17*w^3 - 5*w^2 - 21*w + 5],
[547, 547, 2*w^4 + w^3 - 9*w^2 - 2*w + 4],
[547, 547, 2*w^5 + w^4 - 10*w^3 - 3*w^2 + 12*w - 3],
[547, 547, -w^5 - w^4 + 6*w^3 + 4*w^2 - 10*w - 3],
[587, 587, 3*w^5 - w^4 - 16*w^3 + 7*w^2 + 19*w - 10],
[587, 587, -2*w^5 - 2*w^4 + 9*w^3 + 7*w^2 - 7*w - 3],
[587, 587, w^5 - 2*w^4 - 6*w^3 + 10*w^2 + 8*w - 7],
[587, 587, 2*w^5 + w^4 - 12*w^3 - w^2 + 16*w - 5],
[587, 587, -3*w^5 + 17*w^3 - w^2 - 22*w + 5],
[587, 587, w^5 + w^4 - 7*w^3 - 3*w^2 + 13*w],
[631, 631, -w^5 + 2*w^4 + 8*w^3 - 10*w^2 - 14*w + 9],
[631, 631, 3*w^5 + w^4 - 16*w^3 - w^2 + 20*w - 6],
[631, 631, -w^5 + 2*w^4 + 7*w^3 - 9*w^2 - 10*w + 6],
[631, 631, 2*w^5 + 2*w^4 - 11*w^3 - 6*w^2 + 14*w - 1],
[631, 631, -w^5 - w^4 + 6*w^3 + 5*w^2 - 8*w - 3],
[631, 631, 3*w^5 - 16*w^3 + 2*w^2 + 20*w - 6],
[673, 673, -w^5 - w^4 + 5*w^3 + w^2 - 5*w + 5],
[673, 673, -w^4 + 6*w^2 - 6],
[673, 673, 2*w^5 - 13*w^3 + 3*w^2 + 21*w - 8],
[673, 673, w^5 - 2*w^4 - 5*w^3 + 9*w^2 + 6*w - 6],
[673, 673, -3*w^5 + w^4 + 16*w^3 - 5*w^2 - 18*w + 5],
[673, 673, -2*w^4 - 3*w^3 + 8*w^2 + 10*w - 5],
[757, 757, 2*w^4 - w^3 - 8*w^2 + 3*w + 5],
[757, 757, -3*w^5 + 18*w^3 - 3*w^2 - 26*w + 10],
[757, 757, -w^5 + 6*w^3 - w^2 - 6*w + 4],
[757, 757, -2*w^5 - w^4 + 10*w^3 + 4*w^2 - 11*w - 4],
[757, 757, -3*w^5 + 15*w^3 - w^2 - 15*w + 1],
[757, 757, w^5 + w^4 - 5*w^3 - 4*w^2 + 7*w + 4],
[797, 797, w^5 - w^4 - 6*w^3 + 3*w^2 + 9*w - 3],
[797, 797, -w^5 - w^4 + 7*w^3 + 4*w^2 - 11*w],
[797, 797, 2*w^5 + w^4 - 10*w^3 - 4*w^2 + 10*w + 4],
[797, 797, 3*w^5 + w^4 - 17*w^3 - w^2 + 22*w - 5],
[797, 797, -2*w^5 + w^4 + 11*w^3 - 4*w^2 - 14*w],
[797, 797, 2*w^5 - 12*w^3 + w^2 + 18*w - 6],
[839, 839, w^5 + w^4 - 5*w^3 - w^2 + 4*w - 5],
[839, 839, -4*w^5 + w^4 + 22*w^3 - 6*w^2 - 27*w + 8],
[839, 839, 2*w^5 + w^4 - 13*w^3 - w^2 + 21*w - 6],
[839, 839, w^5 - w^4 - 7*w^3 + 4*w^2 + 9*w - 2],
[839, 839, w^5 + 2*w^4 - 2*w^3 - 8*w^2 - 5*w + 5],
[839, 839, -w^5 + 2*w^4 + 5*w^3 - 10*w^2 - 6*w + 8],
[841, 29, 2*w^5 - 13*w^3 + 2*w^2 + 19*w - 6],
[841, 29, -3*w^5 + 17*w^3 - 3*w^2 - 21*w + 7],
[841, 29, 3*w^5 - 16*w^3 + 3*w^2 + 18*w - 7],
[881, 881, 3*w^5 + w^4 - 17*w^3 - w^2 + 22*w - 7],
[881, 881, 2*w^5 - w^4 - 12*w^3 + 6*w^2 + 15*w - 8],
[881, 881, -w^5 + 7*w^3 - 2*w^2 - 11*w + 8],
[881, 881, -2*w^5 - w^4 + 10*w^3 + 2*w^2 - 10*w],
[881, 881, 2*w^5 - w^4 - 11*w^3 + 6*w^2 + 13*w - 4],
[881, 881, 2*w^5 - 12*w^3 + w^2 + 16*w - 6],
[883, 883, 2*w^5 + w^4 - 13*w^3 - 2*w^2 + 19*w - 4],
[883, 883, -2*w^5 + w^4 + 14*w^3 - 6*w^2 - 22*w + 8],
[883, 883, w^5 + w^4 - 3*w^3 - 5*w^2 + 5],
[883, 883, -4*w^5 + 22*w^3 - 4*w^2 - 27*w + 10],
[883, 883, -4*w^5 + w^4 + 23*w^3 - 8*w^2 - 30*w + 12],
[883, 883, 3*w^5 + 2*w^4 - 16*w^3 - 6*w^2 + 19*w - 1],
[967, 967, -w^5 + 2*w^4 + 7*w^3 - 9*w^2 - 13*w + 5],
[967, 967, -3*w^5 + 16*w^3 - 3*w^2 - 20*w + 9],
[967, 967, 2*w^5 - 13*w^3 + 2*w^2 + 18*w - 6],
[967, 967, 2*w^5 - 2*w^4 - 12*w^3 + 9*w^2 + 15*w - 8],
[967, 967, 3*w^5 - 16*w^3 + 4*w^2 + 18*w - 9],
[967, 967, 2*w^5 + 2*w^4 - 11*w^3 - 6*w^2 + 13*w]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

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

heckeEigenvaluesArray := [-2, -9, -3, -6, -6, -3, 1, 6, 7, -8, 1, 1, -8, 10, -11, -12, -9, -15, -9, 0, -6, 18, -3, -10, 8, 11, 2, -16, 5, -6, 21, 15, -12, 0, 3, 20, 5, -19, -5, 7, 16, -14, 10, -23, 21, 12, -15, -21, 18, 0, 0, 18, 30, -21, 12, 9, -11, 13, 19, -8, -23, -32, 4, 13, -26, -2, -20, -29, -33, -39, -24, -30, 36, 36, -19, 26, -10, -19, 38, -25, -30, 18, 24, -27, 12, -6, -1, -19, 8, 17, -19, 20, -30, 36, -6, -30, -9, -27, -22, 20, -4, -7, 26, 35, -15, 24, 24, -18, 18, -42, -1, -16, -25, -37, 29, 14, 46, -26, 28, 1, -44, 1, -29, 16, 7, -38, -14, -20, 6, -12, -54, -54, 0, -27, 36, -51, -24, 21, 3, 30, -14, -23, -5, 18, 42, -15, 15, 30, -54, 52, 34, 4, -44, -8, 46, -4, -10, 32, 14, -4, 47];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {41,41,-w^4 - 2*w^3 + 4*w^2 + 6*w - 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;