/*
  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, -5, 2, 8, -4, -2, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {79, 79, -3*w^5 + 4*w^4 + 13*w^3 - 14*w^2 - 9*w + 5}>;

primesArray := [
[9, 3, -w^2 + 2],
[19, 19, -w^3 + 4*w],
[19, 19, w^3 - 3*w - 1],
[29, 29, w^5 - w^4 - 6*w^3 + 4*w^2 + 8*w - 3],
[29, 29, w^4 - w^3 - 4*w^2 + 2*w + 2],
[31, 31, -w^4 + 4*w^2 + w - 3],
[41, 41, 2*w^5 - 2*w^4 - 10*w^3 + 7*w^2 + 11*w - 4],
[49, 7, -2*w^5 + 3*w^4 + 10*w^3 - 12*w^2 - 11*w + 6],
[59, 59, w^5 - 2*w^4 - 5*w^3 + 8*w^2 + 7*w - 5],
[59, 59, w^5 - w^4 - 4*w^3 + 3*w^2 + 3*w - 3],
[59, 59, 2*w^5 - 3*w^4 - 9*w^3 + 11*w^2 + 9*w - 4],
[61, 61, -w^5 + w^4 + 5*w^3 - 3*w^2 - 7*w + 1],
[64, 2, -2],
[71, 71, -2*w^5 + 2*w^4 + 10*w^3 - 8*w^2 - 11*w + 6],
[79, 79, -3*w^5 + 4*w^4 + 13*w^3 - 14*w^2 - 9*w + 5],
[79, 79, -w^4 - w^3 + 5*w^2 + 4*w - 3],
[79, 79, -2*w^5 + 2*w^4 + 9*w^3 - 7*w^2 - 8*w + 4],
[81, 3, 3*w^5 - 5*w^4 - 14*w^3 + 19*w^2 + 13*w - 8],
[89, 89, 2*w^5 - 2*w^4 - 9*w^3 + 6*w^2 + 8*w - 1],
[101, 101, -w^4 - w^3 + 5*w^2 + 3*w - 3],
[101, 101, -w^5 + 2*w^4 + 6*w^3 - 8*w^2 - 9*w + 4],
[109, 109, -2*w^5 + 4*w^4 + 8*w^3 - 14*w^2 - 6*w + 5],
[109, 109, -3*w^5 + 4*w^4 + 13*w^3 - 14*w^2 - 10*w + 7],
[125, 5, w^5 - w^4 - 5*w^3 + 4*w^2 + 5*w],
[131, 131, -w^2 - w + 3],
[131, 131, w^5 - 3*w^4 - 4*w^3 + 11*w^2 + 3*w - 4],
[131, 131, w^5 - 2*w^4 - 5*w^3 + 9*w^2 + 6*w - 5],
[131, 131, -2*w^5 + 4*w^4 + 8*w^3 - 15*w^2 - 6*w + 7],
[139, 139, w^4 - 3*w^3 - 3*w^2 + 9*w],
[149, 149, 2*w^5 - 3*w^4 - 9*w^3 + 12*w^2 + 7*w - 5],
[149, 149, w^5 - 2*w^4 - 4*w^3 + 8*w^2 + 2*w - 2],
[151, 151, 3*w^5 - 4*w^4 - 13*w^3 + 14*w^2 + 10*w - 5],
[151, 151, -3*w^5 + 4*w^4 + 14*w^3 - 15*w^2 - 12*w + 6],
[151, 151, 4*w^5 - 5*w^4 - 18*w^3 + 18*w^2 + 15*w - 9],
[151, 151, 2*w^5 - 2*w^4 - 9*w^3 + 7*w^2 + 8*w - 5],
[169, 13, -2*w^5 + w^4 + 11*w^3 - 3*w^2 - 13*w + 1],
[179, 179, w^3 - w^2 - 2*w + 3],
[181, 181, w^5 - 2*w^4 - 6*w^3 + 7*w^2 + 10*w - 3],
[181, 181, -2*w^4 + w^3 + 8*w^2 - 3*w - 2],
[191, 191, w^5 - w^4 - 6*w^3 + 5*w^2 + 9*w - 4],
[191, 191, -w^3 + 2*w^2 + 3*w - 4],
[191, 191, -2*w^5 + w^4 + 11*w^3 - 3*w^2 - 12*w],
[191, 191, w^4 - 2*w^3 - 4*w^2 + 7*w],
[199, 199, -3*w^5 + 4*w^4 + 15*w^3 - 14*w^2 - 16*w + 6],
[199, 199, -2*w^5 + 3*w^4 + 10*w^3 - 12*w^2 - 12*w + 7],
[211, 211, -2*w^5 + 4*w^4 + 9*w^3 - 15*w^2 - 8*w + 5],
[211, 211, -3*w^5 + 5*w^4 + 13*w^3 - 18*w^2 - 10*w + 8],
[229, 229, -3*w^5 + 4*w^4 + 13*w^3 - 14*w^2 - 10*w + 6],
[229, 229, w^4 - 4*w^2 - 1],
[229, 229, w^5 - w^4 - 3*w^3 + 3*w^2 - 3*w - 2],
[239, 239, -2*w^5 + 3*w^4 + 9*w^3 - 11*w^2 - 8*w + 8],
[241, 241, w^5 - 3*w^4 - 4*w^3 + 10*w^2 + 3*w - 2],
[241, 241, w^5 - w^4 - 5*w^3 + 3*w^2 + 4*w + 1],
[251, 251, -3*w^5 + 3*w^4 + 15*w^3 - 10*w^2 - 17*w + 3],
[271, 271, -2*w^5 + 4*w^4 + 9*w^3 - 14*w^2 - 8*w + 2],
[271, 271, -2*w^5 + 3*w^4 + 8*w^3 - 9*w^2 - 4*w],
[271, 271, w^5 - 2*w^4 - 3*w^3 + 7*w^2 - 2*w - 4],
[289, 17, 4*w^5 - 6*w^4 - 17*w^3 + 22*w^2 + 12*w - 9],
[311, 311, -2*w^5 + 3*w^4 + 7*w^3 - 9*w^2 - 3*w + 2],
[331, 331, 3*w^5 - 5*w^4 - 14*w^3 + 19*w^2 + 14*w - 12],
[349, 349, -4*w^5 + 6*w^4 + 18*w^3 - 21*w^2 - 17*w + 7],
[349, 349, -w^4 + w^3 + 5*w^2 - 3*w - 3],
[359, 359, -2*w^5 + 4*w^4 + 9*w^3 - 15*w^2 - 10*w + 9],
[361, 19, 2*w^5 - 3*w^4 - 9*w^3 + 11*w^2 + 6*w - 4],
[361, 19, -3*w^5 + 5*w^4 + 13*w^3 - 18*w^2 - 10*w + 9],
[379, 379, -4*w^5 + 5*w^4 + 18*w^3 - 17*w^2 - 14*w + 7],
[379, 379, 2*w^5 - 4*w^4 - 10*w^3 + 15*w^2 + 11*w - 5],
[389, 389, -5*w^5 + 7*w^4 + 22*w^3 - 25*w^2 - 17*w + 11],
[389, 389, w^5 - 3*w^4 - 3*w^3 + 12*w^2 - 2*w - 5],
[401, 401, -4*w^5 + 6*w^4 + 18*w^3 - 23*w^2 - 15*w + 10],
[401, 401, -w^5 + 3*w^4 + 4*w^3 - 12*w^2 - w + 4],
[401, 401, 4*w^5 - 5*w^4 - 20*w^3 + 19*w^2 + 21*w - 11],
[401, 401, -3*w^5 + 3*w^4 + 16*w^3 - 12*w^2 - 18*w + 6],
[409, 409, -2*w^4 + 2*w^3 + 7*w^2 - 6*w - 2],
[409, 409, -w^5 + 3*w^4 + 5*w^3 - 13*w^2 - 5*w + 9],
[419, 419, 2*w^5 - 2*w^4 - 10*w^3 + 7*w^2 + 10*w - 6],
[419, 419, -w^5 + 7*w^3 - w^2 - 10*w + 4],
[419, 419, w^5 - 2*w^4 - 5*w^3 + 10*w^2 + 5*w - 9],
[421, 421, 3*w^5 - 3*w^4 - 16*w^3 + 12*w^2 + 18*w - 7],
[431, 431, 4*w^5 - 7*w^4 - 17*w^3 + 25*w^2 + 13*w - 10],
[431, 431, -2*w^5 + 3*w^4 + 7*w^3 - 10*w^2 - w + 5],
[431, 431, 4*w^5 - 6*w^4 - 19*w^3 + 23*w^2 + 17*w - 10],
[439, 439, -2*w^5 + 4*w^4 + 10*w^3 - 15*w^2 - 11*w + 4],
[461, 461, w^5 - 2*w^4 - 3*w^3 + 6*w^2 + 2],
[461, 461, w^5 - w^4 - 5*w^3 + 5*w^2 + 6*w - 7],
[479, 479, -2*w^5 + w^4 + 9*w^3 - 2*w^2 - 8*w],
[479, 479, 2*w^5 - 2*w^4 - 10*w^3 + 8*w^2 + 12*w - 7],
[491, 491, -3*w^5 + 3*w^4 + 14*w^3 - 12*w^2 - 12*w + 7],
[491, 491, -3*w^5 + 3*w^4 + 15*w^3 - 12*w^2 - 14*w + 8],
[491, 491, -2*w^2 + w + 6],
[499, 499, -3*w^5 + 3*w^4 + 14*w^3 - 10*w^2 - 13*w + 6],
[509, 509, -2*w^5 + 2*w^4 + 8*w^3 - 6*w^2 - 5*w],
[521, 521, 3*w^5 - 4*w^4 - 14*w^3 + 15*w^2 + 15*w - 9],
[521, 521, -3*w^5 + 5*w^4 + 14*w^3 - 19*w^2 - 15*w + 9],
[521, 521, -4*w^5 + 5*w^4 + 18*w^3 - 17*w^2 - 16*w + 6],
[529, 23, w^5 - 3*w^4 - 5*w^3 + 11*w^2 + 7*w - 5],
[541, 541, -3*w^5 + 4*w^4 + 14*w^3 - 15*w^2 - 14*w + 10],
[569, 569, -w^5 - w^4 + 6*w^3 + 4*w^2 - 6*w - 1],
[571, 571, -2*w^4 + w^3 + 9*w^2 - 3*w - 4],
[599, 599, -3*w^5 + 3*w^4 + 14*w^3 - 11*w^2 - 11*w + 4],
[601, 601, w^4 - 2*w^2 - w - 2],
[619, 619, 2*w^5 - 3*w^4 - 9*w^3 + 11*w^2 + 10*w - 5],
[619, 619, w^5 - 3*w^4 - 4*w^3 + 12*w^2 + 3*w - 3],
[619, 619, -3*w^5 + 5*w^4 + 13*w^3 - 20*w^2 - 10*w + 10],
[619, 619, w^5 - 2*w^4 - 5*w^3 + 10*w^2 + 5*w - 8],
[619, 619, -4*w^5 + 6*w^4 + 19*w^3 - 23*w^2 - 17*w + 11],
[619, 619, -4*w^5 + 6*w^4 + 17*w^3 - 22*w^2 - 13*w + 10],
[631, 631, w^5 - 3*w^4 - 5*w^3 + 13*w^2 + 5*w - 6],
[641, 641, -w^3 + w^2 + 3*w - 5],
[641, 641, -w^5 + 3*w^4 + 2*w^3 - 10*w^2 + 3*w + 4],
[659, 659, -w^5 + 3*w^4 + 5*w^3 - 12*w^2 - 5*w + 5],
[661, 661, -2*w^5 + 3*w^4 + 8*w^3 - 12*w^2 - 3*w + 8],
[661, 661, 2*w^5 - 3*w^4 - 8*w^3 + 10*w^2 + 6*w - 1],
[691, 691, -w^3 - w^2 + 5*w + 1],
[691, 691, -3*w^5 + 4*w^4 + 13*w^3 - 13*w^2 - 10*w + 6],
[691, 691, 3*w^5 - 4*w^4 - 15*w^3 + 15*w^2 + 14*w - 6],
[701, 701, 3*w^5 - 6*w^4 - 13*w^3 + 23*w^2 + 10*w - 10],
[701, 701, -3*w^5 + 6*w^4 + 13*w^3 - 23*w^2 - 11*w + 9],
[709, 709, 3*w^5 - 6*w^4 - 13*w^3 + 22*w^2 + 11*w - 10],
[709, 709, -w^5 + 5*w^3 - 4*w - 2],
[709, 709, -w^5 + w^4 + 7*w^3 - 6*w^2 - 11*w + 5],
[719, 719, -2*w^5 + 3*w^4 + 9*w^3 - 12*w^2 - 7*w + 4],
[751, 751, w^4 + w^3 - 5*w^2 - 6*w + 4],
[769, 769, 4*w^5 - 5*w^4 - 19*w^3 + 18*w^2 + 18*w - 10],
[769, 769, 4*w^5 - 6*w^4 - 17*w^3 + 22*w^2 + 13*w - 11],
[809, 809, w^5 - w^4 - 4*w^3 + 5*w^2 + w - 7],
[809, 809, 2*w^5 - 5*w^4 - 9*w^3 + 19*w^2 + 10*w - 7],
[809, 809, -w^5 + 5*w^3 + w^2 - 7*w],
[809, 809, -3*w^5 + 4*w^4 + 15*w^3 - 16*w^2 - 15*w + 11],
[811, 811, -2*w^5 + 4*w^4 + 8*w^3 - 13*w^2 - 6*w + 5],
[811, 811, -w^5 + 3*w^4 + 3*w^3 - 9*w^2 - w],
[811, 811, 2*w^5 - 4*w^4 - 9*w^3 + 16*w^2 + 10*w - 9],
[811, 811, 2*w^5 - 4*w^4 - 10*w^3 + 16*w^2 + 10*w - 7],
[821, 821, -w^5 + 2*w^4 + 5*w^3 - 6*w^2 - 5*w - 2],
[829, 829, 3*w^5 - 5*w^4 - 13*w^3 + 19*w^2 + 12*w - 9],
[839, 839, 2*w^5 - 4*w^4 - 7*w^3 + 14*w^2 + w - 6],
[839, 839, 4*w^5 - 5*w^4 - 18*w^3 + 19*w^2 + 14*w - 10],
[841, 29, -2*w^5 + 4*w^4 + 10*w^3 - 15*w^2 - 10*w + 5],
[841, 29, -w^5 + 3*w^4 + 6*w^3 - 12*w^2 - 10*w + 6],
[859, 859, 3*w^5 - 3*w^4 - 15*w^3 + 10*w^2 + 14*w - 2],
[881, 881, w^5 - 2*w^4 - 6*w^3 + 7*w^2 + 10*w - 4],
[911, 911, -4*w^5 + 5*w^4 + 19*w^3 - 20*w^2 - 17*w + 11],
[919, 919, 2*w^5 - 3*w^4 - 10*w^3 + 13*w^2 + 9*w - 10],
[929, 929, -w^5 + w^4 + 5*w^3 - 2*w^2 - 8*w - 1],
[941, 941, -5*w^5 + 7*w^4 + 23*w^3 - 26*w^2 - 20*w + 11],
[961, 31, -3*w^5 + 4*w^4 + 16*w^3 - 16*w^2 - 20*w + 8],
[971, 971, w^5 - 5*w^3 + 5*w - 4],
[991, 991, 3*w^5 - 5*w^4 - 12*w^3 + 18*w^2 + 8*w - 6],
[991, 991, 3*w^5 - 5*w^4 - 14*w^3 + 18*w^2 + 12*w - 7]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x;
K := Rationals(); e := 1;

heckeEigenvaluesArray := [4, -4, 6, 0, -7, 1, 3, 5, -8, 7, 5, -10, -13, 15, -1, 6, -3, 10, -4, 16, 3, -2, -4, 6, 0, 2, -13, 10, 2, -6, 2, -8, 0, -12, -10, 14, -4, 7, 17, -12, 8, -8, 17, 15, -13, -12, 10, 5, 25, 9, 22, 6, 20, 8, -15, -14, 14, 3, 0, -18, -34, -14, -8, -2, 16, -11, -33, -3, -33, 30, -12, 30, 4, 40, 5, -18, 30, 31, -29, 20, -7, -30, -4, 0, -20, 6, -17, 41, 33, -36, -28, -42, 43, 20, -14, -21, 26, 16, -40, -12, 35, 26, -19, -44, -5, 11, -3, -13, -42, 15, 7, -8, 41, -15, 21, 30, 14, 15, 51, -18, -14, -48, -8, -1, 10, -38, 36, 42, 12, -24, -20, 53, 33, -54, -24, 24, 10, 22, -37, -28, 10, 3, 27, 5, 27, 21, -27, -8, -28];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {79, 79, -3*w^5 + 4*w^4 + 13*w^3 - 14*w^2 - 9*w + 5}>] := 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;