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

NN := ideal<ZF | {1024, 16, 2*w^3 - 8*w}>;

primesArray := [
[4, 2, -w],
[4, 2, 1/2*w^3 - 1/2*w^2 - 5/2*w + 1],
[19, 19, 1/2*w^3 - 3/2*w^2 - 1/2*w + 4],
[19, 19, 1/2*w^3 + 1/2*w^2 - 5/2*w - 1],
[25, 5, w^3 - w^2 - 3*w + 1],
[29, 29, -1/2*w^3 + 3/2*w^2 + 3/2*w - 1],
[29, 29, -w^2 + 5],
[31, 31, -w^3 + 2*w^2 + 3*w - 3],
[31, 31, -1/2*w^3 - 1/2*w^2 + 3/2*w + 3],
[41, 41, -1/2*w^3 - 1/2*w^2 + 5/2*w],
[41, 41, -1/2*w^3 + 3/2*w^2 + 1/2*w - 5],
[59, 59, w^3 - w^2 - 5*w + 1],
[59, 59, 3/2*w^3 - 1/2*w^2 - 11/2*w - 1],
[61, 61, 3/2*w^3 - 1/2*w^2 - 11/2*w - 2],
[61, 61, -2*w^2 + w + 7],
[71, 71, 1/2*w^3 - 3/2*w^2 - 5/2*w + 6],
[71, 71, -1/2*w^3 + 5/2*w^2 - 1/2*w - 5],
[71, 71, 1/2*w^3 + 3/2*w^2 - 7/2*w - 5],
[71, 71, -w^3 + 3*w^2 + 2*w - 7],
[81, 3, -3],
[89, 89, -1/2*w^3 + 5/2*w^2 + 3/2*w - 6],
[101, 101, w^2 - 2*w - 5],
[101, 101, -1/2*w^3 - 1/2*w^2 + 7/2*w - 1],
[109, 109, w^3 - 2*w^2 - 3*w + 1],
[109, 109, -5/2*w^3 + 7/2*w^2 + 17/2*w - 8],
[109, 109, 1/2*w^3 + 1/2*w^2 - 3/2*w - 5],
[109, 109, -2*w^3 + 2*w^2 + 8*w - 1],
[121, 11, 3/2*w^3 - 3/2*w^2 - 9/2*w + 1],
[121, 11, -3/2*w^3 + 3/2*w^2 + 9/2*w - 2],
[131, 131, -1/2*w^3 - 3/2*w^2 + 11/2*w + 6],
[131, 131, -1/2*w^3 - 1/2*w^2 + 9/2*w + 2],
[131, 131, -1/2*w^3 - 1/2*w^2 + 9/2*w + 1],
[131, 131, -3/2*w^3 + 3/2*w^2 + 13/2*w - 5],
[149, 149, 5/2*w^3 - 5/2*w^2 - 17/2*w],
[149, 149, -1/2*w^3 + 3/2*w^2 + 3/2*w + 1],
[151, 151, 2*w^3 - 2*w^2 - 7*w + 3],
[151, 151, 5/2*w^3 - 5/2*w^2 - 19/2*w + 1],
[181, 181, -w^3 + w^2 + w - 3],
[181, 181, 1/2*w^3 - 5/2*w^2 + 1/2*w + 8],
[191, 191, 1/2*w^3 + 3/2*w^2 - 9/2*w - 8],
[191, 191, 3/2*w^3 - 3/2*w^2 - 13/2*w - 1],
[211, 211, -1/2*w^3 + 3/2*w^2 - 1/2*w - 5],
[211, 211, -w^3 + 5*w - 1],
[229, 229, -2*w^3 + 4*w^2 + 6*w - 7],
[229, 229, 3/2*w^3 - 3/2*w^2 - 15/2*w + 4],
[239, 239, w^3 + w^2 - 4*w - 1],
[239, 239, -3/2*w^3 + 1/2*w^2 + 15/2*w - 1],
[241, 241, 1/2*w^3 + 1/2*w^2 - 1/2*w - 4],
[241, 241, 3/2*w^3 - 5/2*w^2 - 11/2*w + 3],
[269, 269, -w^3 + 2*w^2 + w - 5],
[269, 269, -3/2*w^3 + 5/2*w^2 + 7/2*w - 8],
[269, 269, 3/2*w^3 - 7/2*w^2 - 7/2*w + 4],
[269, 269, -w^3 - w^2 + 4*w + 7],
[281, 281, -1/2*w^3 + 5/2*w^2 - 1/2*w - 9],
[281, 281, 1/2*w^3 + 3/2*w^2 - 7/2*w - 1],
[289, 17, -w^3 + 2*w^2 + 5*w - 1],
[289, 17, -1/2*w^3 + 3/2*w^2 + 7/2*w - 7],
[331, 331, 3/2*w^3 - 3/2*w^2 - 15/2*w + 2],
[331, 331, -5/2*w^3 + 7/2*w^2 + 17/2*w - 4],
[331, 331, 3*w - 1],
[331, 331, -3/2*w^3 + 1/2*w^2 + 7/2*w + 3],
[349, 349, -1/2*w^3 - 1/2*w^2 + 5/2*w - 3],
[349, 349, -1/2*w^3 + 3/2*w^2 + 1/2*w - 8],
[359, 359, -5/2*w^3 + 9/2*w^2 + 11/2*w - 6],
[359, 359, -1/2*w^3 + 7/2*w^2 + 1/2*w - 6],
[361, 19, -2*w^3 + 2*w^2 + 6*w - 3],
[379, 379, 1/2*w^3 + 5/2*w^2 - 9/2*w - 7],
[379, 379, w^3 - w^2 - 6*w + 3],
[389, 389, 1/2*w^3 - 1/2*w^2 - 5/2*w - 4],
[389, 389, w - 5],
[401, 401, 2*w^2 + w - 7],
[401, 401, -3/2*w^3 + 7/2*w^2 + 11/2*w - 6],
[401, 401, w^3 - 3*w^2 - 3*w + 3],
[401, 401, 2*w^2 - 9],
[409, 409, -1/2*w^3 - 5/2*w^2 + 9/2*w + 8],
[409, 409, 1/2*w^3 + 3/2*w^2 - 3/2*w - 7],
[409, 409, -3/2*w^3 + 7/2*w^2 + 9/2*w - 5],
[409, 409, -1/2*w^3 + 7/2*w^2 - 3/2*w - 7],
[419, 419, 2*w^3 - 11*w - 5],
[419, 419, 5/2*w^3 - 3/2*w^2 - 17/2*w + 2],
[421, 421, -1/2*w^3 + 5/2*w^2 + 7/2*w - 2],
[421, 421, -w^3 + w^2 + w + 5],
[431, 431, 2*w^3 - w^2 - 6*w - 1],
[431, 431, -5/2*w^3 + 7/2*w^2 + 15/2*w - 5],
[439, 439, -w^3 + 7*w - 5],
[439, 439, -w^3 + w + 3],
[449, 449, -2*w^3 + 2*w^2 + 5*w - 3],
[449, 449, -1/2*w^3 - 1/2*w^2 + 11/2*w - 1],
[449, 449, 1/2*w^3 + 1/2*w^2 - 5/2*w - 7],
[449, 449, 5/2*w^3 - 5/2*w^2 - 11/2*w + 5],
[491, 491, -5/2*w^3 + 5/2*w^2 + 17/2*w - 2],
[491, 491, 2*w^3 - 2*w^2 - 5*w + 1],
[499, 499, w^3 + w^2 - 5*w - 1],
[499, 499, -w^3 + 3*w^2 + w - 9],
[521, 521, -w^3 + 4*w^2 + w - 9],
[521, 521, 1/2*w^3 + 5/2*w^2 - 7/2*w - 7],
[541, 541, -3/2*w^3 + 5/2*w^2 + 11/2*w - 1],
[541, 541, 1/2*w^3 + 1/2*w^2 - 1/2*w - 6],
[569, 569, -3/2*w^3 + 5/2*w^2 + 3/2*w - 5],
[569, 569, -2*w^3 + 4*w^2 + 5*w - 13],
[571, 571, -w^3 + 3*w^2 - 7],
[571, 571, -w^3 + 2*w^2 + w - 7],
[599, 599, 1/2*w^3 + 5/2*w^2 - 5/2*w - 11],
[599, 599, 1/2*w^3 + 3/2*w^2 - 11/2*w - 8],
[619, 619, -w - 5],
[619, 619, -1/2*w^3 + 1/2*w^2 + 5/2*w - 6],
[619, 619, -1/2*w^3 + 7/2*w^2 - 1/2*w - 9],
[619, 619, 3*w^2 - 2*w - 7],
[641, 641, -w^3 - 2*w^2 + 5*w + 3],
[641, 641, 2*w^2 - 4*w - 7],
[641, 641, 3/2*w^3 - 9/2*w^2 - 5/2*w + 13],
[641, 641, -3/2*w^3 + 7/2*w^2 + 5/2*w - 9],
[659, 659, -w^3 + 7*w + 1],
[659, 659, 5/2*w^3 - 5/2*w^2 - 19/2*w - 1],
[659, 659, -1/2*w^3 + 7/2*w^2 + 1/2*w - 8],
[659, 659, 2*w^2 + 2*w - 7],
[661, 661, 2*w^3 - 2*w^2 - 10*w + 1],
[661, 661, w^3 - 5*w - 7],
[691, 691, -w^3 + 3*w^2 + 4*w - 3],
[691, 691, -3/2*w^3 + 7/2*w^2 + 1/2*w - 6],
[691, 691, 5/2*w^3 - 1/2*w^2 - 23/2*w - 2],
[691, 691, -1/2*w^3 + 5/2*w^2 + 5/2*w - 10],
[701, 701, 5/2*w^3 - 5/2*w^2 - 23/2*w + 4],
[701, 701, -1/2*w^3 + 1/2*w^2 + 7/2*w - 7],
[701, 701, -2*w^3 + w^2 + 10*w - 3],
[701, 701, -1/2*w^3 + 7/2*w^2 - 3/2*w - 12],
[709, 709, -1/2*w^3 + 5/2*w^2 - 3/2*w - 8],
[709, 709, -3/2*w^3 + 5/2*w^2 + 5/2*w - 7],
[719, 719, -w^3 + 4*w^2 - w - 9],
[719, 719, 3/2*w^3 + 3/2*w^2 - 17/2*w - 5],
[739, 739, 3/2*w^3 + 1/2*w^2 - 11/2*w - 8],
[739, 739, 2*w^3 - 4*w^2 - 5*w + 3],
[761, 761, 3*w^2 - 11],
[761, 761, 5/2*w^3 - 7/2*w^2 - 13/2*w + 4],
[761, 761, -3/2*w^3 + 9/2*w^2 + 9/2*w - 7],
[761, 761, 5/2*w^3 - 3/2*w^2 - 17/2*w - 1],
[821, 821, -w^3 + w^2 + 7*w + 1],
[821, 821, -4*w - 1],
[821, 821, -w^3 + w^2 + 7*w - 5],
[821, 821, 2*w^3 - 2*w^2 - 10*w + 5],
[829, 829, 3*w^3 - 2*w^2 - 11*w + 1],
[829, 829, -5/2*w^3 + 7/2*w^2 + 11/2*w - 5],
[839, 839, 3/2*w^3 + 1/2*w^2 - 9/2*w - 6],
[839, 839, 5/2*w^3 - 9/2*w^2 - 15/2*w + 6],
[841, 29, 5/2*w^3 - 5/2*w^2 - 15/2*w + 4],
[859, 859, -3*w^3 + 3*w^2 + 10*w - 1],
[859, 859, 5/2*w^3 - 5/2*w^2 - 13/2*w],
[881, 881, -3/2*w^3 + 7/2*w^2 + 9/2*w - 3],
[881, 881, -w^3 + 3*w^2 + 2*w - 13],
[881, 881, -5/2*w^3 + 9/2*w^2 + 19/2*w - 8],
[881, 881, -7/2*w^3 + 11/2*w^2 + 25/2*w - 10],
[911, 911, w^3 - w^2 + 2*w - 1],
[911, 911, 7/2*w^3 - 11/2*w^2 - 17/2*w + 4],
[911, 911, -3*w^3 + 2*w^2 + 11*w + 1],
[911, 911, 5/2*w^3 - 7/2*w^2 - 11/2*w + 3],
[919, 919, 1/2*w^3 + 1/2*w^2 + 3/2*w - 5],
[919, 919, -5/2*w^3 + 7/2*w^2 + 21/2*w - 4],
[929, 929, -3/2*w^3 + 9/2*w^2 + 3/2*w - 11],
[929, 929, -1/2*w^3 + 1/2*w^2 + 11/2*w + 1],
[929, 929, -3/2*w^3 + 3/2*w^2 + 17/2*w - 5],
[929, 929, 3/2*w^3 + 3/2*w^2 - 15/2*w - 4],
[941, 941, -1/2*w^3 + 5/2*w^2 - 5/2*w - 7],
[941, 941, -3/2*w^3 + 7/2*w^2 + 5/2*w - 10],
[961, 31, 5/2*w^3 - 5/2*w^2 - 15/2*w + 2],
[971, 971, 5/2*w^3 + 1/2*w^2 - 21/2*w - 4],
[971, 971, 3*w^3 - 15*w - 5],
[971, 971, -2*w^3 + 2*w^2 + 3*w - 5],
[971, 971, -5/2*w^3 + 5/2*w^2 + 21/2*w - 10],
[991, 991, -3/2*w^3 + 7/2*w^2 + 9/2*w - 2],
[991, 991, 1/2*w^3 + 3/2*w^2 - 3/2*w - 10]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^2 - 48;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [0, 1, e, 1/2*e + 2, -e - 2, -1/2*e, 6, 1/2*e + 6, 1/2*e - 2, -e - 2, e - 2, -e, e + 8, -2, 6, -e - 4, -e + 4, 1/2*e - 2, -2*e, 3/2*e + 4, 3/2*e - 4, -e + 2, 14, e + 10, 1/2*e - 4, 1/2*e + 12, -10, 2, e - 10, -e, 12, -e, 1/2*e - 14, -e + 2, 2*e + 6, e + 4, 5/2*e - 2, -3/2*e - 4, -2*e - 2, 3/2*e + 2, 2*e + 8, -2*e + 4, -3/2*e + 2, -3*e + 2, 1/2*e - 12, e + 12, -e + 4, e + 14, -22, 6, 1/2*e - 20, -2*e + 6, 14, -3/2*e + 8, 2*e + 10, -4*e + 2, -2*e + 2, 2*e + 4, 4, 2*e - 4, -e - 8, -2, -10, 1/2*e - 26, 16, -4*e + 10, -4, e - 16, 7/2*e - 8, -2*e - 2, e - 26, -4*e - 6, -22, 10, -2*e - 6, -1/2*e - 12, -2*e + 26, -1/2*e - 28, -5/2*e - 10, -3*e, -e - 14, 3/2*e, e + 12, -e + 28, -4*e + 8, e + 20, -3*e - 18, -3/2*e, -2*e + 2, -4*e + 2, -3*e + 8, -e - 24, -e - 16, -2*e + 12, -5/2*e + 12, -3/2*e - 24, 7/2*e, -3*e - 14, -7/2*e + 16, 2*e + 2, 1/2*e + 2, -e - 24, 4*e, -e + 36, -28, 3*e - 8, -e - 32, -2*e - 4, 5/2*e + 16, e + 14, 2*e - 6, -6, 4*e + 12, 4*e - 4, 1/2*e - 22, 3*e + 24, 2*e + 30, 6, -3/2*e + 34, -2*e + 20, 9/2*e + 2, -2*e + 28, -7/2*e - 4, 6*e - 2, 2*e + 6, -3*e + 18, -2*e + 6, -e + 42, -3*e - 20, e + 12, -3*e - 24, -1/2*e - 2, -2*e + 18, -e - 18, -38, -e - 18, -2, -5*e + 10, 4*e - 2, 22, 1/2*e - 4, -3*e + 34, e - 12, e - 28, 3*e - 10, 4*e + 20, 11/2*e - 10, 2*e + 10, 2*e - 22, -5/2*e - 12, -7*e + 6, e - 28, -9/2*e - 6, e + 28, -4*e, 2*e + 16, -2*e + 40, 3/2*e + 36, 9/2*e + 8, 2*e - 6, -2*e - 30, -e + 10, 7/2*e + 16, -11/2*e + 16, 2*e - 36, -e - 24, 3*e + 16, -e - 24, -16, -3*e - 36];
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 | {4, 2, 1/2*w^3 - 1/2*w^2 - 5/2*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;