/*
  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![11, 6, -8, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {31, 31, w^3 - 6*w + 1}>;

primesArray := [
[9, 3, 1/3*w^3 - 8/3*w - 2/3],
[9, 3, -w + 1],
[11, 11, w],
[11, 11, -1/3*w^3 + 2/3*w + 5/3],
[11, 11, 1/3*w^3 - 8/3*w + 1/3],
[11, 11, -2/3*w^3 + 13/3*w + 4/3],
[16, 2, 2],
[25, 5, 2/3*w^3 - 10/3*w - 1/3],
[29, 29, -w - 3],
[29, 29, -1/3*w^3 + 8/3*w - 10/3],
[31, 31, w^3 - 6*w + 1],
[31, 31, w^3 - 6*w - 2],
[41, 41, 2/3*w^3 + w^2 - 13/3*w - 10/3],
[41, 41, 2/3*w^3 - 13/3*w + 5/3],
[59, 59, 2/3*w^3 - 13/3*w + 8/3],
[59, 59, w^3 + w^2 - 6*w - 4],
[79, 79, 2/3*w^3 + w^2 - 10/3*w - 19/3],
[79, 79, 1/3*w^3 + w^2 - 2/3*w - 17/3],
[89, 89, 4/3*w^3 - 23/3*w - 5/3],
[89, 89, 1/3*w^3 + 2*w^2 - 5/3*w - 32/3],
[101, 101, 5/3*w^3 + w^2 - 31/3*w - 16/3],
[101, 101, 2/3*w^3 - w^2 - 4/3*w + 8/3],
[109, 109, -w^3 - 2*w^2 + 7*w + 7],
[109, 109, -1/3*w^3 + 2*w^2 + 2/3*w - 16/3],
[131, 131, w^3 - w^2 - 7*w + 5],
[131, 131, -2/3*w^3 - w^2 + 4/3*w + 13/3],
[139, 139, -2/3*w^3 - w^2 + 13/3*w + 4/3],
[139, 139, w^2 - w - 7],
[151, 151, -5/3*w^3 + 28/3*w - 5/3],
[151, 151, -1/3*w^3 + w^2 + 2/3*w - 16/3],
[151, 151, w^3 + w^2 - 6*w - 3],
[151, 151, -w^3 + w^2 + 7*w - 6],
[179, 179, 5/3*w^3 + w^2 - 28/3*w - 19/3],
[179, 179, -1/3*w^3 + w^2 + 11/3*w - 4/3],
[181, 181, -4/3*w^3 - 2*w^2 + 20/3*w + 29/3],
[181, 181, 1/3*w^3 + w^2 - 2/3*w - 20/3],
[181, 181, w^3 + w^2 - 5*w - 7],
[181, 181, 5/3*w^3 + w^2 - 28/3*w - 16/3],
[191, 191, 2*w - 1],
[191, 191, 2/3*w^3 - 16/3*w - 1/3],
[199, 199, 1/3*w^3 - 11/3*w + 4/3],
[199, 199, 1/3*w^3 - 11/3*w - 2/3],
[211, 211, -2/3*w^3 - w^2 + 19/3*w + 16/3],
[211, 211, 2/3*w^3 + w^2 - 19/3*w - 7/3],
[229, 229, 1/3*w^3 - 11/3*w + 1/3],
[239, 239, -5/3*w^3 - w^2 + 25/3*w + 19/3],
[239, 239, -4/3*w^3 + w^2 + 20/3*w - 7/3],
[251, 251, -2/3*w^3 + w^2 + 16/3*w - 14/3],
[251, 251, -1/3*w^3 - w^2 - 1/3*w + 14/3],
[269, 269, 2/3*w^3 + w^2 - 7/3*w - 19/3],
[269, 269, 1/3*w^3 + 2*w^2 - 2/3*w - 32/3],
[281, 281, w^3 - w^2 - 5*w + 1],
[281, 281, 4/3*w^3 + w^2 - 20/3*w - 23/3],
[289, 17, 2/3*w^3 + w^2 - 16/3*w - 4/3],
[289, 17, 1/3*w^3 + w^2 - 11/3*w - 20/3],
[331, 331, 1/3*w^3 + 2*w^2 - 8/3*w - 23/3],
[331, 331, 2/3*w^3 + 2*w^2 - 13/3*w - 28/3],
[349, 349, 4/3*w^3 - 17/3*w + 1/3],
[349, 349, -5/3*w^3 + 28/3*w - 2/3],
[361, 19, -1/3*w^3 + 5/3*w + 14/3],
[361, 19, 4/3*w^3 - 20/3*w - 5/3],
[379, 379, 4/3*w^3 + 2*w^2 - 23/3*w - 41/3],
[379, 379, -5/3*w^3 + 31/3*w - 5/3],
[389, 389, 5/3*w^3 + w^2 - 25/3*w - 16/3],
[389, 389, -4/3*w^3 + w^2 + 20/3*w - 10/3],
[401, 401, -1/3*w^3 - 3*w^2 + 5/3*w + 26/3],
[401, 401, 1/3*w^3 + w^2 - 11/3*w - 23/3],
[401, 401, -2/3*w^3 - 3*w^2 + 10/3*w + 52/3],
[401, 401, 2/3*w^3 + w^2 - 16/3*w - 1/3],
[409, 409, -2*w^3 - w^2 + 11*w + 2],
[409, 409, -5/3*w^3 - 2*w^2 + 28/3*w + 37/3],
[409, 409, 1/3*w^3 - 11/3*w - 17/3],
[409, 409, 5/3*w^3 + 3*w^2 - 25/3*w - 37/3],
[419, 419, -1/3*w^3 + 3*w^2 - 1/3*w - 37/3],
[419, 419, 2*w^3 + 3*w^2 - 12*w - 13],
[421, 421, -1/3*w^3 + 3*w^2 + 2/3*w - 31/3],
[421, 421, -5/3*w^3 - w^2 + 25/3*w + 13/3],
[421, 421, -w^3 + 7*w - 4],
[421, 421, -4/3*w^3 + w^2 + 20/3*w - 13/3],
[439, 439, -4/3*w^3 + w^2 + 14/3*w - 7/3],
[439, 439, 4/3*w^3 - 17/3*w - 5/3],
[439, 439, -7/3*w^3 - w^2 + 41/3*w + 17/3],
[439, 439, 5/3*w^3 - 28/3*w - 4/3],
[449, 449, w^3 + 3*w^2 - 6*w - 17],
[449, 449, 1/3*w^3 + 3*w^2 - 8/3*w - 26/3],
[449, 449, -4/3*w^3 + 26/3*w + 17/3],
[449, 449, 7/3*w^3 + w^2 - 38/3*w - 8/3],
[461, 461, -5/3*w^3 + w^2 + 28/3*w - 14/3],
[461, 461, -2/3*w^3 - 2*w^2 + 10/3*w + 43/3],
[461, 461, -2*w^3 - w^2 + 12*w + 4],
[461, 461, -w^3 + 8*w - 5],
[479, 479, w^3 + w^2 - 7*w - 2],
[479, 479, -4/3*w^3 - w^2 + 23/3*w + 5/3],
[499, 499, -4/3*w^3 - 2*w^2 + 23/3*w + 17/3],
[499, 499, -4/3*w^3 - w^2 + 29/3*w + 8/3],
[521, 521, 2*w^3 - 11*w + 2],
[521, 521, 4/3*w^3 - w^2 - 14/3*w + 1/3],
[569, 569, w^3 + 3*w^2 - 6*w - 9],
[569, 569, 4/3*w^3 + w^2 - 26/3*w - 32/3],
[569, 569, -w^3 + 6*w - 6],
[569, 569, -1/3*w^3 - 3*w^2 + 8/3*w + 50/3],
[571, 571, 2/3*w^3 + 3*w^2 - 10/3*w - 46/3],
[571, 571, 2*w^2 - 15],
[599, 599, -w^3 + 2*w^2 + 4*w - 10],
[599, 599, -w^2 - 3*w + 8],
[601, 601, 5/3*w^3 + 3*w^2 - 28/3*w - 34/3],
[601, 601, 1/3*w^3 - 3*w^2 - 2/3*w + 43/3],
[619, 619, 2/3*w^3 - w^2 - 13/3*w + 2/3],
[619, 619, 2/3*w^3 + w^2 - 7/3*w - 25/3],
[619, 619, 2*w^3 + w^2 - 11*w - 6],
[619, 619, -2/3*w^3 + 13/3*w - 17/3],
[631, 631, 2/3*w^3 + 2*w^2 - 19/3*w - 31/3],
[631, 631, w^3 + 2*w^2 - 8*w - 6],
[631, 631, 2/3*w^3 + 2*w^2 - 4/3*w - 31/3],
[631, 631, -2/3*w^3 + 2*w^2 + 16/3*w - 23/3],
[659, 659, -2/3*w^3 + 1/3*w + 16/3],
[659, 659, 5/3*w^3 - 34/3*w - 13/3],
[661, 661, w^3 - 8*w + 2],
[661, 661, 4/3*w^3 + w^2 - 23/3*w + 1/3],
[661, 661, 5/3*w^3 + w^2 - 31/3*w - 10/3],
[661, 661, -3*w - 1],
[691, 691, -4/3*w^3 + 29/3*w + 2/3],
[691, 691, 1/3*w^3 + 4/3*w - 5/3],
[691, 691, -5/3*w^3 + w^2 + 25/3*w - 5/3],
[691, 691, 2*w^3 + w^2 - 10*w - 7],
[701, 701, -2/3*w^3 - 3*w^2 + 19/3*w + 46/3],
[701, 701, w^3 - 2*w^2 - 6*w + 8],
[701, 701, 4/3*w^3 + 2*w^2 - 17/3*w - 29/3],
[701, 701, 5/3*w^3 + 3*w^2 - 28/3*w - 52/3],
[709, 709, 1/3*w^3 - 14/3*w + 10/3],
[709, 709, -2/3*w^3 + 19/3*w + 7/3],
[719, 719, 3*w^2 - w - 10],
[719, 719, -4/3*w^3 - 3*w^2 + 23/3*w + 47/3],
[739, 739, 5/3*w^3 + w^2 - 34/3*w - 10/3],
[739, 739, 4/3*w^3 + 2*w^2 - 23/3*w - 14/3],
[761, 761, 7/3*w^3 + 2*w^2 - 41/3*w - 26/3],
[761, 761, 2*w^3 + 3*w^2 - 12*w - 15],
[769, 769, -2/3*w^3 + 19/3*w - 8/3],
[769, 769, -4/3*w^3 + 2*w^2 + 17/3*w - 31/3],
[769, 769, 1/3*w^3 - 14/3*w - 5/3],
[769, 769, -1/3*w^3 + 3*w^2 + 2/3*w - 34/3],
[809, 809, -4/3*w^3 + w^2 + 26/3*w - 7/3],
[809, 809, w^3 + w^2 - 3*w - 7],
[811, 811, 4/3*w^3 + w^2 - 17/3*w - 23/3],
[811, 811, -1/3*w^3 - 2*w^2 + 2/3*w + 38/3],
[821, 821, 8/3*w^3 + 3*w^2 - 43/3*w - 34/3],
[821, 821, 2/3*w^3 + 2*w^2 - 7/3*w - 34/3],
[821, 821, -w^3 - w^2 + 4*w + 8],
[821, 821, 2/3*w^3 + 4*w^2 - 7/3*w - 58/3],
[829, 829, 4/3*w^3 + 3*w^2 - 26/3*w - 50/3],
[829, 829, 7/3*w^3 - 41/3*w - 5/3],
[839, 839, 2*w^3 + w^2 - 10*w - 3],
[839, 839, 5/3*w^3 - w^2 - 25/3*w + 17/3],
[841, 29, 5/3*w^3 - 25/3*w - 7/3],
[859, 859, -7/3*w^3 - w^2 + 32/3*w + 29/3],
[859, 859, 2*w^3 + w^2 - 14*w - 6],
[859, 859, 1/3*w^3 + 2*w^2 - 2/3*w - 47/3],
[859, 859, -5/3*w^3 - 2*w^2 + 31/3*w + 49/3],
[881, 881, w^3 + 4*w^2 - 4*w - 20],
[881, 881, 8/3*w^3 + 2*w^2 - 43/3*w - 34/3],
[919, 919, -w^3 + 3*w - 7],
[919, 919, 1/3*w^3 + 3*w^2 - 5/3*w - 35/3],
[919, 919, -5/3*w^3 + 31/3*w - 23/3],
[919, 919, 2/3*w^3 + 3*w^2 - 10/3*w - 43/3],
[929, 929, w^2 - 2*w - 9],
[929, 929, w^3 + w^2 - 7*w + 1],
[961, 31, 2/3*w^3 - 10/3*w - 19/3],
[971, 971, -w^3 - 3*w^2 + 7*w + 16],
[971, 971, -5/3*w^3 + w^2 + 25/3*w - 8/3],
[971, 971, 2*w^3 + w^2 - 10*w - 6],
[971, 971, 1/3*w^3 + 2*w^2 - 8/3*w - 47/3]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^2 + 4*x - 1;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, e, -1/2*e + 5/2, -1/2*e - 5/2, 2*e + 5, -3/2*e - 9/2, -1/2*e - 15/2, -1/2*e - 3/2, -1/2*e + 1/2, -2*e - 5, 1, -2*e - 7, -7/2*e - 19/2, 4*e + 8, -1/2*e - 27/2, -2*e - 4, -9/2*e - 31/2, 5*e + 11, 7/2*e + 17/2, -3/2*e + 7/2, 3*e + 2, -2*e + 7, -7/2*e + 1/2, -9/2*e - 13/2, -5/2*e + 23/2, 3/2*e - 21/2, 9*e + 21, -3*e - 8, e - 8, -1/2*e - 35/2, 3/2*e - 27/2, 4*e + 3, -5*e - 17, -1/2*e - 21/2, -3/2*e - 39/2, 5*e - 2, -5/2*e + 11/2, 6*e + 8, 2*e - 16, -2*e + 16, -24, -7*e - 8, -3*e - 14, 5*e + 12, 5*e + 24, -23/2*e - 49/2, 15/2*e + 57/2, 5*e + 17, 4*e + 20, 5/2*e - 33/2, 11/2*e + 19/2, 3*e + 20, 9/2*e + 21/2, 7*e + 24, 13/2*e + 51/2, -e + 2, 6*e + 31, -11/2*e - 57/2, -15/2*e - 25/2, 11/2*e + 47/2, 1, 1/2*e + 5/2, 2*e - 7, 19/2*e + 29/2, -e - 4, -21/2*e - 47/2, -4*e - 19, -13/2*e - 1/2, -5*e - 26, -7*e - 27, 2*e - 3, 15/2*e + 11/2, -23, 6*e - 14, 21/2*e + 65/2, -27/2*e - 53/2, 15*e + 36, 7*e + 2, -e - 21, 3*e + 28, -7*e + 6, 7/2*e - 7/2, 5*e - 5, -9*e - 14, -11/2*e - 29/2, 5*e + 4, -6*e - 13, -13/2*e - 45/2, e + 8, 10*e + 16, -31/2*e - 61/2, 8*e + 25, 9/2*e - 29/2, 5*e + 24, 10*e + 24, -11/2*e - 61/2, 31/2*e + 43/2, 3*e + 29, 15/2*e - 25/2, -15, -19/2*e - 17/2, e - 28, -7*e + 1, -10*e - 3, -5*e - 13, 5*e + 40, 2*e - 11, 17/2*e + 3/2, -11/2*e - 33/2, 3*e + 34, 6*e - 10, -7/2*e - 21/2, 31/2*e + 65/2, 23/2*e + 5/2, -15/2*e - 11/2, 3*e + 11, -9*e - 3, -6*e - 5, 5/2*e + 15/2, -4*e + 12, -3*e + 6, -13/2*e - 67/2, -8*e + 11, 9/2*e + 53/2, 17/2*e + 69/2, -23/2*e - 21/2, 5*e - 13, -11/2*e - 33/2, -6*e + 13, -3*e - 40, -13*e - 5, 43/2*e + 91/2, 17/2*e + 9/2, 5/2*e - 17/2, -5/2*e - 17/2, -1/2*e + 19/2, -25/2*e - 79/2, 27/2*e + 19/2, -5/2*e + 1/2, 5*e + 10, 7*e - 13, 13*e + 10, -12*e - 20, 6*e + 22, -29/2*e - 23/2, 4*e + 13, -9*e - 19, -7*e + 10, -4*e - 23, -10*e + 7, 7/2*e - 67/2, 27/2*e + 75/2, -15/2*e - 29/2, -3*e - 26, -3/2*e + 1/2, 21, -1/2*e - 85/2, -15*e - 44, 11*e + 29, -21/2*e - 3/2, e + 36, 19/2*e + 107/2, -13/2*e - 33/2, 18*e + 43, -35, 12*e + 19, -3*e - 2, -16*e - 17, 3/2*e + 87/2, 11*e + 50, -27/2*e - 59/2];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {31, 31, w^3 - 6*w + 1}>] := -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;