/*
  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, 0, -5, 0, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {13, 13, w^3 - 4*w + 2}>;

primesArray := [
[3, 3, w],
[4, 2, w^2 - w - 1],
[9, 3, -w^2 + 2],
[13, 13, w^3 - 4*w + 2],
[13, 13, -w^3 + 4*w + 2],
[17, 17, w^3 - 3*w - 1],
[17, 17, -w^3 + 3*w - 1],
[29, 29, w^3 + w^2 - 4*w - 2],
[29, 29, w^3 - w^2 - 4*w + 2],
[43, 43, w^2 + w - 4],
[43, 43, w^2 - w - 4],
[53, 53, w^3 - 2*w - 2],
[53, 53, -w^3 + 2*w - 2],
[79, 79, w^3 - w^2 - 4*w + 1],
[79, 79, -w^3 - w^2 + 4*w + 1],
[101, 101, -w^3 + w^2 + 3*w - 5],
[101, 101, w^3 + w^2 - 3*w - 5],
[103, 103, 2*w^2 - w - 4],
[103, 103, 2*w^2 + w - 4],
[107, 107, 2*w^2 + w - 5],
[107, 107, 3*w^2 - 3*w - 4],
[107, 107, 2*w^2 - 2*w - 1],
[107, 107, 2*w^2 - w - 5],
[113, 113, w^3 - w^2 - 5*w + 1],
[113, 113, -w^3 - w^2 + 5*w + 1],
[121, 11, w^3 - 2*w^2 - 3*w + 2],
[121, 11, -2*w^3 + w^2 + 7*w - 4],
[127, 127, -2*w^3 + 7*w - 2],
[127, 127, 2*w^3 - 7*w - 2],
[131, 131, w^3 - w^2 - 1],
[131, 131, -w^3 + w^2 + 4*w + 1],
[131, 131, -2*w^3 + 3*w^2 + 4*w - 2],
[131, 131, -2*w^3 + w^2 + 4*w + 2],
[139, 139, w^2 - 2*w - 4],
[139, 139, w^2 + 2*w - 4],
[173, 173, 2*w^2 - 3*w - 5],
[173, 173, 2*w^2 + 3*w - 5],
[179, 179, -w^2 - 3*w + 4],
[179, 179, 2*w^3 - w^2 - 9*w + 2],
[179, 179, -2*w^3 - w^2 + 9*w + 2],
[179, 179, w^2 - 3*w - 4],
[199, 199, -3*w^3 + w^2 + 12*w - 4],
[199, 199, -3*w^3 - w^2 + 12*w + 4],
[211, 211, 2*w^3 - 7*w + 1],
[211, 211, -2*w^3 + 7*w + 1],
[233, 233, -w^2 - w - 2],
[233, 233, -w^2 + w - 2],
[257, 257, -3*w^2 - w + 10],
[257, 257, 3*w^2 - w - 10],
[263, 263, -w^3 + w^2 + 6*w + 1],
[263, 263, w^2 + 2*w - 5],
[263, 263, w^2 - 2*w - 5],
[263, 263, w^3 + w^2 - 6*w + 1],
[269, 269, -2*w^3 - w^2 + 6*w + 1],
[269, 269, 2*w^3 - w^2 - 6*w + 1],
[277, 277, w^3 - w^2 - 2],
[277, 277, -w^3 + 2*w^2 + 3*w - 8],
[277, 277, w^3 + 2*w^2 - 3*w - 8],
[277, 277, -w^3 - w^2 - 2],
[283, 283, -2*w^3 + w^2 + 8*w - 1],
[283, 283, 2*w^3 + w^2 - 8*w - 1],
[289, 17, w^2 - 7],
[313, 313, w^3 + 2*w^2 - 4*w - 2],
[313, 313, 2*w^3 - w^2 - 6*w + 2],
[313, 313, -2*w^3 - w^2 + 6*w + 2],
[313, 313, -w^3 + 2*w^2 + 4*w - 2],
[337, 337, -w^3 - 2*w^2 + 7*w + 2],
[337, 337, -2*w^3 + 4*w + 1],
[337, 337, 2*w^3 - 4*w + 1],
[337, 337, -w^3 + 2*w^2 + 7*w - 2],
[367, 367, -w^3 + 3*w^2 - 2*w - 5],
[367, 367, 2*w^3 - 3*w^2 - 2*w + 4],
[373, 373, -2*w^3 + 10*w + 1],
[373, 373, -3*w^3 - w^2 + 14*w + 5],
[373, 373, 3*w^3 - w^2 - 14*w + 5],
[373, 373, 2*w^3 - 10*w + 1],
[389, 389, 2*w^3 - 6*w - 1],
[389, 389, -2*w^3 + 6*w - 1],
[419, 419, 2*w^3 - 11*w - 4],
[419, 419, 2*w^2 - 2*w - 7],
[419, 419, 2*w^2 + 2*w - 7],
[419, 419, -2*w^3 + 11*w - 4],
[439, 439, 3*w^2 - w - 14],
[439, 439, -3*w^2 - w + 14],
[443, 443, w^3 + 3*w^2 - 4*w - 10],
[443, 443, 3*w^3 + w^2 - 14*w - 4],
[443, 443, 3*w^3 - w^2 - 14*w + 4],
[443, 443, -w^3 + 3*w^2 + 4*w - 10],
[503, 503, -w - 5],
[503, 503, 3*w^3 + w^2 - 12*w - 1],
[503, 503, 3*w^3 - w^2 - 12*w + 1],
[503, 503, w - 5],
[521, 521, w^3 + 2*w^2 - 5*w - 2],
[521, 521, -w^3 + 2*w^2 + 5*w - 2],
[523, 523, 3*w^2 - 2*w - 10],
[523, 523, w^3 + 3*w^2 - 6*w - 11],
[529, 23, 2*w^2 - 11],
[529, 23, 3*w^2 - 10],
[547, 547, -2*w^3 + w^2 + 6*w - 7],
[547, 547, 2*w^3 + w^2 - 6*w - 7],
[563, 563, -2*w^3 + w^2 + 4*w - 4],
[563, 563, -3*w^3 + 10*w - 4],
[563, 563, 3*w^3 - 10*w - 4],
[563, 563, 2*w^3 + w^2 - 4*w - 4],
[569, 569, w^3 + 2*w^2 - 6*w - 10],
[569, 569, -w^3 + 2*w^2 + 6*w - 10],
[571, 571, -2*w^3 + 3*w^2 + 5*w - 2],
[571, 571, -2*w^3 + 2*w^2 + 6*w + 1],
[599, 599, 3*w^3 - 11*w - 2],
[599, 599, 2*w^3 + w^2 - 6*w - 8],
[599, 599, 2*w^3 - w^2 - 6*w + 8],
[599, 599, 3*w^3 - 11*w + 2],
[601, 601, -w^3 - 2*w^2 + 6*w + 2],
[601, 601, 4*w^3 - w^2 - 16*w + 5],
[601, 601, 4*w^3 + w^2 - 16*w - 5],
[601, 601, w^3 - 2*w^2 - 6*w + 2],
[607, 607, -w^3 + w^2 + 4*w - 8],
[607, 607, -w^3 - w^2 + 4*w + 8],
[625, 5, -5],
[641, 641, w^2 + 4*w - 1],
[641, 641, w^2 - 4*w - 1],
[647, 647, w^3 + 3*w^2 - 4*w - 8],
[647, 647, -w^3 + 7*w + 2],
[647, 647, w^3 - 7*w + 2],
[647, 647, -w^3 + 3*w^2 + 4*w - 8],
[653, 653, 2*w^3 - 3*w^2 - 12*w + 10],
[653, 653, 2*w^3 + 3*w^2 - 12*w - 10],
[677, 677, 3*w^2 + w - 8],
[677, 677, 3*w^2 - w - 8],
[701, 701, 2*w^3 - 2*w^2 - 7*w + 2],
[701, 701, -2*w^3 - 2*w^2 + 7*w + 2],
[727, 727, -2*w^3 - 2*w^2 + 11*w + 7],
[727, 727, -2*w^3 + 2*w^2 + 11*w - 7],
[751, 751, -4*w^2 + 3*w + 7],
[751, 751, w^3 - 2*w^2 - w - 2],
[797, 797, 3*w^3 - 11*w + 1],
[797, 797, 3*w^3 - 11*w - 1],
[809, 809, -w^3 + 7*w - 1],
[809, 809, w^3 - 7*w - 1],
[823, 823, -2*w^3 + 4*w^2 + 9*w - 14],
[823, 823, 2*w^3 + 4*w^2 - 9*w - 14],
[841, 29, 3*w^2 - 7],
[857, 857, w^3 + w^2 - w - 5],
[857, 857, -w^3 + w^2 + w - 5],
[859, 859, 5*w^3 + w^2 - 20*w - 2],
[859, 859, -5*w^3 + w^2 + 20*w - 2],
[881, 881, w^3 + 3*w^2 - 2*w - 10],
[881, 881, -w^3 + 3*w^2 + 2*w - 10],
[883, 883, 2*w^3 - w^2 - 8*w - 2],
[883, 883, 2*w^3 + w^2 - 8*w + 2],
[907, 907, 3*w^3 - 2*w^2 - 13*w + 5],
[907, 907, 3*w^3 + 2*w^2 - 13*w - 5],
[919, 919, 3*w^3 - w^2 - 15*w + 1],
[919, 919, 3*w^3 + 3*w^2 - 10*w - 5],
[953, 953, w^3 + 2*w^2 - 2*w - 8],
[953, 953, -w^3 + 2*w^2 + 2*w - 8],
[991, 991, -w^3 - w^2 + 4*w - 4],
[991, 991, w^3 - w^2 - 4*w - 4],
[997, 997, w^3 + 4*w^2 - 3*w - 16],
[997, 997, -3*w^3 + 3*w^2 + 11*w - 13],
[997, 997, 3*w^3 + 3*w^2 - 11*w - 13],
[997, 997, w^3 - 4*w^2 - 3*w + 16]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

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

heckeEigenvaluesArray := [e, e - 1, -e - 1, 1, -e - 4, -2*e^2 + e + 4, e^2 - 3*e - 4, e^2 - 4, e^2 - 4, 3*e, -2*e^2 - 4*e + 4, -4*e^2 + 3*e + 18, 2*e - 1, 4*e^2 - 3*e - 9, -4*e^2 - 2*e + 11, -6*e^2 - 2*e + 15, 4*e^2 + 4*e - 9, -3*e^2 - 6*e + 9, 3*e^2 - 14, -4*e^2 + 5*e + 17, 4*e^2 - 5*e - 17, -4*e^2 - 4*e + 3, -2*e^2 - 3*e + 2, -e^2 - 3*e - 5, 4*e^2 - 17, 7*e^2 + 5*e - 18, -2*e^2 + 3*e + 13, 2*e^2 + 7*e - 6, -2, 3*e^2 + 5*e - 8, 4*e^2 - e - 18, -5*e^2 - 3*e + 13, -7*e^2 - e + 16, -e^2 - 10*e + 4, -5*e^2 + 5*e + 16, -5*e^2 + 3*e + 21, 6*e^2 - 3*e - 25, 3*e^2 - 8, 6*e - 9, 3*e^2 + 2*e - 17, 7*e^2 - 9, 5*e^2 - 3*e - 19, -3*e^2 - 2*e + 1, 7*e^2 + 7*e - 23, -3*e^2 + e + 1, 5*e^2 + 9*e - 13, 5*e^2 + 9*e - 13, 4*e^2 - 3*e - 14, -4*e^2 - 2*e + 6, e^2 - e + 17, -e^2 - e - 11, e^2 + 7*e + 3, 2*e^2 + e - 12, -2*e^2 + 7*e + 7, 4*e^2 + 9, 11*e^2 + 5*e - 25, -2*e^2 - 12*e + 10, 6*e^2 + e - 17, 10*e + 3, -4*e^2 + 14*e + 18, -2*e^2 + 7*e + 21, -6*e^2 - 11*e + 20, 3*e^2 - e - 2, -11*e^2 + 3*e + 42, -2*e^2 + 4*e - 7, 2*e - 12, -2*e^2 - 4*e, 9*e^2 + e - 37, -5*e^2 - 4*e + 6, 10*e^2 - 6*e - 39, -2*e^2 - e - 24, 6*e^2 - e - 26, -12*e^2 + 5*e + 41, 6*e^2 - 6*e - 32, 11*e^2 - 2*e - 26, -11*e^2 - 7*e + 19, -8*e^2 - 8*e + 14, -7*e^2 - 3*e + 39, 4*e - 5, -4*e^2 - 3*e + 10, 4*e^2 - 5*e - 28, -13*e^2 + 3*e + 45, e^2 + 3*e - 26, -e^2 - e + 32, 6*e^2 - 9, 11*e^2 - 5*e - 45, 9*e^2 + 4*e - 12, 14*e^2 - e - 29, 4*e^2 + 9*e - 28, -7*e^2 - 17*e + 30, 8*e^2 + 6*e - 13, 3*e^2 + 7*e + 1, -7*e^2 - 10*e + 12, 10*e^2 + 7*e - 10, -7*e^2 - 2*e + 5, 13*e^2 - 4*e - 36, -9*e^2 - 2*e + 15, -10*e^2 + 2*e + 19, e^2 + 2*e - 13, 14*e^2 + 4*e - 45, 9*e^2 + 13*e - 32, -6*e^2 - 5*e + 9, 2*e^2 + 8*e - 18, e^2 + e + 13, 14*e^2 + 2*e - 31, 7*e^2 - 9*e - 33, 6*e^2 - 7*e - 17, 4*e^2 + 15*e - 9, 12*e^2 - 29, -15*e^2 - 9*e + 27, 5*e^2 + 5*e + 15, -14*e^2 - 18*e + 42, -19*e^2 - 6*e + 35, 16*e^2 - 5*e - 52, -2*e + 6, 3*e - 2, 10*e^2 + 5*e - 43, 5*e^2 - 12*e - 24, -4*e^2 + 7*e + 4, -16*e^2 + e + 38, -17*e^2 - 2*e + 49, 11*e^2 + 6*e - 15, 2*e^2 + e - 25, -7*e^2 - e + 6, -2*e^2 - 12*e - 12, -e^2 - 2*e + 26, -12*e^2 - 13*e + 25, 2*e^2 - e - 44, -e^2 + 5*e, -6*e^2 - 6*e, -13*e^2 - e + 27, 14*e^2 - 8*e - 46, -4*e^2 - 13*e - 2, -11*e^2 + 5*e + 38, -12*e^2 + e + 31, 7*e^2 - 4*e - 58, -2*e^2 - 2*e + 45, 7*e^2 + 19*e - 27, -4*e^2 + 9*e - 10, 18*e^2 + 13*e - 56, -9*e^2 - 8*e + 26, 9*e^2 + 10*e - 4, 9*e^2 - e - 32, 2*e^2 + 18*e - 12, 6*e^2 - 12*e - 52, e^2 + 2*e + 7, 11*e^2 - 2*e - 16, 23*e^2 + 10*e - 62, -5*e^2 - 10*e + 17, -10*e^2 + 16*e + 33, 11*e^2 + 22*e - 44, -10*e^2 - 7*e + 22, -12*e^2 - 3*e + 9, 4*e^2 + 11*e - 2, 2*e^2 + 23*e + 6, -3*e^2 + 5*e + 7, -3*e^2 - 18*e + 2, 7*e^2 + 4*e + 7, e^2 - 3*e + 18, -e + 28, -18*e^2 - 8*e + 36, -10*e^2 + 10];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {13, 13, w^3 - 4*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;