/*
  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![2, 0, -5, 0, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {16, 4, -w^2 + 2}>;

primesArray := [
[2, 2, w],
[2, 2, w + 1],
[13, 13, -w^2 + w + 3],
[13, 13, w^2 + w - 3],
[19, 19, -w^3 + 3*w + 1],
[19, 19, -w^3 + 3*w - 1],
[43, 43, -w^2 + w - 1],
[43, 43, w^2 + w + 1],
[49, 7, w^3 + w^2 - 6*w - 3],
[49, 7, w^3 - w^2 - 6*w + 3],
[53, 53, 2*w^3 - w^2 - 9*w + 3],
[53, 53, 2*w^3 + w^2 - 9*w - 3],
[59, 59, w^3 - w^2 - 4*w + 1],
[59, 59, -w^3 - w^2 + 4*w + 1],
[67, 67, 3*w^3 - 13*w + 1],
[67, 67, -w^3 + w^2 + 6*w - 5],
[81, 3, -3],
[83, 83, -2*w^3 - w^2 + 9*w + 7],
[83, 83, 4*w^3 - 18*w - 1],
[89, 89, -2*w^3 + 10*w + 1],
[89, 89, -w^2 + w + 5],
[89, 89, w^2 + w - 5],
[89, 89, 2*w^3 - 10*w + 1],
[101, 101, -2*w^3 + 8*w + 1],
[101, 101, 2*w^3 - 8*w + 1],
[103, 103, w^3 + 2*w^2 - 5*w - 7],
[103, 103, w^3 + w^2 - 6*w - 7],
[103, 103, -w^2 - 3*w + 1],
[103, 103, -w^3 + 2*w^2 + 5*w - 7],
[127, 127, -w^3 + 2*w^2 + 5*w - 5],
[127, 127, w^3 + 3*w^2 - 4*w - 13],
[127, 127, -w^3 + 3*w^2 + 4*w - 13],
[127, 127, w^3 + 2*w^2 - 5*w - 5],
[149, 149, -2*w^3 - w^2 + 7*w - 1],
[149, 149, 2*w^3 - w^2 - 7*w - 1],
[151, 151, -w^3 + w^2 + 2*w - 3],
[151, 151, 2*w^3 - 10*w + 3],
[151, 151, -2*w^3 + 10*w + 3],
[151, 151, w^3 + w^2 - 2*w - 3],
[157, 157, w^3 + 2*w^2 - 3*w - 1],
[157, 157, -w^3 + 2*w^2 + 3*w - 1],
[169, 13, 2*w^2 - 3],
[179, 179, 2*w^3 + w^2 - 7*w - 3],
[179, 179, -2*w^3 + w^2 + 7*w - 3],
[229, 229, -w^3 + 2*w^2 + 3*w - 7],
[229, 229, w^3 + 2*w^2 - 3*w - 7],
[251, 251, -2*w^3 + 2*w^2 + 8*w - 9],
[251, 251, -2*w^3 - 2*w^2 + 8*w + 9],
[263, 263, 2*w^3 - w^2 - 3*w + 1],
[263, 263, -2*w^3 + 2*w^2 + 10*w - 5],
[263, 263, -2*w^3 - 2*w^2 + 10*w + 5],
[263, 263, -2*w^3 - w^2 + 3*w + 1],
[289, 17, 2*w^2 - 5],
[293, 293, 4*w^3 - w^2 - 19*w + 7],
[293, 293, -4*w^3 - w^2 + 19*w + 7],
[307, 307, -w^3 + 3*w - 5],
[307, 307, w^3 - 3*w - 5],
[331, 331, 4*w^3 - 2*w^2 - 18*w + 7],
[331, 331, 5*w^3 - 3*w^2 - 22*w + 11],
[349, 349, -3*w^3 + 13*w + 3],
[349, 349, -3*w^3 + 13*w - 3],
[361, 19, 2*w^2 - 11],
[373, 373, w^3 + 2*w^2 + w - 3],
[373, 373, -w^3 + 2*w^2 - w - 3],
[383, 383, -w^3 + 5*w + 5],
[383, 383, -3*w^3 + 2*w^2 + 11*w - 5],
[383, 383, 3*w^3 + 2*w^2 - 11*w - 5],
[383, 383, w^3 - 5*w + 5],
[389, 389, w^3 - 4*w^2 + w + 7],
[389, 389, -2*w^3 - w^2 + 7*w + 7],
[421, 421, -w^3 - w^2 - 2*w + 1],
[421, 421, w^3 - w^2 + 2*w + 1],
[433, 433, 4*w^3 - 2*w^2 - 16*w + 5],
[433, 433, -4*w + 1],
[433, 433, 4*w + 1],
[433, 433, -5*w^3 + 3*w^2 + 24*w - 13],
[443, 443, -w^3 + w^2 + 3],
[443, 443, -4*w^2 + 6*w + 7],
[457, 457, 3*w^3 + w^2 - 12*w - 3],
[457, 457, -2*w^3 + 6*w - 1],
[457, 457, 2*w^3 - 6*w - 1],
[457, 457, 3*w^3 - w^2 - 12*w + 3],
[461, 461, w^3 - 3*w^2 - 2*w + 7],
[461, 461, w^3 + 3*w^2 - 2*w - 7],
[463, 463, -2*w^3 + 3*w^2 + 9*w - 11],
[463, 463, 2*w^3 - w^2 - 5*w + 1],
[463, 463, -2*w^3 - w^2 + 5*w + 1],
[463, 463, -2*w^3 - 3*w^2 + 9*w + 11],
[467, 467, w^3 - 7*w + 1],
[467, 467, -w^3 + 7*w + 1],
[491, 491, 2*w^2 - 2*w - 7],
[491, 491, 2*w^2 + 2*w - 7],
[509, 509, w^3 + 3*w^2 - 4*w - 11],
[509, 509, -w^3 + 3*w^2 + 4*w - 11],
[523, 523, 2*w^3 - w^2 - 11*w + 3],
[523, 523, 2*w^3 + w^2 - 11*w - 3],
[529, 23, 3*w^2 - w - 15],
[529, 23, -3*w^2 - w + 15],
[557, 557, 3*w^2 + w - 13],
[557, 557, -3*w^2 + w + 13],
[563, 563, -3*w^3 + 3*w^2 + 12*w - 7],
[563, 563, 2*w^3 - 2*w^2 - 8*w + 3],
[569, 569, w^3 + 3*w^2 - 6*w - 9],
[569, 569, 4*w^3 - w^2 - 17*w + 1],
[569, 569, -4*w^3 - w^2 + 17*w + 1],
[569, 569, -w^3 + 3*w^2 + 6*w - 9],
[587, 587, -w^3 + w^2 + 2*w - 5],
[587, 587, w^3 + w^2 - 2*w - 5],
[593, 593, 3*w^3 - 2*w^2 - 15*w + 5],
[593, 593, -w^3 - 2*w^2 + 5*w + 1],
[593, 593, w^3 - 2*w^2 - 5*w + 1],
[593, 593, -3*w^3 - 2*w^2 + 15*w + 5],
[599, 599, w^3 + w^2 - 3],
[599, 599, -w^2 - w - 3],
[599, 599, -w^2 + w - 3],
[599, 599, -w^3 + w^2 - 3],
[613, 613, -w^3 + w^2 + 6*w - 9],
[613, 613, w^3 + w^2 - 6*w - 9],
[625, 5, -5],
[659, 659, -2*w^3 + 4*w^2 + 4*w - 1],
[659, 659, 2*w^3 + 4*w^2 - 4*w - 1],
[661, 661, w^3 + 3*w^2 - 6*w - 7],
[661, 661, -2*w^3 + 6*w^2 - 7],
[701, 701, -w^3 - 2*w^2 + 3*w + 9],
[701, 701, -w^3 + 2*w^2 + 3*w - 9],
[733, 733, w^3 + 2*w^2 - 7*w - 1],
[733, 733, -4*w^3 + w^2 + 19*w - 11],
[739, 739, -w^3 - w^2 + 2*w + 9],
[739, 739, -w^3 + w^2 + 2*w - 9],
[757, 757, 3*w^3 - w^2 - 10*w + 5],
[757, 757, -3*w^3 - w^2 + 10*w + 5],
[773, 773, -4*w^3 + 5*w^2 + 19*w - 23],
[773, 773, -7*w^3 + 6*w^2 + 33*w - 29],
[797, 797, 3*w^3 - 13*w + 5],
[797, 797, 3*w^3 - 13*w - 5],
[829, 829, -2*w^3 - w^2 + 9*w - 1],
[829, 829, 2*w^3 - w^2 - 9*w - 1],
[859, 859, 3*w^3 - w^2 - 14*w + 1],
[859, 859, -3*w^3 - w^2 + 14*w + 1],
[863, 863, -5*w^3 + 21*w + 3],
[863, 863, -3*w^3 - w^2 + 12*w - 1],
[863, 863, 3*w^3 - w^2 - 12*w - 1],
[863, 863, -5*w^3 + 21*w - 3],
[883, 883, -2*w^3 + 4*w^2 + 8*w - 17],
[883, 883, 2*w^3 + 4*w^2 - 8*w - 17],
[953, 953, -w^3 - w^2 + 8*w + 3],
[953, 953, 4*w^3 - 2*w^2 - 16*w + 7],
[953, 953, -3*w^3 + w^2 + 16*w - 7],
[953, 953, w^3 - w^2 - 8*w + 3],
[961, 31, -4*w^3 + 2*w^2 + 12*w + 3],
[961, 31, 3*w^3 - w^2 - 10*w - 1],
[971, 971, 3*w^3 - 4*w^2 - 13*w + 15],
[971, 971, -4*w^3 + 3*w^2 + 19*w - 17],
[977, 977, -2*w^3 + w^2 + 9*w + 3],
[977, 977, -4*w^3 + w^2 + 19*w - 9],
[977, 977, 3*w^3 - 15*w + 5],
[977, 977, 2*w^3 + w^2 - 9*w + 3]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^6 - 12*x^4 + 35*x^2 - 8;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [0, e, -1/2*e^5 + 5*e^3 - 21/2*e, -1/2*e^5 + 5*e^3 - 21/2*e, e^3 - 5*e, e^3 - 5*e, -e^4 + 9*e^2 - 12, -e^4 + 9*e^2 - 12, 1/2*e^5 - 5*e^3 + 21/2*e, 1/2*e^5 - 5*e^3 + 21/2*e, -2*e^2 + 8, -2*e^2 + 8, -e^4 + 5*e^2 + 8, -e^4 + 5*e^2 + 8, -e^5 + 11*e^3 - 30*e, -e^5 + 11*e^3 - 30*e, 2*e^2, e^4 - 9*e^2 + 12, e^4 - 9*e^2 + 12, -1/2*e^5 + 7*e^3 - 49/2*e, 1/2*e^5 - 7*e^3 + 41/2*e, 1/2*e^5 - 7*e^3 + 41/2*e, -1/2*e^5 + 7*e^3 - 49/2*e, 1/2*e^5 - 7*e^3 + 49/2*e, 1/2*e^5 - 7*e^3 + 49/2*e, 4*e^2 - 12, e^5 - 10*e^3 + 17*e, e^5 - 10*e^3 + 17*e, 4*e^2 - 12, 0, e^5 - 14*e^3 + 45*e, e^5 - 14*e^3 + 45*e, 0, -2*e^4 + 14*e^2 - 6, -2*e^4 + 14*e^2 - 6, -e^5 + 10*e^3 - 17*e, 2*e^4 - 14*e^2 + 4, 2*e^4 - 14*e^2 + 4, -e^5 + 10*e^3 - 17*e, 2*e^4 - 16*e^2 + 12, 2*e^4 - 16*e^2 + 12, 2*e^4 - 16*e^2 + 24, -2*e^5 + 19*e^3 - 37*e, -2*e^5 + 19*e^3 - 37*e, 1/2*e^5 - 7*e^3 + 49/2*e, 1/2*e^5 - 7*e^3 + 49/2*e, e^5 - 13*e^3 + 48*e, e^5 - 13*e^3 + 48*e, -e^5 + 8*e^3 - 15*e, 2*e^4 - 14*e^2 + 20, 2*e^4 - 14*e^2 + 20, -e^5 + 8*e^3 - 15*e, -2*e^4 + 14*e^2 + 6, -2*e^4 + 18*e^2 - 26, -2*e^4 + 18*e^2 - 26, -e^4 + 9*e^2 - 20, -e^4 + 9*e^2 - 20, e^4 - 9*e^2 + 20, e^4 - 9*e^2 + 20, 6*e^2 - 24, 6*e^2 - 24, 2*e^4 - 18*e^2 + 42, -1/2*e^5 + e^3 + 35/2*e, -1/2*e^5 + e^3 + 35/2*e, -2*e^4 + 10*e^2 + 8, -e^5 + 14*e^3 - 45*e, -e^5 + 14*e^3 - 45*e, -2*e^4 + 10*e^2 + 8, 3/2*e^5 - 11*e^3 + 23/2*e, 3/2*e^5 - 11*e^3 + 23/2*e, 3/2*e^5 - 19*e^3 + 103/2*e, 3/2*e^5 - 19*e^3 + 103/2*e, 3/2*e^5 - 15*e^3 + 39/2*e, 1/2*e^5 - 5*e^3 + 21/2*e, 1/2*e^5 - 5*e^3 + 21/2*e, 3/2*e^5 - 15*e^3 + 39/2*e, -e^5 + 17*e^3 - 60*e, -e^5 + 17*e^3 - 60*e, -1/2*e^5 + 9*e^3 - 53/2*e, 3/2*e^5 - 15*e^3 + 55/2*e, 3/2*e^5 - 15*e^3 + 55/2*e, -1/2*e^5 + 9*e^3 - 53/2*e, -2*e^4 + 12*e^2 + 4, -2*e^4 + 12*e^2 + 4, 2*e^4 - 22*e^2 + 20, 2*e^5 - 20*e^3 + 34*e, 2*e^5 - 20*e^3 + 34*e, 2*e^4 - 22*e^2 + 20, 3*e^3 - 23*e, 3*e^3 - 23*e, e^5 - 11*e^3 + 14*e, e^5 - 11*e^3 + 14*e, -2, -2, 2*e^5 - 17*e^3 + 19*e, 2*e^5 - 17*e^3 + 19*e, -3/2*e^5 + 13*e^3 - 43/2*e, -3/2*e^5 + 13*e^3 - 43/2*e, 1/2*e^5 + e^3 - 47/2*e, 1/2*e^5 + e^3 - 47/2*e, -e^4 + 5*e^2 + 16, -e^4 + 5*e^2 + 16, 2*e^4 - 12*e^2 + 4, -2*e^4 + 8*e^2 + 20, -2*e^4 + 8*e^2 + 20, 2*e^4 - 12*e^2 + 4, 3*e^5 - 29*e^3 + 46*e, 3*e^5 - 29*e^3 + 46*e, 2*e^2, -2*e^4 + 16*e^2 - 28, -2*e^4 + 16*e^2 - 28, 2*e^2, -6*e^3 + 46*e, -2*e^4 + 14*e^2 + 12, -2*e^4 + 14*e^2 + 12, -6*e^3 + 46*e, 4*e^4 - 24*e^2 - 6, 4*e^4 - 24*e^2 - 6, 2*e^4 - 18*e^2 + 50, -e^4 + 13*e^2 - 24, -e^4 + 13*e^2 - 24, 2*e^4 - 6*e^2 - 22, 2*e^4 - 6*e^2 - 22, 3/2*e^5 - 15*e^3 + 47/2*e, 3/2*e^5 - 15*e^3 + 47/2*e, 2*e^4 - 18*e^2 + 30, 2*e^4 - 18*e^2 + 30, e^5 - 9*e^3 + 20*e, e^5 - 9*e^3 + 20*e, -5/2*e^5 + 33*e^3 - 217/2*e, -5/2*e^5 + 33*e^3 - 217/2*e, -3/2*e^5 + 17*e^3 - 123/2*e, -3/2*e^5 + 17*e^3 - 123/2*e, -4*e^4 + 28*e^2 - 26, -4*e^4 + 28*e^2 - 26, -2*e^2 + 16, -2*e^2 + 16, e^4 - 9*e^2 + 4, e^4 - 9*e^2 + 4, 4*e^5 - 46*e^3 + 122*e, 6*e^4 - 50*e^2 + 60, 6*e^4 - 50*e^2 + 60, 4*e^5 - 46*e^3 + 122*e, 3*e^5 - 35*e^3 + 108*e, 3*e^5 - 35*e^3 + 108*e, 3/2*e^5 - 17*e^3 + 123/2*e, -1/2*e^5 + 15*e^3 - 129/2*e, -1/2*e^5 + 15*e^3 - 129/2*e, 3/2*e^5 - 17*e^3 + 123/2*e, -3/2*e^5 + 13*e^3 - 11/2*e, -3/2*e^5 + 13*e^3 - 11/2*e, e^4 - 13*e^2 + 56, e^4 - 13*e^2 + 56, 6*e^2 - 36, 4*e^4 - 42*e^2 + 72, 4*e^4 - 42*e^2 + 72, 6*e^2 - 36];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {2,2,w}>] := 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;