/*
  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![3, 5, -4, -2, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {9, 3, w^3 - 2*w^2 - 3*w + 3}>;

primesArray := [
[3, 3, w],
[3, 3, w - 2],
[3, 3, w - 1],
[16, 2, 2],
[23, 23, -w^3 + 4*w + 1],
[25, 5, w^3 - 2*w^2 - 2*w + 2],
[25, 5, w^3 - w^2 - 3*w + 1],
[29, 29, w^3 - 2*w^2 - 4*w + 4],
[29, 29, w^3 - w^2 - 5*w + 1],
[43, 43, -w^3 + 2*w^2 + 3*w - 2],
[43, 43, w^3 - w^2 - 4*w + 2],
[61, 61, -w^2 + 2*w + 4],
[61, 61, w^2 - 5],
[79, 79, 3*w^3 - 7*w^2 - 8*w + 16],
[79, 79, -w^3 + w^2 + 6*w - 4],
[101, 101, -w^3 + 3*w^2 + w - 7],
[101, 101, w^3 - 4*w - 4],
[103, 103, 2*w^3 - w^2 - 8*w - 1],
[103, 103, 2*w^3 - 5*w^2 - 4*w + 8],
[107, 107, 2*w^2 - 3*w - 4],
[107, 107, 3*w - 5],
[107, 107, w^3 - 4*w^2 + 10],
[107, 107, 2*w^2 - w - 5],
[113, 113, -w^3 + 4*w^2 + w - 11],
[113, 113, -2*w^3 + 4*w^2 + 5*w - 5],
[113, 113, 2*w^3 - 2*w^2 - 7*w + 2],
[113, 113, w^3 + w^2 - 6*w - 7],
[121, 11, w^3 - 3*w^2 - 4*w + 8],
[121, 11, w^3 - 7*w - 2],
[127, 127, w^2 - 4*w + 2],
[127, 127, w^3 - w^2 - 2*w - 2],
[127, 127, -w^3 + 2*w^2 + w - 4],
[127, 127, -2*w^3 + 5*w^2 + 3*w - 8],
[131, 131, 2*w^3 - 2*w^2 - 9*w + 2],
[131, 131, -w^3 - w^2 + 7*w + 5],
[157, 157, w^3 - w^2 - 7*w + 8],
[157, 157, w^3 - w^2 - 5*w + 4],
[169, 13, 2*w^2 - 2*w - 5],
[173, 173, -w^3 + 7*w - 1],
[173, 173, -w^3 - w^2 + 6*w + 4],
[179, 179, -2*w^3 + 4*w^2 + 5*w - 8],
[179, 179, -2*w^3 + 2*w^2 + 7*w + 1],
[181, 181, -w^3 + 4*w^2 - 7],
[181, 181, w^2 - 4*w + 5],
[191, 191, 3*w^3 - 7*w^2 - 9*w + 20],
[191, 191, -w^3 + 4*w + 5],
[191, 191, -w^3 + 3*w^2 + w - 8],
[191, 191, 3*w^3 - 3*w^2 - 12*w + 2],
[199, 199, 2*w^2 - 7],
[199, 199, 3*w^3 - 6*w^2 - 9*w + 13],
[233, 233, w^3 - 3*w^2 - 4*w + 11],
[233, 233, 3*w - 7],
[251, 251, 3*w^3 - 7*w^2 - 7*w + 13],
[251, 251, 4*w^3 - w^2 - 20*w - 11],
[251, 251, -w^3 + w^2 + 5*w - 7],
[251, 251, -2*w^3 + 4*w^2 + 8*w - 11],
[257, 257, w^2 - 2*w - 7],
[257, 257, w^2 - 8],
[269, 269, 2*w^3 - 3*w^2 - 8*w + 2],
[269, 269, 2*w^3 - 3*w^2 - 8*w + 7],
[283, 283, -2*w^3 + 4*w^2 + 4*w - 7],
[283, 283, 2*w^3 - 2*w^2 - 6*w - 1],
[289, 17, -w^2 + w - 2],
[289, 17, w^2 - w - 7],
[311, 311, -w^3 + 2*w^2 + 3*w - 8],
[311, 311, -w^3 + w^2 + 4*w + 4],
[313, 313, -w^3 + 2*w^2 + 5*w - 5],
[313, 313, w^3 - w^2 - 6*w + 1],
[337, 337, 3*w - 1],
[337, 337, 3*w - 2],
[347, 347, -2*w^3 - w^2 + 11*w + 11],
[347, 347, -2*w^3 + 7*w^2 + 3*w - 19],
[367, 367, 4*w^3 - 8*w^2 - 13*w + 19],
[367, 367, 4*w^3 - 4*w^2 - 17*w - 2],
[373, 373, w^2 - 3*w - 5],
[373, 373, w^2 + w - 7],
[433, 433, 2*w^3 - 3*w^2 - 5*w + 1],
[433, 433, -2*w^3 + 3*w^2 + 5*w - 5],
[443, 443, w^3 - 7*w + 2],
[443, 443, 3*w^3 - 6*w^2 - 12*w + 20],
[491, 491, -2*w^3 + 3*w^2 + 8*w - 8],
[491, 491, 2*w^3 - 3*w^2 - 11*w + 14],
[521, 521, w^3 + w^2 - 8*w - 7],
[521, 521, w^3 - 5*w^2 + 2*w + 10],
[521, 521, w^3 + 2*w^2 - 5*w - 8],
[521, 521, w^3 - 4*w^2 - 3*w + 13],
[523, 523, 3*w^3 - 5*w^2 - 11*w + 11],
[523, 523, w^3 + w^2 - 6*w - 1],
[529, 23, 3*w^2 - 3*w - 10],
[547, 547, w^3 - 3*w^2 - 4*w + 14],
[547, 547, -2*w^3 + 2*w^2 + 7*w - 5],
[547, 547, w^3 - 4*w + 4],
[547, 547, 3*w^3 - 9*w^2 - 6*w + 22],
[563, 563, w^3 - 2*w^2 - 6*w + 8],
[563, 563, -3*w^3 + 4*w^2 + 11*w - 7],
[563, 563, 3*w^3 - 5*w^2 - 10*w + 5],
[563, 563, 3*w^3 - 8*w^2 - 3*w + 10],
[571, 571, 2*w^3 - 3*w^2 - 5*w + 4],
[571, 571, -2*w^3 + 3*w^2 + 5*w - 2],
[599, 599, 2*w^3 - 2*w^2 - 5*w + 1],
[599, 599, -2*w^3 + 4*w^2 + 3*w - 4],
[641, 641, -2*w^2 + 7*w - 7],
[641, 641, 2*w^2 + w - 8],
[641, 641, 2*w^2 - 5*w - 5],
[641, 641, w^3 - 4*w^2 - 2*w + 16],
[647, 647, -w^3 + 3*w^2 - 2*w - 4],
[647, 647, w^3 - w - 4],
[653, 653, -2*w^3 + 4*w^2 + 9*w - 13],
[653, 653, 3*w^3 - 7*w^2 - 6*w + 11],
[673, 673, w^3 - 2*w^2 - 5*w + 11],
[673, 673, -2*w^3 + 2*w^2 + 10*w - 5],
[673, 673, 2*w^3 - 4*w^2 - 8*w + 5],
[673, 673, 2*w^3 - 7*w^2 - w + 13],
[677, 677, -w^3 + 2*w^2 + 6*w - 2],
[677, 677, 3*w^3 - 7*w^2 - 8*w + 13],
[677, 677, 3*w^3 - 2*w^2 - 13*w - 1],
[677, 677, -w^3 + w^2 + 7*w - 5],
[701, 701, w^2 - 5*w - 1],
[701, 701, -w^3 + 4*w^2 + 3*w - 7],
[701, 701, -w^3 - w^2 + 8*w + 1],
[701, 701, w^2 + 3*w - 5],
[719, 719, -2*w^3 + w^2 + 7*w + 5],
[719, 719, 2*w^3 - 5*w^2 - 3*w + 11],
[727, 727, 2*w^2 - 4*w - 11],
[727, 727, -3*w^3 + w^2 + 10*w + 5],
[751, 751, 4*w^3 - 12*w^2 - 7*w + 28],
[751, 751, 2*w^3 - 6*w^2 - 5*w + 20],
[757, 757, -w^3 + 2*w^2 - 5],
[757, 757, w^3 - w^2 - w - 4],
[797, 797, -4*w^3 + 7*w^2 + 13*w - 11],
[797, 797, 4*w^3 - 5*w^2 - 17*w + 2],
[797, 797, -4*w^3 + 7*w^2 + 15*w - 16],
[797, 797, 4*w^3 - 5*w^2 - 15*w + 5],
[809, 809, -w^3 + 4*w^2 + 3*w - 16],
[809, 809, 3*w^3 - 8*w^2 - 10*w + 26],
[841, 29, 3*w^2 - 3*w - 8],
[857, 857, 4*w^3 - 2*w^2 - 21*w - 7],
[857, 857, -w^3 + 2*w^2 - w + 4],
[881, 881, -w^3 + 2*w^2 + 7*w - 10],
[881, 881, w^3 - 5*w^2 - w + 13],
[881, 881, w^3 + 2*w^2 - 8*w - 8],
[881, 881, -w^3 + w^2 + 8*w + 2],
[887, 887, w^3 - w^2 - 5*w - 5],
[887, 887, -w^3 + 2*w^2 + 4*w - 10],
[907, 907, -3*w^3 + 3*w^2 + 9*w - 5],
[907, 907, 3*w^3 - 6*w^2 - 6*w + 4],
[919, 919, 4*w^3 - 3*w^2 - 16*w - 2],
[919, 919, -4*w^3 + 9*w^2 + 10*w - 17],
[937, 937, -3*w^3 + w^2 + 12*w + 7],
[937, 937, -3*w^3 + 8*w^2 + 5*w - 17],
[953, 953, -2*w^3 + 6*w^2 + 2*w - 13],
[953, 953, -w^3 + 3*w^2 + 4*w - 2],
[953, 953, w^3 - 7*w + 4],
[953, 953, 2*w^3 - 8*w - 7],
[961, 31, 3*w^3 - 4*w^2 - 13*w + 1],
[961, 31, 3*w^3 - 5*w^2 - 12*w + 13],
[991, 991, 3*w^3 - w^2 - 15*w - 4],
[991, 991, -w^3 - 3*w^2 + 7*w + 5],
[991, 991, w^3 - 6*w^2 + 2*w + 8],
[991, 991, 3*w^3 - 8*w^2 - 8*w + 17]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^3 + 2*x^2 - 5*x - 4;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [1, 1, e, e^2 + e - 3, e^2 + 2*e, -2*e - 2, -e^2 + 6, -e - 2, -2*e + 2, 2*e^2 + 4*e - 4, -2*e^2 - e + 8, e^2 + 4*e - 6, -2*e^2 - 5*e + 6, -3*e^2 - 6*e + 8, 2*e, 3*e^2 + 6*e - 6, 4*e^2 + 4*e - 14, 2*e^2 + e - 4, -2*e^2 - 4*e + 8, -2*e^2 - 4*e + 12, -e^2 - 6*e + 4, -2*e^2 - 4*e + 12, -2*e^2 - 4*e + 12, -2*e^2 - 2*e + 10, -4*e^2 - 4*e + 14, -e^2 + 4*e + 6, 2*e^2 + 5*e - 10, 4*e - 2, -2*e^2 - 8*e + 6, 4*e^2 + 2*e - 16, 3*e^2 + 4*e - 8, -16, -e - 12, -2*e^2 + 20, 4*e^2 + 3*e - 16, -2*e^2 - 9*e + 10, 2*e + 6, -2*e^2 - 5*e + 6, 6*e + 6, -4*e^2 - e + 26, 4*e^2 - e - 24, -3*e^2 - 2*e + 20, 3*e^2 - 10, e^2 + 6*e - 2, 10*e + 8, e^2 + 6*e + 8, -2*e^2 + 8, -4*e^2 - 10*e + 8, -6*e^2 - 4*e + 24, 2*e^2 + 9*e - 12, -6*e + 6, 4*e^2 + e - 14, -2*e^2 - 4, -3*e^2 + 12, -4*e^2 + 28, e^2 + 4*e + 4, -e^2 - 6*e + 10, -2*e^2 + 2, -2*e^2 - 2*e + 22, -4*e^2 - 11*e + 18, -4*e^2 - 8*e + 20, 2*e^2 - 6*e - 12, -6*e^2 - 16*e + 18, 2*e^2 - 2*e - 22, -9*e - 12, 2*e^2 - 8, 4*e^2 + 4*e - 18, -2*e - 2, 4*e^2 + 8*e - 10, 2*e^2 + 4*e - 34, 2*e^2 - 5*e - 24, -2*e^2 - 12*e - 4, -4*e^2 - 6*e + 24, 5*e^2 + 6*e - 24, -2*e - 14, -4*e^2 - 8*e + 2, 4*e^2 + 12*e - 2, 6*e^2 + 12*e - 34, -8*e^2 - 8*e + 28, -5*e^2 - 16*e + 12, 10*e^2 + 18*e - 28, -4*e^2 - 6*e + 4, 6*e^2 + 14*e - 26, -8*e - 22, 6*e^2 + 4*e - 22, 12*e^2 + 12*e - 42, -6*e^2 + 44, 5*e^2 + 10*e - 28, -4*e^2 - 2*e + 26, 2*e^2 + 3*e, 2*e - 36, -5*e^2 + 6*e + 28, 6*e^2 + 12*e - 28, 6*e^2 + 10*e - 4, 8*e^2 + 6*e - 20, -2*e^2 + 2*e + 12, -2*e^2 + 20, 2*e^2 + 6*e + 12, 3*e^2 + 8*e - 12, -10*e^2 - 18*e + 40, -6*e - 24, -4*e^2 - 2*e - 6, -4*e^2 + 2*e + 14, 4*e + 14, 5*e^2 + 4*e - 30, 6*e - 24, -2*e^2 + 32, -e^2 - 2*e - 6, 8*e^2 + 22*e - 30, -e^2 - 2*e - 22, -6*e^2 + 2*e + 42, 6*e^2 + 6*e - 22, -11*e^2 - 12*e + 34, 2*e^2 - 4*e - 22, -12*e^2 - 10*e + 46, 4*e^2 + 12*e - 10, -4*e^2 + 8*e + 26, 6*e^2 + 4*e - 22, 6*e^2 + 16*e - 34, -4*e^2 - 9*e + 10, -e^2 + 8*e + 2, -8*e^2 - 14*e + 40, 8*e^2 - 2*e - 48, -2*e^2 - 9*e + 4, -16*e, -6*e^2 - 4*e, -6*e^2 - 12*e + 32, 6*e^2 + 2*e - 30, -2*e^2 - 2*e - 30, -2*e^2 - 5*e + 22, -3*e^2 - 8*e + 14, 14*e + 22, -4*e^2 + 34, -2*e^2 - 12*e - 10, 4*e^2 - 3*e + 2, -2*e - 14, 4*e^2 - 2*e - 26, -4*e^2 - 16*e + 14, -2*e^2 + 14, 8*e^2 + 16*e - 30, -2*e^2 + 12*e + 2, -4*e^2 + 4*e + 30, -2*e^2 + 12*e + 8, 8*e^2 + e - 36, 4*e^2 - 12*e - 44, -10*e^2 - 14*e + 44, -2*e^2 - 9*e - 20, -2*e - 8, 4*e^2 + 12*e - 2, -4*e^2 - 11*e + 2, 6*e + 30, 4*e^2 + 12*e - 22, 6*e^2 + 19*e - 10, -10*e^2 - 9*e + 34, -4*e^2 - 4*e + 34, -5*e^2 + 6*e + 10, 2*e^2 - 12*e, 11*e^2 + 2*e - 56, 6*e^2 + 14*e - 24, -6*e^2 - 16*e];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {3, 3, w}>] := -1;
ALEigenvalues[ideal<ZF | {3, 3, w - 2}>] := -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;