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

NN := ideal<ZF | {49, 7, 2*w^3 - 4*w^2 - 4*w + 1}>;

primesArray := [
[4, 2, w^3 - 2*w^2 - 2*w + 1],
[7, 7, -w^3 + 3*w^2 + w - 3],
[7, 7, -w^2 + w + 2],
[17, 17, -w^3 + 3*w^2 - 3],
[17, 17, -w^3 + w^2 + 4*w],
[25, 5, -w^3 + 3*w^2 + 2*w - 2],
[25, 5, -2*w^3 + 4*w^2 + 5*w - 1],
[41, 41, -w^3 + 2*w^2 + 4*w - 2],
[47, 47, -2*w^3 + 5*w^2 + 4*w - 4],
[47, 47, 2*w^3 - 4*w^2 - 5*w],
[49, 7, w^2 - 4*w - 1],
[71, 71, 2*w - 3],
[71, 71, -w^3 + w^2 + 6*w - 2],
[73, 73, -w^3 + 3*w^2 + 3*w - 5],
[73, 73, 2*w^3 - 5*w^2 - 5*w + 4],
[73, 73, -w^3 + 3*w^2 - 5],
[73, 73, w^3 - w^2 - 4*w + 2],
[79, 79, 2*w^3 - 3*w^2 - 6*w + 2],
[79, 79, 2*w^3 - 3*w^2 - 5*w],
[81, 3, -3],
[89, 89, -w^3 + w^2 + 5*w - 3],
[89, 89, w - 4],
[97, 97, -3*w^3 + 7*w^2 + 6*w - 5],
[97, 97, -2*w^3 + 3*w^2 + 7*w - 2],
[103, 103, -w - 3],
[103, 103, -w^3 + 2*w^2 + 3*w - 5],
[103, 103, -w^3 + 4*w^2 - w - 5],
[103, 103, -3*w^3 + 6*w^2 + 7*w - 3],
[113, 113, w^3 - 4*w^2 + w + 4],
[113, 113, 2*w^2 - 3*w - 4],
[113, 113, 3*w^3 - 6*w^2 - 7*w + 4],
[113, 113, -2*w^3 + 6*w^2 + 2*w - 7],
[137, 137, -w^3 + 4*w^2 - 4],
[137, 137, w^3 - 4*w^2 + 6],
[151, 151, -3*w^3 + 8*w^2 + 3*w - 4],
[151, 151, -2*w^3 + 2*w^2 + 7*w + 4],
[167, 167, -2*w^3 + 3*w^2 + 6*w - 3],
[167, 167, -2*w^3 + 5*w^2 + 2*w - 6],
[191, 191, w^3 - 2*w^2 - w + 5],
[191, 191, -w^3 + 5*w^2 - 3*w - 3],
[193, 193, -w^3 + 2*w^2 + 5*w - 2],
[193, 193, -2*w^3 + 4*w^2 + 7*w - 4],
[199, 199, -2*w^3 + 5*w^2 + 4*w - 2],
[199, 199, w^2 - 5],
[223, 223, -w^3 + 8*w - 2],
[223, 223, -2*w^3 + 2*w^2 + 9*w],
[223, 223, -3*w^3 + 5*w^2 + 11*w - 1],
[223, 223, -w^3 + 4*w^2 + w - 8],
[233, 233, -3*w^3 + 7*w^2 + 6*w - 4],
[233, 233, w^3 - w^2 - 2*w - 3],
[239, 239, -w^3 + 6*w],
[239, 239, 2*w^3 - 5*w^2 - 4*w + 1],
[257, 257, -3*w^3 + 7*w^2 + 7*w - 5],
[257, 257, 3*w^3 - 7*w^2 - 5*w + 1],
[263, 263, w^3 - w^2 - 4*w - 5],
[263, 263, -2*w^3 + 3*w^2 + 9*w - 4],
[281, 281, -w^3 + 3*w^2 + 4*w - 6],
[281, 281, 3*w^3 - 7*w^2 - 8*w + 5],
[289, 17, -3*w^3 + 6*w^2 + 6*w - 4],
[311, 311, -3*w^3 + 5*w^2 + 11*w - 3],
[311, 311, w - 5],
[313, 313, -2*w^3 + 6*w^2 + w - 9],
[313, 313, -w^3 + 3*w^2 + 3*w - 7],
[359, 359, 3*w^3 - 6*w^2 - 8*w],
[359, 359, -w^2 + 3*w - 4],
[359, 359, w^3 - w^2 - 5*w - 5],
[359, 359, w^3 - 2*w^2 - 4],
[383, 383, -4*w^3 + 8*w^2 + 9*w - 3],
[383, 383, -w^3 + 5*w^2 - 3*w - 7],
[401, 401, -3*w^3 + 8*w^2 + 3*w - 9],
[401, 401, -3*w^3 + 7*w^2 + 6*w - 3],
[401, 401, w^3 - w^2 - 2*w - 4],
[401, 401, -2*w^3 + 2*w^2 + 7*w - 1],
[439, 439, -4*w^3 + 9*w^2 + 7*w - 6],
[439, 439, 3*w^3 - 5*w^2 - 7*w + 1],
[449, 449, -4*w^3 + 9*w^2 + 9*w - 6],
[449, 449, 2*w^3 - 2*w^2 - 11*w - 2],
[449, 449, -w^3 + 5*w^2 - 3*w - 11],
[449, 449, -3*w^3 + 5*w^2 + 10*w - 5],
[457, 457, -2*w^3 + 2*w^2 + 9*w - 1],
[457, 457, w^3 - 4*w^2 + 3*w + 5],
[463, 463, 3*w^2 - 5*w - 6],
[463, 463, -w^3 + 5*w^2 - 3*w - 5],
[479, 479, -2*w^3 + 3*w^2 + 9*w - 2],
[479, 479, -w^3 + w^2 + 7*w - 1],
[487, 487, -2*w^2 + 5*w + 6],
[487, 487, -w^3 + 7*w - 2],
[487, 487, -w^3 + 6*w - 2],
[487, 487, -w^3 + 4*w^2 - 2*w - 8],
[503, 503, w^3 - 4*w - 6],
[503, 503, -3*w^3 + 8*w^2 + 4*w - 4],
[521, 521, 3*w^2 - 4*w - 7],
[521, 521, 2*w^3 - 7*w^2 + 6],
[529, 23, -w^3 + 2*w^2 + 2*w - 6],
[529, 23, w^3 - 2*w^2 - 2*w - 4],
[569, 569, 2*w^3 - 7*w^2 - 2*w + 7],
[569, 569, 4*w^3 - 7*w^2 - 10*w + 4],
[569, 569, 3*w^3 - 4*w^2 - 13*w],
[569, 569, 4*w^3 - 8*w^2 - 11*w + 3],
[577, 577, 4*w^2 - 7*w - 10],
[577, 577, w^3 + w^2 - 6*w - 9],
[593, 593, w^3 - 4*w^2 + 2*w - 2],
[593, 593, 3*w^3 - 10*w^2 + w + 11],
[601, 601, -2*w^3 + 4*w^2 + 8*w - 3],
[601, 601, -2*w^3 + 4*w^2 + 8*w - 5],
[617, 617, 5*w^3 - 12*w^2 - 11*w + 11],
[617, 617, 3*w^2 - 4*w - 5],
[617, 617, -2*w^3 + 7*w^2 - 8],
[617, 617, 5*w^3 - 10*w^2 - 13*w + 11],
[631, 631, -3*w^2 + 8*w + 5],
[631, 631, 2*w^3 - 7*w^2 - w + 10],
[631, 631, 5*w^3 - 11*w^2 - 10*w + 9],
[631, 631, 2*w^3 - w^2 - 12*w],
[641, 641, -3*w^3 + 8*w^2 + 5*w - 5],
[641, 641, 2*w^2 - w - 7],
[647, 647, -2*w^3 + 2*w^2 + 11*w - 1],
[647, 647, 2*w^3 - 2*w^2 - 11*w - 1],
[647, 647, -3*w^3 + 9*w^2 + 4*w - 10],
[647, 647, w^3 - 5*w^2 + 6],
[673, 673, -5*w^3 + 11*w^2 + 12*w - 8],
[673, 673, 3*w^3 - 5*w^2 - 10*w + 6],
[719, 719, -3*w^3 + 4*w^2 + 10*w - 2],
[719, 719, -3*w^3 + 8*w^2 + 2*w - 8],
[727, 727, -w^3 + 5*w^2 - 2*w - 6],
[727, 727, -w^3 + 5*w^2 - 2*w - 7],
[743, 743, -5*w^3 + 10*w^2 + 13*w - 4],
[743, 743, -4*w^3 + 9*w^2 + 10*w - 6],
[743, 743, 2*w^3 - 2*w^2 - 10*w + 3],
[743, 743, w^2 + 2*w - 5],
[751, 751, -w^3 + 4*w^2 - 3*w - 6],
[751, 751, 2*w^3 - 2*w^2 - 9*w + 2],
[761, 761, 4*w^3 - 9*w^2 - 6*w + 4],
[761, 761, 3*w^3 - 3*w^2 - 12*w - 1],
[761, 761, 3*w^3 - 9*w^2 + 8],
[761, 761, 4*w^3 - 7*w^2 - 10*w + 1],
[769, 769, 4*w^3 - 7*w^2 - 10*w + 3],
[769, 769, -4*w^3 + 7*w^2 + 14*w - 5],
[769, 769, 4*w^3 - 9*w^2 - 6*w + 6],
[769, 769, -w^2 + 6*w],
[809, 809, -2*w^3 + 2*w^2 + 7*w - 3],
[809, 809, -3*w^3 + 8*w^2 + 3*w - 11],
[823, 823, 3*w^3 - 8*w^2 - 7*w + 6],
[823, 823, -2*w^3 + 6*w^2 + 5*w - 10],
[839, 839, -3*w^3 + 5*w^2 + 12*w - 1],
[839, 839, -w^3 + w^2 + 8*w - 4],
[887, 887, -4*w^3 + 9*w^2 + 5*w - 6],
[887, 887, -5*w^3 + 9*w^2 + 13*w - 5],
[919, 919, -4*w^3 + 6*w^2 + 14*w - 3],
[919, 919, -5*w^3 + 12*w^2 + 9*w - 8],
[929, 929, -2*w^3 + 3*w^2 + 10*w - 5],
[929, 929, -2*w^3 + 3*w^2 + 10*w],
[937, 937, -3*w^3 + 7*w^2 + 8*w - 3],
[937, 937, -w^3 + 3*w^2 + 4*w - 8],
[961, 31, -4*w^3 + 8*w^2 + 8*w - 5],
[961, 31, -4*w^3 + 8*w^2 + 8*w - 3],
[967, 967, 4*w^3 - 8*w^2 - 7*w + 1],
[967, 967, -5*w^3 + 10*w^2 + 11*w - 3],
[977, 977, 3*w^3 - 8*w^2 - w + 7],
[977, 977, -4*w^3 + 6*w^2 + 13*w - 3],
[991, 991, w^2 - w - 8],
[991, 991, -3*w^3 + 5*w^2 + 12*w - 5]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^3 - 2*x^2 - 9*x + 6;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, 1, 1, 0, 0, -2*e, -2*e, -2*e^2 + 2*e + 10, -2*e + 4, -2*e + 4, 2*e + 4, 2*e^2 - 4*e - 12, 2*e^2 - 4*e - 12, -2*e^2 + 4*e + 14, -2*e^2 + 4*e + 14, 2*e - 8, 2*e - 8, -2*e, -2*e, -2*e^2 + 2*e + 14, -2*e^2 + 4*e + 12, -2*e^2 + 4*e + 12, 2*e^2 - 2*e - 14, 2*e^2 - 2*e - 14, -4, -4, 2*e^2 - 2*e - 8, 2*e^2 - 2*e - 8, -2*e^2 + 20, -2*e^2 + 20, -2*e^2 + 2*e + 10, -2*e^2 + 2*e + 10, -2*e + 4, -2*e + 4, 8, 8, -4*e + 8, -4*e + 8, 4*e - 8, 4*e - 8, 2*e^2 - 8*e - 14, 2*e^2 - 8*e - 14, 4*e, 4*e, 8, 2*e^2 + 2*e - 16, 2*e^2 + 2*e - 16, 8, 2*e^2 - 4*e - 12, 2*e^2 - 4*e - 12, -12, -12, -2*e^2 + 8*e + 16, -2*e^2 + 8*e + 16, -2*e^2 + 2*e + 16, -2*e^2 + 2*e + 16, 2*e^2 - 6*e - 26, 2*e^2 - 6*e - 26, 4*e + 18, 6*e^2 - 8*e - 32, 6*e^2 - 8*e - 32, 2*e^2 - 6*e - 30, 2*e^2 - 6*e - 30, -8*e + 4, 4*e^2 - 28, 4*e^2 - 28, -8*e + 4, -4*e^2 + 4*e + 32, -4*e^2 + 4*e + 32, -2*e^2 - 2*e + 12, 2*e^2 - 4*e - 6, 2*e^2 - 4*e - 6, -2*e^2 - 2*e + 12, 2*e + 16, 2*e + 16, -2*e^2 - 2*e + 6, -12*e + 6, -12*e + 6, -2*e^2 - 2*e + 6, 2*e^2 + 8*e - 28, 2*e^2 + 8*e - 28, 2*e^2 - 4*e - 16, 2*e^2 - 4*e - 16, 2*e - 16, 2*e - 16, -16, -16, 8*e - 20, 8*e - 20, 4*e - 20, 4*e - 20, -2*e^2 + 10*e + 18, -2*e^2 + 10*e + 18, -2*e^2 + 12*e + 16, -2*e^2 + 40, 8*e - 10, 2*e^2 + 4*e - 4, 8*e - 10, 2*e^2 + 4*e - 4, -2*e^2 + 8*e + 12, -2*e^2 + 8*e + 12, -2*e + 28, -2*e + 28, 4*e^2 - 8*e - 16, 4*e^2 - 8*e - 16, 4*e^2 - 2*e - 30, 2*e + 14, 2*e + 14, 4*e^2 - 2*e - 30, -16, 6*e^2 - 8*e - 36, 6*e^2 - 8*e - 36, -16, 2*e^2 - 2, 2*e^2 - 2, 4*e^2 - 8*e - 24, -4*e - 16, -4*e - 16, 4*e^2 - 8*e - 24, -2*e^2 + 8*e + 36, -2*e^2 + 8*e + 36, -4*e^2 + 16*e + 20, -4*e^2 + 16*e + 20, 8*e + 4, 8*e + 4, -4*e^2 + 8*e + 48, 4*e^2 - 16*e - 32, -4*e^2 + 8*e + 48, 4*e^2 - 16*e - 32, 4*e^2 - 4*e - 12, 4*e^2 - 4*e - 12, -4*e^2 - 10*e + 48, 4*e^2 - 16*e - 14, 4*e^2 - 16*e - 14, -4*e^2 - 10*e + 48, -4*e^2 + 2*e + 32, -2*e^2 + 6*e - 20, -4*e^2 + 2*e + 32, -2*e^2 + 6*e - 20, 4*e^2 - 2*e, 4*e^2 - 2*e, 2*e^2, 2*e^2, 2*e^2 + 2*e - 48, 2*e^2 + 2*e - 48, -2*e^2 + 10*e + 24, -2*e^2 + 10*e + 24, 2*e^2 - 48, 2*e^2 - 48, -2*e^2 - 8*e + 12, -2*e^2 - 8*e + 12, -4*e - 14, -4*e - 14, -2*e^2 + 12*e + 16, 2*e + 52, 4*e^2 - 32, 4*e^2 - 32, 4*e^2 + 2*e - 32, 4*e^2 + 2*e - 32, -4*e^2 + 8*e + 8, -4*e^2 + 8*e + 8];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {7, 7, -w^3 + 3*w^2 + w - 3}>] := -1;
ALEigenvalues[ideal<ZF | {7, 7, -w^2 + 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;