/*
  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![9, 6, -7, -2, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {23, 23, -1/3*w^3 + 2/3*w^2 + 1/3*w - 2}>;

primesArray := [
[4, 2, -1/3*w^3 + 2/3*w^2 + 4/3*w - 1],
[7, 7, 1/3*w^3 - 2/3*w^2 - 7/3*w + 1],
[7, 7, w - 2],
[7, 7, w - 1],
[9, 3, w],
[9, 3, -1/3*w^3 + 2/3*w^2 + 7/3*w - 2],
[23, 23, -1/3*w^3 + 2/3*w^2 + 1/3*w - 2],
[23, 23, -2/3*w^3 + 4/3*w^2 + 11/3*w - 4],
[31, 31, -2/3*w^3 + 1/3*w^2 + 14/3*w + 2],
[41, 41, -1/3*w^3 + 5/3*w^2 + 4/3*w - 7],
[41, 41, -1/3*w^3 + 5/3*w^2 + 4/3*w - 4],
[47, 47, 1/3*w^3 - 5/3*w^2 - 1/3*w + 5],
[47, 47, w^2 - w - 4],
[73, 73, -w^3 + 3*w^2 + 4*w - 8],
[73, 73, -2/3*w^3 + 7/3*w^2 - 1/3*w - 4],
[73, 73, -w^3 + w^2 + 7*w + 1],
[73, 73, 2/3*w^3 - 4/3*w^2 - 14/3*w + 5],
[79, 79, w^2 - 5],
[79, 79, -2/3*w^3 + 7/3*w^2 + 8/3*w - 6],
[97, 97, -2/3*w^3 + 7/3*w^2 + 5/3*w - 4],
[97, 97, 1/3*w^3 + 1/3*w^2 - 7/3*w - 5],
[103, 103, w^3 - 3*w^2 - 2*w + 5],
[103, 103, -4/3*w^3 + 11/3*w^2 + 13/3*w - 10],
[113, 113, -1/3*w^3 + 5/3*w^2 + 4/3*w - 3],
[113, 113, w^3 - 2*w^2 - 5*w + 1],
[113, 113, -1/3*w^3 + 5/3*w^2 + 4/3*w - 1],
[113, 113, -1/3*w^3 + 5/3*w^2 + 4/3*w - 10],
[127, 127, -2/3*w^3 + 1/3*w^2 + 14/3*w + 1],
[127, 127, -2/3*w^3 + 7/3*w^2 + 2/3*w - 6],
[137, 137, -1/3*w^3 + 5/3*w^2 - 2/3*w - 6],
[137, 137, 1/3*w^3 + 1/3*w^2 - 10/3*w - 1],
[151, 151, -w^3 + 2*w^2 + 3*w - 5],
[151, 151, 1/3*w^3 + 1/3*w^2 - 10/3*w - 7],
[169, 13, -2/3*w^3 + 7/3*w^2 + 5/3*w - 2],
[169, 13, 1/3*w^3 + 1/3*w^2 - 7/3*w - 7],
[191, 191, 1/3*w^3 - 2/3*w^2 + 2/3*w - 2],
[191, 191, 2*w - 5],
[191, 191, -w^3 + 2*w^2 + 6*w - 2],
[191, 191, 2/3*w^3 - 4/3*w^2 - 14/3*w - 1],
[239, 239, 1/3*w^3 - 2/3*w^2 - 7/3*w - 3],
[239, 239, w - 5],
[241, 241, 1/3*w^3 + 1/3*w^2 - 10/3*w],
[241, 241, 4/3*w^3 - 17/3*w^2 + 2/3*w + 9],
[241, 241, -1/3*w^3 + 5/3*w^2 - 2/3*w - 7],
[241, 241, -4/3*w^3 + 14/3*w^2 + 13/3*w - 14],
[257, 257, -w - 4],
[257, 257, -1/3*w^3 + 2/3*w^2 + 7/3*w - 6],
[263, 263, -4/3*w^3 + 11/3*w^2 + 16/3*w - 8],
[263, 263, 2/3*w^3 - 1/3*w^2 - 8/3*w - 3],
[281, 281, -1/3*w^3 - 1/3*w^2 + 10/3*w - 2],
[281, 281, -1/3*w^3 + 5/3*w^2 - 5/3*w - 5],
[281, 281, -1/3*w^3 + 5/3*w^2 - 2/3*w - 9],
[281, 281, -2/3*w^3 + 1/3*w^2 + 17/3*w],
[289, 17, -w^3 + 2*w^2 + 4*w - 4],
[289, 17, w^3 - 2*w^2 - 4*w + 2],
[311, 311, -2/3*w^3 + 10/3*w^2 - 4/3*w - 9],
[311, 311, -2/3*w^3 - 2/3*w^2 + 20/3*w + 5],
[313, 313, -1/3*w^3 + 5/3*w^2 - 2/3*w - 8],
[313, 313, -1/3*w^3 - 1/3*w^2 + 10/3*w - 1],
[313, 313, w^3 - 4*w^2 - 2*w + 10],
[313, 313, 1/3*w^3 + 4/3*w^2 - 10/3*w - 8],
[337, 337, -4/3*w^3 + 5/3*w^2 + 19/3*w + 4],
[337, 337, 5/3*w^3 - 13/3*w^2 - 17/3*w + 5],
[353, 353, -4/3*w^3 + 8/3*w^2 + 25/3*w - 3],
[353, 353, -1/3*w^3 - 1/3*w^2 + 16/3*w + 6],
[353, 353, -2/3*w^3 + 10/3*w^2 + 2/3*w - 13],
[353, 353, -5/3*w^3 + 7/3*w^2 + 29/3*w - 3],
[367, 367, -5/3*w^3 + 4/3*w^2 + 35/3*w + 4],
[367, 367, 2/3*w^3 - 7/3*w^2 - 8/3*w + 4],
[367, 367, -4/3*w^3 + 14/3*w^2 + 1/3*w - 8],
[367, 367, w^2 - 7],
[383, 383, 4/3*w^3 - 14/3*w^2 - 10/3*w + 15],
[383, 383, -5/3*w^3 + 13/3*w^2 + 20/3*w - 12],
[383, 383, 2*w^2 - 2*w - 13],
[383, 383, w^3 - w^2 - 8*w - 1],
[401, 401, -4/3*w^3 + 11/3*w^2 + 22/3*w - 11],
[401, 401, 5/3*w^3 - 10/3*w^2 - 29/3*w + 5],
[409, 409, -w^3 + 2*w^2 + 3*w - 2],
[409, 409, -1/3*w^3 + 8/3*w^2 - 2/3*w - 6],
[409, 409, 1/3*w^3 - 8/3*w^2 + 2/3*w + 12],
[409, 409, 4/3*w^3 - 8/3*w^2 - 19/3*w + 4],
[433, 433, -w^3 + 3*w^2 - 1],
[433, 433, -1/3*w^3 + 5/3*w^2 - 8/3*w - 5],
[433, 433, -w^3 + w^2 + 8*w - 2],
[433, 433, 5/3*w^3 - 7/3*w^2 - 32/3*w - 2],
[439, 439, w^2 - 10],
[439, 439, -2/3*w^3 + 4/3*w^2 + 17/3*w - 7],
[449, 449, -2/3*w^3 + 4/3*w^2 + 11/3*w - 8],
[449, 449, -4/3*w^3 + 5/3*w^2 + 22/3*w + 2],
[457, 457, 2/3*w^3 - 7/3*w^2 - 14/3*w + 6],
[457, 457, 2/3*w^3 - 7/3*w^2 - 14/3*w + 9],
[463, 463, w^3 - 3*w^2 - 6*w + 11],
[463, 463, -1/3*w^3 + 5/3*w^2 + 10/3*w - 4],
[479, 479, -1/3*w^3 + 5/3*w^2 - 2/3*w + 1],
[479, 479, 5/3*w^3 - 10/3*w^2 - 23/3*w + 9],
[487, 487, -2/3*w^3 + 7/3*w^2 + 2/3*w - 8],
[487, 487, -2/3*w^3 + 1/3*w^2 + 14/3*w - 1],
[503, 503, w^3 - 4*w^2 - 2*w + 16],
[503, 503, -1/3*w^3 + 5/3*w^2 - 2/3*w - 11],
[521, 521, -4/3*w^3 + 11/3*w^2 + 10/3*w - 7],
[521, 521, 4/3*w^3 - 5/3*w^2 - 22/3*w],
[529, 23, -1/3*w^3 + 2/3*w^2 + 4/3*w - 6],
[569, 569, -1/3*w^3 + 8/3*w^2 + 1/3*w - 10],
[569, 569, -2/3*w^3 + 10/3*w^2 + 5/3*w - 10],
[577, 577, -2/3*w^3 + 10/3*w^2 + 5/3*w - 12],
[577, 577, -1/3*w^3 + 8/3*w^2 + 1/3*w - 8],
[593, 593, w^3 - w^2 - 5*w - 5],
[593, 593, 4/3*w^3 - 11/3*w^2 - 13/3*w + 4],
[599, 599, 2*w^2 - 3*w - 7],
[599, 599, 2/3*w^3 - 1/3*w^2 - 8/3*w - 4],
[599, 599, -1/3*w^3 + 8/3*w^2 - 5/3*w - 9],
[599, 599, -4/3*w^3 + 11/3*w^2 + 16/3*w - 7],
[607, 607, 4/3*w^3 - 8/3*w^2 - 22/3*w + 9],
[607, 607, -2/3*w^3 + 4/3*w^2 + 2/3*w - 5],
[625, 5, -5],
[631, 631, -1/3*w^3 + 2/3*w^2 + 13/3*w - 5],
[631, 631, -2/3*w^3 + 4/3*w^2 + 17/3*w - 1],
[641, 641, -2/3*w^3 + 7/3*w^2 + 14/3*w - 7],
[641, 641, 2/3*w^3 - 7/3*w^2 - 14/3*w + 8],
[647, 647, -2/3*w^3 - 5/3*w^2 + 26/3*w + 8],
[647, 647, -1/3*w^3 + 8/3*w^2 + 1/3*w - 9],
[647, 647, -2/3*w^3 + 10/3*w^2 + 5/3*w - 11],
[647, 647, -2/3*w^3 + 13/3*w^2 - 10/3*w - 13],
[673, 673, -4/3*w^3 + 5/3*w^2 + 25/3*w - 2],
[673, 673, -w^3 + 3*w^2 + w - 7],
[727, 727, -2/3*w^3 + 7/3*w^2 + 2/3*w - 11],
[727, 727, -2/3*w^3 + 1/3*w^2 + 14/3*w - 4],
[743, 743, w^2 - w - 10],
[743, 743, 1/3*w^3 - 5/3*w^2 - 1/3*w - 1],
[769, 769, -w^3 + 2*w^2 + 7*w - 4],
[769, 769, 3*w - 2],
[809, 809, -1/3*w^3 + 8/3*w^2 - 2/3*w - 8],
[809, 809, -1/3*w^3 + 8/3*w^2 - 2/3*w - 10],
[823, 823, -w^3 + 2*w^2 + 7*w - 5],
[823, 823, 3*w - 1],
[839, 839, 4/3*w^3 - 2/3*w^2 - 25/3*w - 8],
[839, 839, 2/3*w^3 - 7/3*w^2 - 14/3*w + 12],
[857, 857, -1/3*w^3 + 5/3*w^2 - 5/3*w - 7],
[857, 857, -2/3*w^3 + 1/3*w^2 + 17/3*w - 2],
[863, 863, -2/3*w^3 + 7/3*w^2 - 7/3*w - 2],
[863, 863, 5/3*w^3 - 7/3*w^2 - 35/3*w + 1],
[881, 881, -2/3*w^3 + 1/3*w^2 + 14/3*w + 8],
[881, 881, -2/3*w^3 + 13/3*w^2 - 4/3*w - 18],
[881, 881, -5/3*w^3 + 7/3*w^2 + 35/3*w + 3],
[881, 881, -2/3*w^3 + 7/3*w^2 + 2/3*w + 1],
[887, 887, -1/3*w^3 + 11/3*w^2 - 8/3*w - 7],
[887, 887, -1/3*w^3 + 11/3*w^2 - 8/3*w - 18],
[911, 911, 5/3*w^3 - 4/3*w^2 - 29/3*w - 3],
[911, 911, 2*w^3 - 6*w^2 - 5*w + 13],
[929, 929, 2*w^3 - 4*w^2 - 9*w + 2],
[929, 929, 5/3*w^3 - 10/3*w^2 - 17/3*w],
[929, 929, -4/3*w^3 + 14/3*w^2 + 13/3*w - 11],
[929, 929, 1/3*w^3 + 4/3*w^2 - 7/3*w - 9],
[937, 937, -w^3 + 4*w^2 + w - 13],
[937, 937, 2/3*w^3 + 2/3*w^2 - 17/3*w - 3],
[953, 953, -2*w^3 + w^2 + 15*w + 8],
[953, 953, -5/3*w^3 + 19/3*w^2 - 1/3*w - 11],
[961, 31, 4/3*w^3 - 8/3*w^2 - 16/3*w + 5],
[967, 967, -4/3*w^3 + 2/3*w^2 + 31/3*w + 11],
[967, 967, 7/3*w^3 - 5/3*w^2 - 46/3*w - 9],
[977, 977, 2*w^3 - 5*w^2 - 8*w + 10],
[977, 977, -w^3 + 3*w^2 + 2*w - 14],
[977, 977, -4/3*w^3 + 8/3*w^2 + 25/3*w - 11],
[977, 977, -w^3 + w^2 + 6*w - 7],
[983, 983, -5/3*w^3 + 16/3*w^2 + 8/3*w - 12],
[983, 983, -1/3*w^3 + 11/3*w^2 - 11/3*w - 15],
[983, 983, 3*w^2 - 2*w - 16],
[983, 983, -5/3*w^3 + 16/3*w^2 + 11/3*w - 14],
[991, 991, 4/3*w^3 - 8/3*w^2 - 22/3*w + 1],
[991, 991, 2/3*w^3 - 4/3*w^2 - 2/3*w - 3]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^2 - 8;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [-2, -e, 0, e, -1/2*e, -1/2*e, 1, 0, 0, -1/2*e, 3/2*e, e, -e, 5/2*e, 4, -4, 9/2*e, -3*e, e, -3/2*e, 1/2*e, 4, -4, 3/2*e, 11/2*e, -3/2*e, 1/2*e, -12, -4, 8, 16, 12, -12, 9/2*e, 9/2*e, -7*e, -6*e, -7*e, 0, -12, -4, -28, 21/2*e, -12, -15/2*e, -8, 0, -8*e, 10*e, -20, 22, 20, 6, -22, -14, 8, -24, 0, 16, 1/2*e, 13/2*e, 3/2*e, -9/2*e, 5/2*e, 6, -3/2*e, -26, -28, -6*e, 4, -4*e, 0, 24, 7*e, 13*e, -19/2*e, 13/2*e, -16, -15/2*e, 9/2*e, -16, 15/2*e, 30, -2, 15/2*e, -5*e, 9*e, -36, -12, -9/2*e, 7/2*e, -10*e, 8*e, 24, 16, 4, 20, -4, -36, -10, -10, -2, -7/2*e, -11/2*e, -3/2*e, 1/2*e, -17/2*e, 19/2*e, 15*e, 0, -e, -16*e, -4, 12, -14, -12*e, 6*e, -11/2*e, -19/2*e, 12, 10*e, 2*e, 12, 44, -12, -48, 40, -32, 8, 7/2*e, -9/2*e, 19/2*e, 3/2*e, -2*e, 2*e, -8*e, -14*e, 0, -48, 7*e, 19*e, 24, -35/2*e, 29/2*e, 48, -4*e, -6*e, 28, -60, -30, 34, -13/2*e, -1/2*e, 12, -20, 6, 6, 48, -48, 56, -52, -18, 12, -34, -48, 6*e, -10*e, -56, -18*e, -2*e];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {23, 23, -1/3*w^3 + 2/3*w^2 + 1/3*w - 2}>] := -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;