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

NN := ideal<ZF | {13, 13, -w^3 + 3*w}>;

primesArray := [
[7, 7, w^5 - 5*w^3 + 4*w],
[13, 13, -w^3 + 3*w],
[25, 5, w^4 - w^3 - 4*w^2 + 2*w + 2],
[31, 31, -w^5 + 5*w^3 - 5*w + 1],
[31, 31, -w^4 + w^3 + 5*w^2 - 3*w - 3],
[37, 37, w^4 - 5*w^2 + w + 3],
[43, 43, -2*w^5 + w^4 + 9*w^3 - 3*w^2 - 6*w],
[47, 47, w^5 - w^4 - 4*w^3 + 4*w^2 - 2],
[49, 7, -w^4 + 4*w^2 + w - 3],
[53, 53, -w^4 + w^3 + 4*w^2 - 3*w - 4],
[59, 59, w^4 - 5*w^2 - w + 5],
[61, 61, w^3 - w^2 - 3*w],
[64, 2, -2],
[67, 67, -2*w^5 + w^4 + 9*w^3 - 3*w^2 - 7*w],
[67, 67, 2*w^5 - w^4 - 10*w^3 + 4*w^2 + 9*w - 1],
[73, 73, w^5 - w^4 - 4*w^3 + 5*w^2 + w - 3],
[73, 73, 3*w^5 - 3*w^4 - 13*w^3 + 11*w^2 + 7*w - 3],
[83, 83, -2*w^5 + 3*w^4 + 9*w^3 - 11*w^2 - 6*w + 2],
[97, 97, -2*w^4 + w^3 + 9*w^2 - 3*w - 4],
[101, 101, -2*w^5 + 3*w^4 + 9*w^3 - 11*w^2 - 6*w + 4],
[101, 101, w^5 - 2*w^4 - 4*w^3 + 9*w^2 + w - 4],
[107, 107, -2*w^5 + w^4 + 10*w^3 - 4*w^2 - 9*w + 2],
[107, 107, -w^5 + w^4 + 6*w^3 - 4*w^2 - 8*w + 2],
[113, 113, w^5 - w^4 - 4*w^3 + 3*w^2 + 2*w + 2],
[121, 11, w^5 - w^4 - 5*w^3 + 3*w^2 + 5*w - 2],
[127, 127, 2*w^5 - 3*w^4 - 9*w^3 + 12*w^2 + 5*w - 4],
[131, 131, w^5 - w^4 - 6*w^3 + 4*w^2 + 8*w - 1],
[131, 131, 2*w^5 - 3*w^4 - 7*w^3 + 12*w^2 - w - 4],
[149, 149, -2*w^5 + 2*w^4 + 8*w^3 - 8*w^2 - 3*w + 4],
[151, 151, w^5 - 2*w^4 - 4*w^3 + 8*w^2 + 3*w - 3],
[157, 157, w^5 - w^4 - 4*w^3 + 5*w^2 - w - 2],
[163, 163, -2*w^5 + 2*w^4 + 9*w^3 - 6*w^2 - 6*w],
[163, 163, w^5 - 2*w^4 - 4*w^3 + 8*w^2 - 4],
[163, 163, -2*w^5 + w^4 + 9*w^3 - 4*w^2 - 7*w + 4],
[173, 173, 2*w^4 - w^3 - 8*w^2 + 2*w + 4],
[173, 173, -w^5 + 3*w^4 + 4*w^3 - 14*w^2 - w + 9],
[179, 179, -w^5 + 2*w^4 + 4*w^3 - 8*w^2 + 6],
[191, 191, 2*w^4 - 2*w^3 - 9*w^2 + 6*w + 5],
[193, 193, -w^5 + w^4 + 5*w^3 - 4*w^2 - 6*w + 2],
[199, 199, -2*w^5 + 3*w^4 + 9*w^3 - 11*w^2 - 6*w + 3],
[211, 211, -w^5 + 5*w^3 - 2*w - 2],
[211, 211, -2*w^5 + w^4 + 9*w^3 - 4*w^2 - 5*w + 1],
[211, 211, w^5 - 5*w^3 - 2*w^2 + 4*w + 4],
[223, 223, w^5 - 5*w^3 - w^2 + 7*w + 1],
[227, 227, -w^5 + w^4 + 6*w^3 - 4*w^2 - 7*w + 1],
[227, 227, w^3 - 2*w^2 - 4*w + 3],
[229, 229, -w^5 + 6*w^3 - 7*w + 1],
[233, 233, 3*w^5 - 2*w^4 - 15*w^3 + 8*w^2 + 13*w - 3],
[233, 233, -w^5 + 4*w^3 + w^2 - 2*w - 3],
[233, 233, -w^3 + w^2 + 4*w - 1],
[257, 257, w^5 - 2*w^4 - 5*w^3 + 8*w^2 + 4*w - 2],
[257, 257, w^5 + w^4 - 6*w^3 - 4*w^2 + 6*w + 1],
[263, 263, 3*w^5 - 3*w^4 - 13*w^3 + 10*w^2 + 7*w - 1],
[269, 269, w^5 - w^4 - 4*w^3 + 5*w^2 - 3],
[283, 283, w^5 - w^4 - 5*w^3 + 5*w^2 + 5*w - 1],
[293, 293, -2*w^4 + w^3 + 8*w^2 - 3*w - 4],
[293, 293, 2*w^5 - 4*w^4 - 8*w^3 + 17*w^2 + 3*w - 8],
[307, 307, w^5 - 7*w^3 + 11*w - 1],
[311, 311, w^4 - 2*w^3 - 4*w^2 + 8*w + 1],
[313, 313, -w^5 + 5*w^3 + w^2 - 3*w - 3],
[317, 317, -w^5 + 4*w^3 - w^2 - w + 4],
[317, 317, -3*w^5 + 4*w^4 + 13*w^3 - 16*w^2 - 6*w + 8],
[331, 331, w^5 - 7*w^3 + 10*w],
[331, 331, 3*w^4 - 2*w^3 - 13*w^2 + 7*w + 7],
[331, 331, w^5 - 4*w^3 + w^2 + w - 3],
[343, 7, -4*w^5 + 3*w^4 + 19*w^3 - 10*w^2 - 15*w + 2],
[347, 347, 2*w^5 - 2*w^4 - 10*w^3 + 7*w^2 + 7*w],
[349, 349, 3*w^5 - 2*w^4 - 14*w^3 + 7*w^2 + 9*w + 1],
[349, 349, -w^5 + 2*w^4 + 3*w^3 - 9*w^2 + w + 7],
[359, 359, 2*w^5 - 2*w^4 - 10*w^3 + 7*w^2 + 10*w - 4],
[359, 359, w^5 - 7*w^3 + 11*w + 1],
[373, 373, w^5 - 6*w^3 + 6*w - 2],
[383, 383, -4*w^5 + 4*w^4 + 18*w^3 - 14*w^2 - 12*w + 3],
[389, 389, w^5 + w^4 - 5*w^3 - 5*w^2 + 6*w + 4],
[397, 397, 2*w^5 - w^4 - 9*w^3 + 5*w^2 + 6*w - 5],
[397, 397, -2*w^5 + 3*w^4 + 8*w^3 - 10*w^2 - 3*w + 3],
[409, 409, -w^5 + w^4 + 5*w^3 - 3*w^2 - 6*w - 3],
[421, 421, -3*w^5 + w^4 + 14*w^3 - 3*w^2 - 10*w - 2],
[431, 431, 2*w^4 - w^3 - 10*w^2 + 3*w + 7],
[433, 433, -3*w^5 + 2*w^4 + 14*w^3 - 6*w^2 - 11*w + 2],
[433, 433, 2*w^5 - 3*w^4 - 10*w^3 + 11*w^2 + 10*w - 2],
[439, 439, -2*w^5 + 2*w^4 + 10*w^3 - 8*w^2 - 7*w + 3],
[439, 439, 3*w^5 - 4*w^4 - 13*w^3 + 16*w^2 + 8*w - 5],
[439, 439, -w^5 + w^4 + 5*w^3 - 6*w^2 - 4*w + 6],
[461, 461, w^5 + w^4 - 5*w^3 - 4*w^2 + 6*w + 1],
[467, 467, w^5 + w^4 - 4*w^3 - 6*w^2 + w + 5],
[479, 479, w^5 - w^4 - 5*w^3 + 3*w^2 + 2*w + 1],
[487, 487, 3*w^5 - 3*w^4 - 13*w^3 + 12*w^2 + 7*w - 3],
[499, 499, 2*w^5 - 2*w^4 - 11*w^3 + 7*w^2 + 12*w - 1],
[503, 503, w^5 - 2*w^4 - 5*w^3 + 9*w^2 + 5*w - 3],
[509, 509, 3*w^5 - 3*w^4 - 15*w^3 + 11*w^2 + 13*w - 2],
[509, 509, 2*w^5 - 2*w^4 - 10*w^3 + 7*w^2 + 11*w - 2],
[521, 521, 3*w^5 - 3*w^4 - 14*w^3 + 10*w^2 + 11*w - 3],
[521, 521, -w^5 + 2*w^4 + 5*w^3 - 6*w^2 - 5*w],
[523, 523, -2*w^5 + 2*w^4 + 10*w^3 - 7*w^2 - 11*w + 1],
[523, 523, -2*w^5 + w^4 + 10*w^3 - 3*w^2 - 11*w],
[523, 523, -2*w^5 + w^4 + 10*w^3 - 4*w^2 - 10*w + 4],
[547, 547, -2*w^4 + 2*w^3 + 10*w^2 - 6*w - 7],
[557, 557, 3*w^5 - 3*w^4 - 14*w^3 + 10*w^2 + 9*w - 1],
[569, 569, 3*w^5 - 4*w^4 - 13*w^3 + 14*w^2 + 8*w - 2],
[571, 571, w^4 + w^3 - 5*w^2 - 2*w + 5],
[577, 577, 3*w^4 - 2*w^3 - 13*w^2 + 7*w + 6],
[577, 577, w^5 + w^4 - 5*w^3 - 4*w^2 + 4*w + 4],
[587, 587, 2*w^5 - w^4 - 10*w^3 + 5*w^2 + 9*w - 4],
[599, 599, -2*w^5 + 3*w^4 + 10*w^3 - 12*w^2 - 9*w + 3],
[599, 599, w^5 + 2*w^4 - 6*w^3 - 9*w^2 + 8*w + 6],
[601, 601, -2*w^5 + 3*w^4 + 8*w^3 - 11*w^2 - 5*w + 4],
[607, 607, w^5 - 2*w^4 - 5*w^3 + 8*w^2 + 3*w - 1],
[613, 613, w^5 - 2*w^4 - 5*w^3 + 10*w^2 + 5*w - 7],
[613, 613, -w^5 + 2*w^4 + 3*w^3 - 7*w^2],
[617, 617, -3*w^5 + 3*w^4 + 12*w^3 - 11*w^2 - 5*w + 4],
[625, 5, -2*w^5 + 2*w^4 + 9*w^3 - 6*w^2 - 5*w + 1],
[631, 631, -2*w^5 + w^4 + 9*w^3 - 3*w^2 - 8*w + 1],
[641, 641, w^5 - 5*w^3 - w^2 + 5*w + 5],
[659, 659, -w^5 + 3*w^4 + 5*w^3 - 12*w^2 - 5*w + 3],
[661, 661, 4*w^5 - 3*w^4 - 19*w^3 + 11*w^2 + 14*w - 2],
[677, 677, -2*w^5 + 2*w^4 + 9*w^3 - 7*w^2 - 6*w - 2],
[683, 683, -w^5 + w^4 + 4*w^3 - 5*w^2 + w + 4],
[683, 683, -2*w^4 + w^3 + 9*w^2 - 5*w - 4],
[701, 701, -2*w^5 + 3*w^4 + 10*w^3 - 11*w^2 - 10*w + 4],
[701, 701, -3*w^5 + w^4 + 15*w^3 - 3*w^2 - 14*w + 1],
[709, 709, 2*w^4 - 2*w^3 - 9*w^2 + 5*w + 7],
[719, 719, w^4 - 2*w^3 - 4*w^2 + 7*w + 4],
[719, 719, w^3 + 2*w^2 - 4*w - 5],
[727, 727, 2*w^5 - 3*w^4 - 8*w^3 + 11*w^2 + 4*w - 1],
[729, 3, -3],
[743, 743, 3*w^5 - 2*w^4 - 15*w^3 + 6*w^2 + 12*w],
[743, 743, -w^4 + w^3 + 6*w^2 - 4*w - 3],
[769, 769, 2*w^5 - 3*w^4 - 8*w^3 + 12*w^2 + 3*w - 2],
[769, 769, -3*w^5 + 4*w^4 + 13*w^3 - 15*w^2 - 8*w + 3],
[787, 787, 2*w^3 - w^2 - 5*w],
[797, 797, 3*w^5 - w^4 - 15*w^3 + 4*w^2 + 12*w - 1],
[827, 827, 3*w^4 - w^3 - 13*w^2 + 2*w + 8],
[839, 839, -2*w^5 + w^4 + 10*w^3 - 4*w^2 - 11*w + 2],
[853, 853, -2*w^4 + w^3 + 10*w^2 - 4*w - 6],
[853, 853, -w^5 + w^4 + 6*w^3 - 3*w^2 - 7*w - 3],
[853, 853, -2*w^5 + 3*w^4 + 9*w^3 - 11*w^2 - 8*w + 3],
[857, 857, w^5 - 2*w^4 - 3*w^3 + 10*w^2 - 2*w - 7],
[859, 859, -4*w^5 + 3*w^4 + 18*w^3 - 10*w^2 - 12*w],
[863, 863, w^5 - 3*w^4 - 3*w^3 + 14*w^2 - 2*w - 9],
[877, 877, -w^5 + 7*w^3 - 12*w - 1],
[881, 881, 2*w^4 - 2*w^3 - 9*w^2 + 8*w + 6],
[881, 881, 3*w^5 - w^4 - 15*w^3 + 4*w^2 + 12*w - 2],
[887, 887, -w^5 - w^4 + 5*w^3 + 5*w^2 - 2*w - 4],
[919, 919, 4*w^5 - 4*w^4 - 18*w^3 + 14*w^2 + 11*w - 3],
[919, 919, -w^5 + 4*w^3 - w - 3],
[929, 929, -w^5 + 2*w^4 + 4*w^3 - 10*w^2 + 5],
[937, 937, 3*w^5 - 4*w^4 - 12*w^3 + 15*w^2 + 5*w - 3],
[947, 947, 3*w^5 - 3*w^4 - 13*w^3 + 10*w^2 + 8*w - 4],
[953, 953, 2*w^5 - 9*w^3 - w^2 + 6*w + 4],
[953, 953, -3*w^5 + 3*w^4 + 12*w^3 - 11*w^2 - 5*w + 2],
[967, 967, -2*w^5 + 9*w^3 + 2*w^2 - 7*w - 4],
[977, 977, 3*w^5 - w^4 - 14*w^3 + 3*w^2 + 11*w - 1],
[977, 977, w^4 - w^3 - 6*w^2 + 3*w + 3],
[997, 997, 3*w^5 - 2*w^4 - 15*w^3 + 7*w^2 + 13*w - 4]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

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

heckeEigenvaluesArray := [-3, 1, 3, -10, -7, -5, -8, -1, -6, 6, -4, -9, -13, 10, -2, 0, -12, -3, -4, 15, -12, -10, -10, 0, -8, -5, -2, 16, 4, -10, 4, -8, 3, -13, -9, 8, 2, -6, 5, 24, 20, -4, 16, 22, 9, -16, 19, -18, 6, 3, -14, 14, -23, -22, -4, -24, 14, -16, -11, 26, -10, 18, 1, 6, -17, 6, 0, 29, -8, 20, 25, 14, 24, -24, -27, -22, -10, 38, -7, -30, -16, -13, -16, 38, -15, 23, -39, -28, 19, 10, 4, -5, 9, 3, -27, -11, 20, 23, 24, 24, -15, 18, 16, 15, 29, 11, -20, 43, -22, -47, 18, -8, 7, 8, 34, -28, 6, 20, -36, -7, 3, -1, -20, -27, -48, -4, 36, 18, -21, -30, 12, -22, 0, 34, -30, -35, 46, 17, 36, -28, -18, 6, 22, -36, -15, -21, 49, 12, 3, -9, -51, -4, -15, -28, 46];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {13, 13, -w^3 + 3*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;