/*
  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, -3, 4, 6, -4, -2, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {37, 37, -w^5 + 3*w^4 + 2*w^3 - 8*w^2 - w + 3}>;

primesArray := [
[7, 7, w^2 - w - 2],
[13, 13, w^4 - 2*w^3 - 3*w^2 + 5*w],
[17, 17, -w^3 + w^2 + 3*w],
[19, 19, w^3 - w^2 - 3*w + 1],
[19, 19, -w^4 + w^3 + 4*w^2 - 2*w - 2],
[23, 23, -w^4 + 2*w^3 + 3*w^2 - 3*w - 2],
[31, 31, w^5 - 2*w^4 - 4*w^3 + 5*w^2 + 5*w - 1],
[37, 37, -w^5 + 3*w^4 + 2*w^3 - 8*w^2 - w + 3],
[37, 37, -w^5 + 2*w^4 + 4*w^3 - 6*w^2 - 4*w + 1],
[47, 47, -w^3 + 2*w^2 + w - 3],
[64, 2, -2],
[67, 67, 2*w - 1],
[79, 79, w^4 - w^3 - 4*w^2 + 2*w],
[79, 79, -w^5 + 2*w^4 + 3*w^3 - 5*w^2 - w + 4],
[97, 97, w^5 - 2*w^4 - 3*w^3 + 6*w^2 - 3],
[97, 97, w^5 - 3*w^4 - 2*w^3 + 9*w^2 - 2*w - 4],
[101, 101, w^5 - 2*w^4 - 3*w^3 + 5*w^2 - w - 2],
[101, 101, w^5 - 3*w^4 - 2*w^3 + 10*w^2 - w - 5],
[103, 103, 2*w^5 - 3*w^4 - 9*w^3 + 8*w^2 + 8*w - 3],
[107, 107, w^2 - 2*w - 3],
[109, 109, 2*w^5 - 3*w^4 - 9*w^3 + 7*w^2 + 9*w - 2],
[113, 113, w^4 - w^3 - 5*w^2 + 3*w + 3],
[121, 11, -2*w^5 + 4*w^4 + 7*w^3 - 10*w^2 - 5*w + 5],
[125, 5, -w^5 + w^4 + 6*w^3 - 3*w^2 - 8*w],
[125, 5, -w^3 + 2*w^2 + 3*w - 2],
[127, 127, -2*w^5 + 4*w^4 + 6*w^3 - 9*w^2 - 2*w + 2],
[131, 131, -w^5 + 2*w^4 + 3*w^3 - 4*w^2 - 2],
[137, 137, w^4 - 2*w^3 - 3*w^2 + 5*w + 3],
[139, 139, -w^5 + 3*w^4 + 2*w^3 - 8*w^2 + w],
[157, 157, -2*w^5 + 4*w^4 + 7*w^3 - 10*w^2 - 6*w + 2],
[157, 157, -2*w^5 + 4*w^4 + 7*w^3 - 11*w^2 - 6*w + 6],
[173, 173, -w^5 + 2*w^4 + 5*w^3 - 7*w^2 - 5*w + 3],
[179, 179, -w^4 + w^3 + 6*w^2 - 4*w - 4],
[181, 181, w^4 - w^3 - 4*w^2 - w + 2],
[181, 181, -w^5 + 3*w^4 + 2*w^3 - 8*w^2 + 2*w],
[181, 181, w^5 - 3*w^4 - 3*w^3 + 9*w^2 + 3*w - 3],
[191, 191, w^4 - w^3 - 4*w^2 + w + 4],
[191, 191, w^5 - 2*w^4 - 5*w^3 + 8*w^2 + 4*w - 3],
[193, 193, -w^5 + w^4 + 5*w^3 - 2*w^2 - 6*w + 1],
[211, 211, -3*w^5 + 6*w^4 + 11*w^3 - 16*w^2 - 9*w + 7],
[223, 223, 2*w^5 - 6*w^4 - 3*w^3 + 17*w^2 - 4*w - 6],
[227, 227, w^5 - 3*w^4 - w^3 + 8*w^2 - 3*w - 4],
[229, 229, -w^5 + 3*w^4 + w^3 - 7*w^2 + 5*w],
[233, 233, -w^5 + 2*w^4 + 2*w^3 - 3*w^2 - 1],
[251, 251, w^5 - 7*w^3 - 2*w^2 + 10*w + 1],
[257, 257, 3*w^5 - 5*w^4 - 12*w^3 + 11*w^2 + 9*w - 1],
[257, 257, -3*w^5 + 6*w^4 + 10*w^3 - 15*w^2 - 4*w + 3],
[257, 257, -w^5 + 3*w^4 + 3*w^3 - 12*w^2 - w + 7],
[263, 263, -2*w^5 + 6*w^4 + 4*w^3 - 17*w^2 + w + 5],
[263, 263, -w^5 + 2*w^4 + 5*w^3 - 6*w^2 - 7*w + 2],
[277, 277, -w^5 + 7*w^3 + 2*w^2 - 8*w - 5],
[277, 277, w^5 - w^4 - 6*w^3 + 3*w^2 + 7*w - 2],
[277, 277, -2*w^5 + 3*w^4 + 10*w^3 - 8*w^2 - 11*w + 2],
[281, 281, -2*w^5 + 4*w^4 + 7*w^3 - 9*w^2 - 5*w + 1],
[283, 283, -2*w^4 + 3*w^3 + 7*w^2 - 7*w - 2],
[283, 283, -3*w^5 + 5*w^4 + 11*w^3 - 10*w^2 - 7*w],
[293, 293, -w^5 + 2*w^4 + 4*w^3 - 7*w^2 - 4*w + 2],
[311, 311, -2*w^5 + 5*w^4 + 6*w^3 - 14*w^2 - 3*w + 3],
[311, 311, w^5 - 4*w^4 + 13*w^2 - 6*w - 6],
[311, 311, -3*w^5 + 6*w^4 + 11*w^3 - 15*w^2 - 10*w + 3],
[311, 311, -w^4 + w^3 + 4*w^2 - 2*w - 5],
[317, 317, -2*w^5 + 4*w^4 + 6*w^3 - 8*w^2 - 3*w + 2],
[317, 317, -w^5 + 8*w^3 - 11*w - 1],
[331, 331, -w^4 + 6*w^2 + w - 5],
[347, 347, -2*w^4 + 4*w^3 + 5*w^2 - 6*w],
[349, 349, 2*w^5 - 5*w^4 - 4*w^3 + 11*w^2 - 3*w],
[349, 349, -2*w^4 + 4*w^3 + 5*w^2 - 8*w + 1],
[349, 349, -2*w^5 + 4*w^4 + 7*w^3 - 9*w^2 - 4*w + 1],
[353, 353, 2*w^5 - 4*w^4 - 8*w^3 + 13*w^2 + 7*w - 6],
[353, 353, -w^5 + 2*w^4 + 2*w^3 - 2*w^2 + w - 4],
[353, 353, -w^5 + w^4 + 4*w^3 - 3*w - 4],
[359, 359, -2*w^5 + 5*w^4 + 7*w^3 - 16*w^2 - 6*w + 6],
[359, 359, w^5 - 2*w^4 - 4*w^3 + 6*w^2 + w - 3],
[367, 367, -w^5 + 2*w^4 + 2*w^3 - 2*w^2 + 2*w - 2],
[367, 367, -w^5 + 3*w^4 + 2*w^3 - 9*w^2 - w + 2],
[373, 373, w^5 - 2*w^4 - 3*w^3 + 6*w^2 - w - 3],
[379, 379, -w^5 + 4*w^4 + w^3 - 14*w^2 + w + 8],
[379, 379, -2*w^5 + 5*w^4 + 6*w^3 - 14*w^2 - 5*w + 4],
[383, 383, -w^5 + w^4 + 6*w^3 - 4*w^2 - 7*w + 1],
[389, 389, 2*w^4 - 3*w^3 - 7*w^2 + 5*w + 4],
[397, 397, -w^5 + 7*w^3 + 3*w^2 - 10*w - 4],
[401, 401, -2*w^5 + 6*w^4 + 4*w^3 - 18*w^2 + w + 9],
[401, 401, -3*w^5 + 7*w^4 + 8*w^3 - 18*w^2 - w + 7],
[409, 409, -2*w^4 + 4*w^3 + 7*w^2 - 8*w - 6],
[409, 409, 2*w^5 - 4*w^4 - 8*w^3 + 12*w^2 + 5*w - 5],
[419, 419, -w^5 + 4*w^4 + w^3 - 14*w^2 + 3*w + 7],
[419, 419, 2*w^5 - 6*w^4 - 4*w^3 + 19*w^2 - 2*w - 9],
[439, 439, w^5 - 4*w^4 - w^3 + 12*w^2 - w - 4],
[443, 443, w^5 - 2*w^4 - 4*w^3 + 7*w^2 + 4*w - 7],
[457, 457, -3*w^5 + 5*w^4 + 12*w^3 - 11*w^2 - 10*w + 2],
[457, 457, 2*w^5 - 3*w^4 - 9*w^3 + 9*w^2 + 7*w - 5],
[457, 457, 3*w^5 - 6*w^4 - 10*w^3 + 15*w^2 + 7*w - 4],
[463, 463, -w^5 + 4*w^4 + w^3 - 14*w^2 + 2*w + 6],
[463, 463, -w^5 + 2*w^4 + 5*w^3 - 6*w^2 - 8*w + 1],
[487, 487, w^5 - 2*w^4 - 4*w^3 + 4*w^2 + 5*w + 3],
[499, 499, w^2 - w - 5],
[499, 499, w^4 - w^3 - 5*w^2 + 4*w + 5],
[523, 523, 2*w^5 - 2*w^4 - 11*w^3 + 4*w^2 + 12*w + 1],
[557, 557, -2*w^5 + 3*w^4 + 9*w^3 - 8*w^2 - 9*w + 5],
[563, 563, w^5 - 4*w^4 - w^3 + 12*w^2 - w - 3],
[563, 563, -3*w^5 + 6*w^4 + 9*w^3 - 14*w^2 - 2*w + 3],
[571, 571, -2*w^4 + 4*w^3 + 5*w^2 - 8*w - 1],
[577, 577, 3*w^5 - 5*w^4 - 12*w^3 + 12*w^2 + 9*w - 1],
[587, 587, 3*w^5 - 7*w^4 - 10*w^3 + 20*w^2 + 7*w - 7],
[587, 587, w^5 - 3*w^4 - 3*w^3 + 9*w^2 + 2*w - 1],
[587, 587, -3*w^5 + 6*w^4 + 11*w^3 - 16*w^2 - 8*w + 2],
[593, 593, w^5 - 3*w^4 - 2*w^3 + 7*w^2 + 2*w + 1],
[613, 613, -2*w^5 + 5*w^4 + 6*w^3 - 15*w^2 - w + 7],
[631, 631, -w^5 + w^4 + 6*w^3 - 4*w^2 - 6*w + 5],
[641, 641, -w^5 + 8*w^3 - w^2 - 10*w + 1],
[641, 641, 2*w^5 - 4*w^4 - 8*w^3 + 10*w^2 + 9*w - 2],
[643, 643, -2*w^4 + 4*w^3 + 7*w^2 - 8*w - 4],
[643, 643, w^5 - w^4 - 5*w^3 + w^2 + 5*w + 5],
[653, 653, w^5 - 3*w^4 - w^3 + 9*w^2 - 5*w - 4],
[653, 653, w^5 - 3*w^4 - w^3 + 7*w^2 - w],
[659, 659, w^5 - 3*w^4 - 3*w^3 + 10*w^2 + w - 3],
[659, 659, w^4 - w^3 - 6*w^2 + 3*w + 4],
[673, 673, -w^5 + 6*w^3 + 3*w^2 - 8*w - 4],
[691, 691, 2*w^5 - 6*w^4 - 4*w^3 + 17*w^2 - w - 7],
[691, 691, 3*w^5 - 6*w^4 - 11*w^3 + 17*w^2 + 8*w - 5],
[701, 701, 2*w^5 - 3*w^4 - 9*w^3 + 9*w^2 + 6*w - 3],
[701, 701, w^5 - 3*w^4 - 2*w^3 + 11*w^2 - 3*w - 8],
[701, 701, -2*w^5 + 2*w^4 + 10*w^3 - 4*w^2 - 10*w + 1],
[709, 709, 2*w^5 - 3*w^4 - 9*w^3 + 9*w^2 + 8*w - 4],
[709, 709, w^2 - 5],
[727, 727, 3*w^5 - 6*w^4 - 10*w^3 + 15*w^2 + 6*w - 4],
[729, 3, -3],
[733, 733, -w^5 + 4*w^4 - 11*w^2 + 3*w + 5],
[733, 733, -2*w^5 + 2*w^4 + 9*w^3 - w^2 - 8*w - 4],
[739, 739, 3*w^5 - 6*w^4 - 10*w^3 + 15*w^2 + 7*w - 3],
[743, 743, w^5 - 4*w^4 - w^3 + 12*w^2 - 3],
[751, 751, -w^5 + 2*w^4 + 5*w^3 - 8*w^2 - 6*w + 3],
[757, 757, 3*w^5 - 5*w^4 - 12*w^3 + 12*w^2 + 8*w - 3],
[761, 761, w^4 - 3*w^3 - 2*w^2 + 9*w - 1],
[769, 769, 2*w^4 - 3*w^3 - 6*w^2 + 4*w + 4],
[773, 773, 3*w^5 - 5*w^4 - 11*w^3 + 10*w^2 + 8*w - 2],
[773, 773, -w^5 + w^4 + 5*w^3 - w^2 - 3*w - 2],
[773, 773, 3*w^5 - 7*w^4 - 9*w^3 + 18*w^2 + 5*w - 5],
[821, 821, 3*w^5 - 6*w^4 - 10*w^3 + 15*w^2 + 7*w - 5],
[821, 821, 2*w^4 - 4*w^3 - 6*w^2 + 7*w + 1],
[823, 823, -3*w^5 + 6*w^4 + 10*w^3 - 16*w^2 - 4*w + 5],
[823, 823, -2*w^5 + 5*w^4 + 5*w^3 - 13*w^2 + w + 5],
[827, 827, -3*w^5 + 5*w^4 + 13*w^3 - 13*w^2 - 11*w + 3],
[827, 827, 2*w^5 - 5*w^4 - 6*w^3 + 16*w^2 + w - 5],
[829, 829, 2*w^5 - 5*w^4 - 5*w^3 + 14*w^2 - 2*w - 5],
[841, 29, w^5 - w^4 - 4*w^3 - w^2 + 4*w + 6],
[853, 853, -2*w^5 + 5*w^4 + 5*w^3 - 13*w^2 - 2*w + 3],
[857, 857, 2*w^5 - 2*w^4 - 10*w^3 + 4*w^2 + 9*w - 1],
[857, 857, w^5 - 2*w^4 - 3*w^3 + 5*w^2 - 5],
[857, 857, -3*w^5 + 5*w^4 + 13*w^3 - 14*w^2 - 10*w + 4],
[863, 863, w^4 - 2*w^3 - 2*w^2 + 2*w - 2],
[877, 877, w^5 - 3*w^4 - 2*w^3 + 11*w^2 - 3*w - 7],
[881, 881, -w^5 + 4*w^4 - 12*w^2 + 3*w + 7],
[883, 883, -4*w^5 + 8*w^4 + 13*w^3 - 19*w^2 - 6*w + 4],
[883, 883, -2*w^5 + 6*w^4 + 3*w^3 - 16*w^2 + 4*w + 6],
[883, 883, -w^5 + 4*w^4 - 13*w^2 + 5*w + 5],
[961, 31, -2*w^5 + 4*w^4 + 8*w^3 - 11*w^2 - 6*w + 1],
[967, 967, -w^5 + 7*w^3 - 7*w],
[967, 967, w^5 - 4*w^4 - w^3 + 14*w^2 - 4*w - 5],
[971, 971, -w^5 + 2*w^4 + 4*w^3 - 7*w^2 - 3*w + 1],
[997, 997, 3*w^5 - 5*w^4 - 11*w^3 + 11*w^2 + 7*w - 1],
[997, 997, 2*w^4 - 3*w^3 - 8*w^2 + 4*w + 5]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

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

heckeEigenvaluesArray := [0, e, -e, -e, 4, -e, -e, -1, 0, 0, 13, 4, e, e, -2, e, 2*e, -2*e, 0, -12, 2, -18, 22, 18, 18, -8, -12, 0, -e, 0, 3*e, -3*e, -e, 10, 0, -22, -e, 0, -2*e, -20, e, 12, -22, e, 12, e, 4*e, 2*e, 0, -e, 26, -10, 10, 4*e, 3*e, -28, 4*e, 0, -2*e, 0, 0, 5*e, 30, -3*e, 36, 34, -2*e, -e, 18, 3*e, -e, 0, e, -16, -4*e, -26, -e, e, 24, 6, e, 18, -e, 2*e, 4*e, 12, 3*e, 8, 3*e, 0, e, 10, -3*e, 2*e, 4*e, 20, 0, -28, 4*e, -36, 36, -e, 2, -36, 36, -4*e, 3*e, -e, -16, -e, 3*e, -3*e, 4, 18, -18, -36, -4*e, -34, -44, -5*e, -30, 18, 5*e, 2*e, -2*e, -16, 22, -34, -7*e, -5*e, 24, -16, -5*e, -6, -50, -6, 6, -42, 6, 54, 32, -3*e, 5*e, -5*e, -5*e, -5*e, -e, 0, 6, 6*e, -24, 46, e, e, 7*e, -2*e, 50, 6*e, 40, 8*e, 26, -3*e];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {37,37,-w^5+3*w^4+2*w^3-8*w^2-w+3}>] := 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;