/*
  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![44, 0, -14, 0, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {16, 2, 2}>;

primesArray := [
[4, 2, -1/2*w^3 + w^2 + 4*w - 8],
[11, 11, 1/2*w^3 - 3/2*w^2 - 4*w + 11],
[11, 11, -1/2*w^2 - w + 2],
[11, 11, -1/2*w^2 + w + 2],
[19, 19, 1/2*w^3 - 1/2*w^2 - 4*w + 5],
[19, 19, 1/2*w^3 + w^2 - 4*w - 9],
[19, 19, 1/2*w^3 - w^2 - 4*w + 9],
[19, 19, 1/2*w^3 + 1/2*w^2 - 4*w - 5],
[25, 5, 1/2*w^2 - 1],
[29, 29, 1/2*w^3 - 4*w - 1],
[29, 29, -1/2*w^3 + 4*w - 1],
[31, 31, -w - 1],
[31, 31, w - 1],
[41, 41, 1/2*w^3 + 1/2*w^2 - 4*w - 2],
[41, 41, -1/2*w^3 + 1/2*w^2 + 4*w - 2],
[49, 7, -1/2*w^3 + 1/2*w^2 + 5*w - 7],
[49, 7, 3/2*w^3 - 7/2*w^2 - 13*w + 29],
[59, 59, -1/2*w^3 - 3/2*w^2 + 3*w + 10],
[59, 59, 1/2*w^3 - 3/2*w^2 - 3*w + 10],
[61, 61, 1/2*w^2 + w - 6],
[61, 61, 1/2*w^2 - w - 6],
[71, 71, w^3 - 5/2*w^2 - 8*w + 20],
[71, 71, -w^3 + 5/2*w^2 + 10*w - 24],
[81, 3, -3],
[101, 101, -1/2*w^3 + 1/2*w^2 + 4*w - 1],
[101, 101, 1/2*w^3 + 1/2*w^2 - 4*w - 1],
[109, 109, 3/2*w^2 + w - 12],
[109, 109, 3/2*w^2 - w - 12],
[149, 149, 1/2*w^3 + 2*w^2 - 4*w - 15],
[149, 149, -1/2*w^3 + 2*w^2 + 4*w - 15],
[151, 151, 1/2*w^3 - 1/2*w^2 - 4*w + 8],
[151, 151, -w^3 + 5/2*w^2 + 8*w - 18],
[151, 151, -w^3 + 3/2*w^2 + 10*w - 18],
[151, 151, -1/2*w^3 - 1/2*w^2 + 4*w + 8],
[179, 179, 1/2*w^3 + w^2 - 4*w - 5],
[179, 179, -1/2*w^3 + w^2 + 4*w - 5],
[181, 181, 1/2*w^3 - 5*w - 3],
[181, 181, w^3 - 1/2*w^2 - 7*w - 1],
[181, 181, -1/2*w^3 - 5/2*w^2 + 4*w + 15],
[181, 181, 1/2*w^3 - 5*w + 3],
[191, 191, w^3 - 2*w^2 - 7*w + 13],
[191, 191, -w^3 - 2*w^2 + 7*w + 13],
[199, 199, -5/2*w^2 - 2*w + 18],
[199, 199, -5/2*w^2 + 2*w + 18],
[241, 241, -3/2*w^3 + 7/2*w^2 + 14*w - 30],
[241, 241, -3/2*w^3 + 5/2*w^2 + 16*w - 30],
[251, 251, 3/2*w^2 - w - 13],
[251, 251, 3/2*w^2 + w - 13],
[269, 269, 1/2*w^3 + 1/2*w^2 - 3*w - 10],
[269, 269, -w^3 + 1/2*w^2 + 7*w - 8],
[269, 269, w^3 + 1/2*w^2 - 7*w - 8],
[269, 269, -1/2*w^3 + 1/2*w^2 + 3*w - 10],
[271, 271, -w^3 + 8*w + 5],
[271, 271, -2*w^2 + 2*w + 15],
[271, 271, 2*w^2 + 2*w - 15],
[271, 271, w^3 - 8*w + 5],
[281, 281, -2*w + 3],
[281, 281, w^3 - 2*w^2 - 12*w + 25],
[311, 311, 1/2*w^3 + 5/2*w^2 - 3*w - 17],
[311, 311, -1/2*w^3 + 5/2*w^2 + 3*w - 17],
[331, 331, -1/2*w^3 + 3/2*w^2 + 5*w - 10],
[331, 331, 1/2*w^3 + 3/2*w^2 - 5*w - 10],
[349, 349, -1/2*w^2 + 3*w - 3],
[349, 349, -w^3 + 5/2*w^2 + 11*w - 25],
[379, 379, 5/2*w^2 - w - 21],
[379, 379, 5/2*w^2 + w - 21],
[389, 389, 3/2*w^2 - w - 14],
[389, 389, 1/2*w^3 - 5/2*w^2 - 3*w + 16],
[389, 389, -1/2*w^3 - 5/2*w^2 + 3*w + 16],
[389, 389, 3/2*w^2 + w - 14],
[409, 409, w^3 + w^2 - 8*w - 7],
[409, 409, -w^3 + w^2 + 8*w - 7],
[419, 419, 3/2*w^3 - 5/2*w^2 - 15*w + 28],
[419, 419, 3/2*w^3 - 7/2*w^2 - 13*w + 28],
[439, 439, 1/2*w^2 + 2*w - 9],
[439, 439, -3/2*w^2 + 2*w + 6],
[439, 439, -3/2*w^2 - 2*w + 6],
[439, 439, 1/2*w^2 - 2*w - 9],
[449, 449, -w^3 + 1/2*w^2 + 6*w - 1],
[449, 449, 1/2*w^2 + 2*w - 5],
[449, 449, 1/2*w^2 - 2*w - 5],
[449, 449, w^3 + 1/2*w^2 - 6*w - 1],
[461, 461, w^3 + 3/2*w^2 - 7*w - 9],
[461, 461, -w^3 + 3/2*w^2 + 7*w - 9],
[491, 491, -1/2*w^3 - 1/2*w^2 + 6*w + 7],
[491, 491, -5/2*w^2 + w + 16],
[491, 491, 5/2*w^2 + w - 16],
[491, 491, 1/2*w^3 - 1/2*w^2 - 6*w + 7],
[499, 499, -1/2*w^3 + 5/2*w^2 + 4*w - 19],
[499, 499, 1/2*w^3 + 5/2*w^2 - 4*w - 19],
[541, 541, -1/2*w^3 + 3/2*w^2 + 4*w - 7],
[541, 541, 1/2*w^3 + 3/2*w^2 - 4*w - 7],
[569, 569, 1/2*w^2 + 2*w - 2],
[569, 569, 1/2*w^2 - 2*w - 2],
[571, 571, -2*w^3 + 9/2*w^2 + 17*w - 39],
[571, 571, 1/2*w^3 - 1/2*w^2 - 5*w - 2],
[571, 571, -1/2*w^3 - 1/2*w^2 + 5*w - 2],
[571, 571, w^3 - 5/2*w^2 - 11*w + 27],
[599, 599, -w^3 + 2*w^2 + 8*w - 13],
[599, 599, w^3 + 2*w^2 - 8*w - 13],
[601, 601, 1/2*w^2 + 2*w - 4],
[601, 601, 1/2*w^2 - 2*w - 4],
[619, 619, -1/2*w^3 + 2*w^2 + 6*w - 13],
[619, 619, 1/2*w^3 + 2*w^2 - 6*w - 13],
[631, 631, -w^3 - 1/2*w^2 + 6*w - 2],
[631, 631, w^3 - 1/2*w^2 - 6*w - 2],
[641, 641, w^3 - 8*w - 1],
[641, 641, -1/2*w^3 - 1/2*w^2 + 5*w - 1],
[641, 641, 1/2*w^3 - 1/2*w^2 - 5*w - 1],
[641, 641, w^3 - 8*w + 1],
[661, 661, -1/2*w^3 + 7/2*w^2 + 2*w - 25],
[661, 661, 3/2*w^3 - 9/2*w^2 - 11*w + 32],
[661, 661, -5/2*w^3 + 13/2*w^2 + 23*w - 56],
[661, 661, 1/2*w^3 + 7/2*w^2 - 2*w - 25],
[691, 691, 1/2*w^3 + 7/2*w^2 - 3*w - 28],
[691, 691, -1/2*w^3 + 7/2*w^2 + 3*w - 28],
[701, 701, -1/2*w^3 + 1/2*w^2 + 5*w - 10],
[701, 701, 3/2*w^3 - 5/2*w^2 - 17*w + 34],
[709, 709, 1/2*w^3 - 5/2*w^2 - 6*w + 17],
[709, 709, w^3 + 5/2*w^2 - 7*w - 14],
[709, 709, -w^3 + 5/2*w^2 + 7*w - 14],
[709, 709, -1/2*w^3 - 5/2*w^2 + 6*w + 17],
[719, 719, -w^3 + 7/2*w^2 + 8*w - 23],
[719, 719, w^3 + 7/2*w^2 - 8*w - 23],
[751, 751, 7/2*w^2 + 2*w - 27],
[751, 751, 7/2*w^2 - 2*w - 27],
[761, 761, -1/2*w^3 + 3/2*w^2 + 3*w - 13],
[761, 761, 1/2*w^3 + 3/2*w^2 - 3*w - 13],
[769, 769, -w^3 + 7/2*w^2 + 8*w - 25],
[769, 769, w^3 + 7/2*w^2 - 8*w - 25],
[809, 809, 2*w^3 - 7/2*w^2 - 20*w + 40],
[809, 809, -1/2*w^2 + 2*w - 4],
[811, 811, -1/2*w^2 - 3*w + 8],
[811, 811, 1/2*w^3 + 3/2*w^2 - 3*w - 18],
[811, 811, -1/2*w^3 + 3/2*w^2 + 3*w - 18],
[811, 811, 1/2*w^2 - 3*w - 8],
[821, 821, 3/2*w^3 - 7/2*w^2 - 16*w + 37],
[821, 821, -3/2*w^3 + 7/2*w^2 + 12*w - 29],
[839, 839, 3*w^2 - 2*w - 23],
[839, 839, 3*w^2 + 2*w - 23],
[841, 29, -5/2*w^2 + 16],
[859, 859, 5/2*w^2 + 3*w - 16],
[859, 859, 5/2*w^2 - 3*w - 16],
[911, 911, 3*w^2 - 2*w - 17],
[911, 911, 3*w^2 + 2*w - 17],
[919, 919, 3*w^2 - w - 23],
[919, 919, -2*w^2 + 3*w + 15],
[919, 919, -2*w^2 - 3*w + 15],
[919, 919, 3*w^2 + w - 23],
[941, 941, 3/2*w^2 - 3*w - 2],
[941, 941, 2*w^3 - 11/2*w^2 - 17*w + 42],
[961, 31, -w^2 + 13],
[971, 971, 1/2*w^3 + 1/2*w^2 - 6*w - 5],
[971, 971, -1/2*w^3 + 1/2*w^2 + 6*w - 5],
[991, 991, w^3 + 3/2*w^2 - 8*w - 8],
[991, 991, -w^3 + 3/2*w^2 + 8*w - 8]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^4 - 40*x^2 + 80;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [0, -2, e, -e, -1/8*e^3 + 9/2*e, -1/4*e^2 + 5, -1/4*e^2 + 5, 1/8*e^3 - 9/2*e, 1/4*e^2 - 9, -1/8*e^3 + 7/2*e, 1/8*e^3 - 7/2*e, -1/8*e^3 + 11/2*e, 1/8*e^3 - 11/2*e, 3/16*e^3 - 29/4*e, -3/16*e^3 + 29/4*e, -1/16*e^3 + 15/4*e, 1/16*e^3 - 15/4*e, -1/4*e^2 + 5, -1/4*e^2 + 5, 1/8*e^2 - 19/2, 1/8*e^2 - 19/2, 3/4*e^2 - 17, 3/4*e^2 - 17, -1/4*e^2 + 13, -3/8*e^3 + 25/2*e, 3/8*e^3 - 25/2*e, 3/8*e^2 - 25/2, 3/8*e^2 - 25/2, 3/8*e^2 - 25/2, 3/8*e^2 - 25/2, 1/2*e^2 - 22, 1/4*e^3 - 7*e, -1/4*e^3 + 7*e, 1/2*e^2 - 22, 1/4*e^3 - 6*e, -1/4*e^3 + 6*e, 1/8*e^3 - 15/2*e, 1/8*e^3 - 11/2*e, -1/8*e^3 + 11/2*e, -1/8*e^3 + 15/2*e, -1/4*e^3 + 7*e, 1/4*e^3 - 7*e, -3/4*e^2 + 25, -3/4*e^2 + 25, 5/16*e^3 - 51/4*e, -5/16*e^3 + 51/4*e, -1/2*e^2 + 12, -1/2*e^2 + 12, 1/8*e^2 - 35/2, 3/8*e^2 - 25/2, 3/8*e^2 - 25/2, 1/8*e^2 - 35/2, -3/8*e^3 + 33/2*e, -1/4*e^2 - 13, -1/4*e^2 - 13, 3/8*e^3 - 33/2*e, -7/16*e^3 + 49/4*e, 7/16*e^3 - 49/4*e, -8, -8, -3/8*e^3 + 23/2*e, 3/8*e^3 - 23/2*e, -3/8*e^3 + 25/2*e, 3/8*e^3 - 25/2*e, -e^2 + 30, -e^2 + 30, -7/8*e^2 + 5/2, 5/8*e^2 - 15/2, 5/8*e^2 - 15/2, -7/8*e^2 + 5/2, -7/16*e^3 + 73/4*e, 7/16*e^3 - 73/4*e, 1/2*e^3 - 23*e, -1/2*e^3 + 23*e, 0, -4*e, 4*e, 0, 7/16*e^3 - 81/4*e, 3/16*e^3 - 37/4*e, -3/16*e^3 + 37/4*e, -7/16*e^3 + 81/4*e, -5/8*e^3 + 51/2*e, 5/8*e^3 - 51/2*e, -5/8*e^3 + 41/2*e, 1/4*e^2 - 17, 1/4*e^2 - 17, 5/8*e^3 - 41/2*e, -30, -30, 5/8*e^3 - 39/2*e, -5/8*e^3 + 39/2*e, -13/16*e^3 + 115/4*e, 13/16*e^3 - 115/4*e, -e^2 + 22, -1/2*e^3 + 19*e, 1/2*e^3 - 19*e, -e^2 + 22, -1/8*e^3 + 19/2*e, 1/8*e^3 - 19/2*e, 7/16*e^3 - 81/4*e, -7/16*e^3 + 81/4*e, -1/8*e^3 + 9/2*e, 1/8*e^3 - 9/2*e, -1/2*e^2 - 2, -1/2*e^2 - 2, -11/16*e^3 + 93/4*e, 7/16*e^3 - 81/4*e, -7/16*e^3 + 81/4*e, 11/16*e^3 - 93/4*e, -7/8*e^2 + 101/2, -1/8*e^2 + 19/2, -1/8*e^2 + 19/2, -7/8*e^2 + 101/2, -3/4*e^2 - 17, -3/4*e^2 - 17, -5/8*e^2 - 1/2, -5/8*e^2 - 1/2, 1/8*e^3 - 15/2*e, -7/8*e^3 + 61/2*e, 7/8*e^3 - 61/2*e, -1/8*e^3 + 15/2*e, -1/8*e^3 + 3/2*e, 1/8*e^3 - 3/2*e, e^2 + 12, e^2 + 12, -3/4*e^2 + 23, -3/4*e^2 + 23, -10, -10, -1/2*e^2 + 20, -1/2*e^2 + 20, 1/4*e^3 - 4*e, 1/2*e^2 - 52, 1/2*e^2 - 52, -1/4*e^3 + 4*e, -3/8*e^2 + 89/2, -3/8*e^2 + 89/2, -3/2*e^2 + 10, -3/2*e^2 + 10, e^2 - 2, 9/8*e^3 - 77/2*e, -9/8*e^3 + 77/2*e, -3/8*e^3 + 9/2*e, 3/8*e^3 - 9/2*e, 5/4*e^2 - 55, 5/8*e^3 - 39/2*e, -5/8*e^3 + 39/2*e, 5/4*e^2 - 55, 1/8*e^3 + 5/2*e, -1/8*e^3 - 5/2*e, e^2 - 58, -9/8*e^3 + 77/2*e, 9/8*e^3 - 77/2*e, 1/2*e^3 - 18*e, -1/2*e^3 + 18*e];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {4, 2, -1/2*w^3 + w^2 + 4*w - 8}>] := -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;