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

NN := ideal<ZF | {21, 21, w^4 - 3*w^3 - w^2 + 6*w - 3}>;

primesArray := [
[3, 3, w - 1],
[5, 5, w^2 - w - 2],
[7, 7, w^4 - 2*w^3 - 3*w^2 + 4*w + 2],
[13, 13, w^3 - 2*w^2 - 2*w + 2],
[29, 29, -w^4 + 3*w^3 + 2*w^2 - 7*w - 1],
[29, 29, -w^2 + 2*w + 3],
[31, 31, w^4 - 2*w^3 - 3*w^2 + 5*w],
[31, 31, w^3 - 2*w^2 - 3*w + 2],
[32, 2, 2],
[43, 43, -w^2 - w + 4],
[53, 53, -w^4 + w^3 + 6*w^2 - 2*w - 5],
[53, 53, -w^4 + 2*w^3 + 4*w^2 - 5*w - 2],
[73, 73, w^4 - w^3 - 6*w^2 + 2*w + 6],
[73, 73, w^3 - w^2 - 4*w + 2],
[81, 3, 2*w^4 - 5*w^3 - 4*w^2 + 9*w + 2],
[83, 83, w^3 - w^2 - 5*w],
[89, 89, -w^4 + 2*w^3 + 4*w^2 - 4*w - 2],
[97, 97, w^4 - 3*w^3 - 2*w^2 + 6*w + 2],
[101, 101, -w^4 + 3*w^3 + 3*w^2 - 7*w - 3],
[103, 103, -w^4 + w^3 + 6*w^2 - w - 7],
[103, 103, 2*w^4 - 4*w^3 - 6*w^2 + 8*w + 1],
[103, 103, w^4 - w^3 - 6*w^2 + 3*w + 4],
[107, 107, -w^4 + 2*w^3 + 3*w^2 - 5*w + 2],
[109, 109, w^3 - 4*w - 1],
[109, 109, -w^2 + 2*w + 4],
[131, 131, w^4 - w^3 - 6*w^2 + w + 6],
[137, 137, w^3 - w^2 - 5*w + 3],
[149, 149, w^4 - 3*w^3 - 3*w^2 + 9*w + 6],
[149, 149, -w^4 + 6*w^2 + 3*w],
[163, 163, 2*w^2 - w - 5],
[167, 167, -2*w^4 + 3*w^3 + 8*w^2 - 4*w - 3],
[169, 13, w^4 - 3*w^3 + 6*w - 3],
[169, 13, w^3 - 4*w - 4],
[173, 173, w^4 - w^3 - 4*w^2 + 2*w + 1],
[173, 173, -w^4 + 3*w^3 + 3*w^2 - 7*w - 5],
[173, 173, w^4 - w^3 - 4*w^2 + w - 1],
[179, 179, w^2 - 3*w - 2],
[191, 191, -w^4 + 3*w^3 + w^2 - 6*w - 1],
[193, 193, -w^3 + 2*w^2 + w - 4],
[193, 193, -w^4 + 3*w^3 + 3*w^2 - 8*w - 1],
[199, 199, w^4 - 3*w^3 - w^2 + 7*w],
[199, 199, -2*w^4 + 5*w^3 + 5*w^2 - 10*w - 2],
[199, 199, w^4 - 3*w^3 + 5*w - 4],
[211, 211, 2*w^3 - 2*w^2 - 7*w - 1],
[211, 211, -2*w^3 + 3*w^2 + 5*w - 2],
[223, 223, -2*w^4 + 3*w^3 + 10*w^2 - 7*w - 9],
[227, 227, w^3 - w^2 - 4*w + 3],
[229, 229, w^4 - 4*w^3 + 10*w - 5],
[229, 229, -2*w^4 + 4*w^3 + 7*w^2 - 9*w - 8],
[239, 239, -2*w^4 + 3*w^3 + 7*w^2 - 3*w - 3],
[241, 241, w^3 - w^2 - 2*w - 2],
[257, 257, -w^4 + w^3 + 6*w^2 - 4*w - 4],
[263, 263, 2*w^3 - 4*w^2 - 5*w + 6],
[269, 269, w^3 - 2*w^2 - 4*w + 4],
[269, 269, -2*w^4 + 2*w^3 + 11*w^2 - 3*w - 10],
[271, 271, w^3 - 5*w],
[281, 281, w^4 - w^3 - 5*w^2 + 3*w + 3],
[281, 281, -2*w^4 + 4*w^3 + 6*w^2 - 6*w - 3],
[281, 281, w^3 - 5*w - 1],
[283, 283, -2*w^4 + 4*w^3 + 7*w^2 - 8*w - 3],
[289, 17, -2*w^4 + 3*w^3 + 8*w^2 - 5*w - 2],
[293, 293, 2*w^4 - 5*w^3 - 7*w^2 + 14*w + 9],
[311, 311, -w^4 + 2*w^3 + 2*w^2 - 4*w + 3],
[313, 313, w^3 - 6*w],
[331, 331, w^4 - 2*w^3 - 2*w^2 + w - 2],
[359, 359, w^4 - w^3 - 5*w^2 + 2*w - 1],
[373, 373, -w^4 + 3*w^3 + 2*w^2 - 8*w - 3],
[373, 373, 2*w^3 - 4*w^2 - 3*w + 3],
[379, 379, -w^4 + 4*w^3 - w^2 - 9*w + 3],
[379, 379, -w^4 + 3*w^3 + w^2 - 6*w - 2],
[379, 379, w^4 + w^3 - 7*w^2 - 6*w + 3],
[383, 383, -w^4 + 2*w^3 + 3*w^2 - 6*w - 2],
[383, 383, w^4 - 5*w^2 - 3*w + 3],
[383, 383, 2*w^4 - 5*w^3 - 3*w^2 + 8*w],
[389, 389, 3*w^4 - 7*w^3 - 8*w^2 + 16*w + 1],
[397, 397, -2*w^4 + 3*w^3 + 8*w^2 - 3*w - 7],
[397, 397, -w^4 + 2*w^3 + 3*w^2 - 4*w - 5],
[397, 397, -2*w^4 + 3*w^3 + 7*w^2 - 4*w - 3],
[409, 409, -w^4 + w^3 + 7*w^2 - 5*w - 9],
[419, 419, w^4 - 2*w^3 - w^2 + 2*w - 2],
[419, 419, 2*w^4 - 3*w^3 - 7*w^2 + 5*w + 2],
[419, 419, -w^4 + 4*w^3 - 9*w - 2],
[433, 433, -w^4 + 3*w^3 - 8*w + 4],
[433, 433, w^4 - 7*w^2 - w + 6],
[439, 439, 3*w^4 - 6*w^3 - 10*w^2 + 11*w + 7],
[449, 449, -2*w^4 + 5*w^3 + 5*w^2 - 11*w - 4],
[457, 457, -2*w^4 + 5*w^3 + 4*w^2 - 10*w - 2],
[463, 463, 2*w^4 - 5*w^3 - 4*w^2 + 8*w + 1],
[467, 467, 2*w^4 - 4*w^3 - 6*w^2 + 6*w + 1],
[479, 479, 3*w^4 - 4*w^3 - 12*w^2 + 4*w + 7],
[479, 479, 2*w^4 - 6*w^3 - 4*w^2 + 14*w + 1],
[491, 491, w^4 - 6*w^2 - w + 4],
[503, 503, 2*w^4 - w^3 - 11*w^2 - 3*w + 8],
[503, 503, 2*w^3 - 5*w^2 - 2*w + 6],
[509, 509, w^4 - 5*w^2 - 6*w],
[509, 509, w^4 - 3*w^3 - 3*w^2 + 9*w + 1],
[523, 523, 3*w^4 - 7*w^3 - 7*w^2 + 12*w + 3],
[523, 523, 3*w^4 - 8*w^3 - 7*w^2 + 17*w + 3],
[523, 523, w^4 - 7*w^2 - 2*w + 4],
[529, 23, -w^4 + w^3 + 7*w^2 - 2*w - 9],
[541, 541, w^4 - 3*w^3 - 3*w^2 + 5*w + 2],
[547, 547, -w^3 + 3*w^2 + 3*w - 4],
[557, 557, 2*w^3 - 4*w^2 - 6*w + 9],
[577, 577, 2*w^4 - 2*w^3 - 11*w^2 + 5*w + 10],
[577, 577, w^4 - 4*w^3 - w^2 + 9*w],
[577, 577, 2*w^4 - 3*w^3 - 9*w^2 + 4*w + 8],
[577, 577, w^4 - 7*w^2 - 3*w + 7],
[577, 577, -3*w^4 + 3*w^3 + 13*w^2 - w - 4],
[593, 593, 3*w - 2],
[601, 601, 2*w^4 - 4*w^3 - 7*w^2 + 5*w + 6],
[601, 601, -w^4 + 3*w^3 + w^2 - 7*w - 1],
[607, 607, w^3 - w^2 - 2*w - 3],
[625, 5, 2*w^4 - 5*w^3 - 8*w^2 + 14*w + 11],
[631, 631, -w^2 + w - 2],
[631, 631, w^4 - 7*w^2 - w + 8],
[643, 643, -2*w^4 + 4*w^3 + 9*w^2 - 10*w - 8],
[647, 647, 3*w^4 - 6*w^3 - 10*w^2 + 12*w + 8],
[653, 653, -3*w^4 + 6*w^3 + 11*w^2 - 12*w - 9],
[653, 653, w^4 - w^3 - 2*w^2 - 3*w - 3],
[653, 653, -2*w^4 + 4*w^3 + 7*w^2 - 6*w - 8],
[701, 701, w^4 - w^3 - 2*w^2 - w - 5],
[719, 719, 2*w^4 - 5*w^3 - 6*w^2 + 14*w + 2],
[727, 727, 2*w^4 - 5*w^3 - 5*w^2 + 9*w + 3],
[733, 733, 2*w^4 - 5*w^3 - 5*w^2 + 13*w + 3],
[739, 739, -2*w^3 + 4*w^2 + 4*w - 5],
[743, 743, w^4 - 2*w^3 - 3*w^2 + 6*w - 3],
[743, 743, 2*w^4 - 2*w^3 - 11*w^2 + 2*w + 10],
[757, 757, -w^4 + 2*w^3 + 5*w^2 - 6*w - 10],
[757, 757, 2*w^4 - 6*w^3 - 2*w^2 + 11*w - 4],
[761, 761, 3*w^3 - 4*w^2 - 9*w + 3],
[769, 769, 3*w^3 - 5*w^2 - 9*w + 7],
[769, 769, -2*w^4 + 4*w^3 + 7*w^2 - 11*w - 2],
[773, 773, 2*w^4 - 5*w^3 - 6*w^2 + 12*w + 1],
[773, 773, 3*w^4 - 4*w^3 - 12*w^2 + 3*w + 5],
[773, 773, 2*w^3 - w^2 - 10*w - 4],
[797, 797, -w^4 + 4*w^3 + 2*w^2 - 12*w - 3],
[811, 811, w^4 - 2*w^3 - 6*w^2 + 8*w + 4],
[823, 823, w^4 - 4*w^3 - w^2 + 10*w + 1],
[823, 823, 2*w^4 - 4*w^3 - 4*w^2 + 4*w - 3],
[827, 827, -w^4 + 2*w^3 + 2*w^2 - w + 3],
[827, 827, 2*w^4 - 4*w^3 - 5*w^2 + 7*w + 2],
[829, 829, -w^4 + 4*w^3 + 2*w^2 - 10*w - 2],
[829, 829, 3*w^3 - 3*w^2 - 11*w - 2],
[829, 829, w^4 - 2*w^3 - 3*w^2 + 2*w - 3],
[839, 839, -w^4 + 5*w^3 - 3*w^2 - 10*w + 5],
[839, 839, -w^4 + w^3 + 5*w^2 - 9],
[853, 853, 2*w^4 - 6*w^3 - 5*w^2 + 15*w + 4],
[859, 859, -3*w^4 + 8*w^3 + 6*w^2 - 18*w + 5],
[863, 863, w^4 - w^3 - 6*w^2 + 5*w + 2],
[877, 877, 3*w^3 - 4*w^2 - 8*w + 1],
[881, 881, 3*w^4 - 7*w^3 - 9*w^2 + 15*w + 3],
[881, 881, -w^3 + 7*w + 1],
[883, 883, -2*w^4 + 6*w^3 + 3*w^2 - 15*w],
[887, 887, w^4 - w^3 - 5*w^2 + w - 1],
[907, 907, 3*w^4 - 5*w^3 - 11*w^2 + 9*w + 5],
[919, 919, w^4 - 2*w^3 - 6*w^2 + 8*w + 7],
[937, 937, 2*w^4 - 2*w^3 - 9*w^2 + 2*w + 3],
[937, 937, -3*w^4 + 8*w^3 + 6*w^2 - 16*w + 1],
[937, 937, 3*w^4 - 5*w^3 - 11*w^2 + 7*w + 4],
[947, 947, w^3 - w^2 - 7*w + 2],
[947, 947, -w^4 + 2*w^3 + 4*w^2 - 7*w - 3],
[967, 967, -w^4 + 4*w^3 - 11*w],
[971, 971, 3*w^4 - 6*w^3 - 8*w^2 + 10*w],
[971, 971, w^4 - 8*w^2 + 2*w + 9],
[971, 971, w^4 - 4*w^3 + 2*w^2 + 7*w - 4],
[977, 977, -w^4 + 4*w^3 - 12*w + 5],
[983, 983, -w^4 + w^3 + 4*w^2 + 3*w - 2],
[983, 983, -2*w^4 + 3*w^3 + 8*w^2 - 6*w - 4],
[997, 997, 2*w^4 - 4*w^3 - 5*w^2 + 5*w - 3],
[997, 997, w^4 + w^3 - 5*w^2 - 9*w - 2]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

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

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

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {3, 3, w - 1}>] := -1;
ALEigenvalues[ideal<ZF | {7, 7, w^4 - 2*w^3 - 3*w^2 + 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;