/*
  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![4, 0, -9, 0, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {36,6,-1/2*w^2 + 1/2*w + 5}>;

primesArray := [
[4, 2, w],
[4, 2, -1/2*w^3 + 9/2*w],
[9, 3, 1/4*w^3 - 11/4*w - 1/2],
[9, 3, 1/4*w^3 - 11/4*w + 1/2],
[25, 5, 1/2*w^3 - 7/2*w],
[29, 29, -1/4*w^3 + 1/2*w^2 + 9/4*w - 1/2],
[29, 29, -1/2*w^2 - 1/2*w + 4],
[29, 29, -1/2*w^2 + 1/2*w + 4],
[29, 29, -1/4*w^3 - 1/2*w^2 + 9/4*w + 1/2],
[49, 7, 3/4*w^3 + 1/2*w^2 - 23/4*w - 3/2],
[49, 7, 1/2*w^3 + 1/2*w^2 - 3*w - 3],
[61, 61, 1/2*w^3 + 1/2*w^2 - 5*w - 1],
[61, 61, -1/4*w^3 - 1/2*w^2 + 13/4*w + 7/2],
[61, 61, -1/4*w^3 + 1/2*w^2 + 13/4*w - 7/2],
[61, 61, -1/2*w^3 + 1/2*w^2 + 5*w - 1],
[79, 79, 3/4*w^3 - 25/4*w - 9/2],
[79, 79, -1/4*w^3 + 1/2*w^2 - 3/4*w - 3/2],
[79, 79, 1/4*w^3 - 3/4*w - 9/2],
[79, 79, 3/2*w^3 + 1/2*w^2 - 13*w - 3],
[101, 101, 1/4*w^3 + w^2 - 7/4*w - 5/2],
[101, 101, 5/4*w^3 - 1/2*w^2 - 45/4*w + 9/2],
[101, 101, 1/4*w^3 + w^2 - 7/4*w - 13/2],
[101, 101, 1/4*w^3 - w^2 - 7/4*w + 5/2],
[121, 11, 3/4*w^3 - 21/4*w - 1/2],
[121, 11, 1/4*w^3 - 7/4*w - 7/2],
[131, 131, -3/4*w^3 + 29/4*w - 1/2],
[131, 131, 1/4*w^3 - 15/4*w - 1/2],
[131, 131, 1/4*w^3 - 15/4*w + 1/2],
[131, 131, 3/4*w^3 - 29/4*w - 1/2],
[139, 139, w^2 + w + 1],
[139, 139, 1/2*w^3 + w^2 - 9/2*w - 10],
[139, 139, 1/2*w^3 - w^2 - 9/2*w + 10],
[139, 139, w^2 - w + 1],
[169, 13, 1/2*w^3 - 11/2*w],
[179, 179, -3/4*w^3 + w^2 + 21/4*w - 13/2],
[179, 179, 1/2*w^3 + 1/2*w^2 - 6*w - 5],
[179, 179, -3/4*w^3 - w^2 + 25/4*w + 7/2],
[179, 179, -3/2*w^2 + 1/2*w + 2],
[181, 181, -1/2*w^2 + 1/2*w - 2],
[181, 181, 1/4*w^3 - 1/2*w^2 - 9/4*w + 13/2],
[181, 181, 5/4*w^3 - 39/4*w + 3/2],
[181, 181, 1/2*w^2 + 1/2*w + 2],
[191, 191, -w^3 + 1/2*w^2 + 19/2*w - 4],
[191, 191, 1/4*w^3 + 1/2*w^2 - 17/4*w - 1/2],
[191, 191, 1/2*w^3 + 1/2*w^2 - 6*w - 3],
[191, 191, -3/4*w^3 + 1/2*w^2 + 31/4*w - 3/2],
[199, 199, 1/4*w^3 + 1/2*w^2 - 5/4*w - 15/2],
[199, 199, -1/4*w^3 + w^2 + 11/4*w - 17/2],
[199, 199, 1/2*w^3 + 1/2*w^2 - 4*w + 3],
[199, 199, -1/4*w^3 - w^2 + 11/4*w + 1/2],
[211, 211, -3/4*w^3 + 1/2*w^2 + 19/4*w - 9/2],
[211, 211, -w^3 - 1/2*w^2 + 15/2*w],
[211, 211, -1/4*w^3 + 1/2*w^2 + 13/4*w - 11/2],
[211, 211, 3/4*w^3 + 1/2*w^2 - 19/4*w - 9/2],
[251, 251, 3/4*w^3 - 1/2*w^2 - 27/4*w + 1/2],
[251, 251, 1/2*w^2 - 3/2*w - 4],
[251, 251, 1/2*w^2 + 3/2*w - 4],
[251, 251, -3/4*w^3 - 1/2*w^2 + 27/4*w + 1/2],
[269, 269, -3/4*w^3 + 17/4*w - 1/2],
[269, 269, 1/2*w^3 + 1/2*w^2 - 4*w - 7],
[269, 269, -5/4*w^3 + 39/4*w + 1/2],
[269, 269, 3/4*w^3 - 17/4*w - 1/2],
[289, 17, -1/4*w^3 + 11/4*w - 9/2],
[289, 17, 1/4*w^3 - 11/4*w - 9/2],
[311, 311, 3/4*w^3 + w^2 - 29/4*w - 19/2],
[311, 311, 3/4*w^3 + w^2 - 25/4*w - 21/2],
[311, 311, 3/4*w^3 - w^2 - 13/4*w + 7/2],
[311, 311, -3/4*w^3 + w^2 + 29/4*w - 19/2],
[361, 19, -1/4*w^3 + 7/4*w - 9/2],
[361, 19, -w^3 + 7*w - 1],
[389, 389, 5/4*w^3 - 3/2*w^2 - 41/4*w + 13/2],
[389, 389, 5/4*w^3 - 47/4*w + 9/2],
[389, 389, 1/2*w^3 + 3/2*w^2 - 2*w - 7],
[389, 389, 5/4*w^3 + 3/2*w^2 - 41/4*w - 13/2],
[419, 419, 5/4*w^3 - 1/2*w^2 - 37/4*w + 5/2],
[419, 419, w^3 - 1/2*w^2 - 13/2*w + 2],
[419, 419, w^3 + 1/2*w^2 - 13/2*w - 2],
[419, 419, 5/4*w^3 + 1/2*w^2 - 37/4*w - 5/2],
[439, 439, -1/4*w^3 - 1/2*w^2 - 7/4*w + 5/2],
[439, 439, 2*w^3 + 1/2*w^2 - 35/2*w - 2],
[439, 439, 2*w^3 - 1/2*w^2 - 35/2*w + 2],
[439, 439, -1/4*w^3 + 1/2*w^2 - 7/4*w - 5/2],
[491, 491, 1/4*w^3 + 3/2*w^2 - 9/4*w - 7/2],
[491, 491, -3/2*w^2 + 1/2*w + 10],
[491, 491, -1/4*w^3 - 5/4*w + 5/2],
[491, 491, 1/4*w^3 - 3/2*w^2 - 9/4*w + 7/2],
[521, 521, 1/2*w^2 - 3/2*w - 6],
[521, 521, -3/4*w^3 - 1/2*w^2 + 27/4*w - 3/2],
[521, 521, 3/4*w^3 - 1/2*w^2 - 27/4*w - 3/2],
[521, 521, 1/2*w^2 + 3/2*w - 6],
[529, 23, 1/2*w^3 - 11/2*w - 6],
[529, 23, -1/2*w^3 + 11/2*w - 6],
[569, 569, 3/4*w^3 + w^2 - 33/4*w - 3/2],
[569, 569, -3/4*w^3 + w^2 + 33/4*w - 15/2],
[569, 569, 3/4*w^3 + w^2 - 33/4*w - 15/2],
[569, 569, -3/4*w^3 + w^2 + 33/4*w - 3/2],
[571, 571, -1/2*w^3 - 3/2*w^2 + 5*w + 9],
[571, 571, 1/4*w^3 - 19/4*w + 1/2],
[571, 571, 1/4*w^3 - 19/4*w - 1/2],
[571, 571, 3/4*w^3 + 3/2*w^2 - 27/4*w - 19/2],
[599, 599, -3/4*w^3 - w^2 + 21/4*w + 1/2],
[599, 599, 1/2*w^3 - 3/2*w^2 - 6*w + 9],
[599, 599, -1/2*w^3 - 3/2*w^2 + 6*w + 9],
[599, 599, 3/4*w^3 - w^2 - 21/4*w + 1/2],
[601, 601, 1/2*w^3 - w^2 - 9/2*w],
[601, 601, -w^2 - w + 9],
[601, 601, -7/4*w^3 + 57/4*w - 7/2],
[601, 601, 1/2*w^3 + w^2 - 9/2*w],
[641, 641, w^3 + 1/2*w^2 - 11/2*w - 2],
[641, 641, -7/4*w^3 + 1/2*w^2 + 55/4*w - 5/2],
[641, 641, 7/4*w^3 + 1/2*w^2 - 55/4*w - 5/2],
[641, 641, -w^3 + 1/2*w^2 + 11/2*w - 2],
[659, 659, 7/4*w^3 - 1/2*w^2 - 59/4*w + 3/2],
[659, 659, 5/2*w^3 - w^2 - 41/2*w + 8],
[659, 659, 5/2*w^3 + w^2 - 41/2*w - 8],
[659, 659, w^3 + 3/2*w^2 - 19/2*w - 12],
[701, 701, 1/2*w^3 - 3/2*w^2 - 3*w + 5],
[701, 701, 3/4*w^3 + 3/2*w^2 - 23/4*w - 17/2],
[701, 701, 3/4*w^3 - 3/2*w^2 - 23/4*w + 17/2],
[701, 701, 1/2*w^3 + 3/2*w^2 - 3*w - 5],
[719, 719, 3/4*w^3 - 3/2*w^2 - 23/4*w + 13/2],
[719, 719, -1/2*w^3 + 3/2*w^2 + 3*w - 7],
[719, 719, 1/2*w^3 + 3/2*w^2 - 3*w - 7],
[719, 719, 3/4*w^3 + 3/2*w^2 - 23/4*w - 13/2],
[751, 751, 1/2*w^3 - 3/2*w - 4],
[751, 751, -3/2*w^3 + 25/2*w + 4],
[751, 751, 3/2*w^3 - 25/2*w + 4],
[751, 751, -1/2*w^3 + 3/2*w - 4],
[809, 809, 3/2*w^2 - 1/2*w - 4],
[809, 809, 1/4*w^3 + 3/2*w^2 - 9/4*w - 19/2],
[809, 809, 1/4*w^3 - 3/2*w^2 - 9/4*w + 19/2],
[809, 809, 3/2*w^2 + 1/2*w - 4],
[829, 829, -3/2*w^3 - 1/2*w^2 + 11*w + 1],
[829, 829, 5/4*w^3 - 1/2*w^2 - 33/4*w + 7/2],
[829, 829, 5/4*w^3 + 1/2*w^2 - 33/4*w - 7/2],
[829, 829, -3/2*w^3 + 1/2*w^2 + 11*w - 1],
[859, 859, -1/2*w^3 - 1/2*w^2 + 7*w + 3],
[859, 859, 5/4*w^3 + 1/2*w^2 - 49/4*w - 3/2],
[859, 859, -5/4*w^3 + 1/2*w^2 + 49/4*w - 3/2],
[859, 859, 1/2*w^3 - 1/2*w^2 - 7*w + 3],
[881, 881, 5/4*w^3 + 2*w^2 - 43/4*w - 29/2],
[881, 881, -1/4*w^3 + 2*w^2 - 1/4*w - 7/2],
[881, 881, -w^3 + 3/2*w^2 + 15/2*w - 12],
[881, 881, 5/4*w^3 - 2*w^2 - 43/4*w + 29/2],
[911, 911, 1/4*w^3 - w^2 - 7/4*w - 5/2],
[911, 911, -1/4*w^3 + w^2 + 7/4*w - 23/2],
[911, 911, -9/4*w^3 + 1/2*w^2 + 73/4*w - 9/2],
[911, 911, 9/4*w^3 + w^2 - 75/4*w - 11/2],
[919, 919, 5/4*w^3 - 1/2*w^2 - 41/4*w - 1/2],
[919, 919, -3/4*w^3 - w^2 + 29/4*w + 3/2],
[919, 919, -3/4*w^3 + w^2 + 29/4*w - 3/2],
[919, 919, -5/4*w^3 - 1/2*w^2 + 41/4*w - 1/2],
[961, 31, 1/2*w^3 - 7/2*w - 6],
[961, 31, 5/4*w^3 - 35/4*w - 1/2],
[971, 971, -7/4*w^3 + 61/4*w - 7/2],
[971, 971, -1/4*w^3 - 5/4*w - 7/2],
[971, 971, 11/4*w^3 + w^2 - 97/4*w - 17/2],
[971, 971, 1/4*w^3 - 2*w^2 - 7/4*w + 7/2],
[991, 991, -5/4*w^3 - w^2 + 43/4*w + 7/2],
[991, 991, w^3 - 1/2*w^2 - 17/2*w - 2],
[991, 991, -w^3 - 1/2*w^2 + 17/2*w - 2],
[991, 991, -5/4*w^3 + w^2 + 43/4*w - 7/2]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

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

heckeEigenvaluesArray := [-1, e, 0, -1, -e + 1, 2*e, 2*e, -3*e + 5, e - 5, -2*e + 10, e - 5, e + 7, -4*e + 2, 2, 2*e + 2, -3*e + 5, -3*e + 5, 2*e - 10, 10, 3*e - 13, e + 7, -6*e + 2, 2, 2*e + 12, 4*e - 8, 8*e - 8, -5*e + 7, -4*e - 8, -6*e + 2, 2*e - 20, 5*e - 5, 4*e, 10, 2*e - 10, -10, 2*e - 10, -2*e + 10, 10, -4*e + 2, -4*e + 2, 2, e + 7, -5*e - 3, -8, -6*e + 12, 2*e - 8, -2*e + 20, -2*e + 10, -e + 25, -4*e, 2, 2*e + 2, 2*e + 12, 3*e + 7, -6*e + 12, 3*e - 3, e + 7, -6*e + 2, -8*e + 10, e + 5, -10, 6*e, 9*e - 15, e + 5, -7*e + 7, -5*e - 3, -6*e + 12, -e + 17, 2*e + 2, -5*e - 3, -5*e + 5, 2*e - 20, -8*e, -2*e - 30, -10*e + 10, -4*e + 10, 3*e + 15, -8*e + 20, -10*e + 10, 0, -3*e - 15, -30, -e - 23, 10*e + 2, -10*e + 22, -10*e - 8, 8*e - 8, 3*e + 17, 12*e - 18, 2*e - 18, -4*e - 20, 3*e - 5, 3*e + 15, -e - 5, -10, 2*e, 8*e - 8, 2*e + 22, 4*e - 8, -6*e + 32, -2*e - 30, -4*e + 20, -12*e, 10*e - 10, 14*e - 8, -6*e + 22, 4*e - 28, -8*e - 18, -10*e - 8, 7*e + 17, 10*e + 2, -4*e + 22, -14*e + 20, 8*e - 10, -8*e + 20, 2*e + 20, -10*e + 2, -4*e + 2, 4*e + 12, 5*e - 13, 12*e - 10, -10*e, 6*e - 30, -2*e + 20, -19*e + 17, -2*e + 32, 4*e + 12, -2*e - 28, 10, -8*e + 10, 2*e + 20, -13*e + 15, -12*e + 10, 12*e + 10, -12*e + 10, 3*e + 25, -5*e - 5, -4*e + 40, 8*e, -12*e + 30, -4*e - 28, 8*e + 2, 4*e - 18, 10*e + 12, 13*e - 13, -3*e + 27, 9*e + 27, -8*e + 32, -4*e - 40, -4*e + 20, 5*e - 25, 8*e - 40, 2, -14*e - 8, 12*e - 18, 9*e + 7, 12, 2*e + 32, 16*e - 18, 2*e + 32, -8, -e - 13];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {4,2,-w}>] := 1;
ALEigenvalues[ideal<ZF | {9,3,-1/4*w^3 + 11/4*w - 1/2}>] := 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;