/*
  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![7, 6, -5, -2, 1];
F<w> := NumberField(g);
ZF := Integers(F);

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

primesArray := [
[4, 2, -w^2 + w + 3],
[7, 7, w],
[7, 7, -w^2 + 2*w + 1],
[7, 7, w^2 - 2],
[7, 7, w - 1],
[23, 23, -w^3 + w^2 + 3*w - 1],
[23, 23, w^3 - 2*w^2 - 2*w + 2],
[31, 31, w^2 - 5],
[31, 31, -w^2 + 2*w + 4],
[41, 41, -w^3 + 2*w^2 + 2*w - 6],
[41, 41, w^3 - w^2 - 3*w - 3],
[47, 47, w^2 - 2*w - 5],
[47, 47, w^2 - 6],
[71, 71, -w^3 + 2*w^2 + 3*w - 1],
[71, 71, -w^3 + w^2 + 4*w - 3],
[79, 79, -w - 3],
[79, 79, w - 4],
[81, 3, -3],
[89, 89, w^2 - 3*w - 2],
[89, 89, w^2 + w - 4],
[97, 97, 2*w^3 - 5*w^2 - 4*w + 9],
[97, 97, -2*w^3 + w^2 + 8*w + 2],
[113, 113, w^3 - 2*w^2 - 4*w + 2],
[113, 113, w^3 - w^2 - 5*w + 3],
[121, 11, 2*w^2 - w - 9],
[121, 11, -2*w^2 + 3*w + 8],
[127, 127, w^3 - 3*w^2 - 2*w + 5],
[127, 127, w^3 - 5*w - 1],
[137, 137, 2*w - 1],
[167, 167, 2*w^3 - 2*w^2 - 7*w - 2],
[167, 167, w^3 - 7*w - 4],
[167, 167, 2*w^3 - w^2 - 9*w - 2],
[167, 167, 2*w^3 - 4*w^2 - 5*w + 9],
[191, 191, 2*w^3 - 4*w^2 - 5*w + 5],
[191, 191, -2*w^3 + 2*w^2 + 7*w - 2],
[193, 193, -2*w^3 + 3*w^2 + 6*w - 6],
[193, 193, -2*w^3 + 2*w^2 + 7*w - 4],
[193, 193, -2*w^3 + 4*w^2 + 5*w - 3],
[193, 193, 2*w^3 - 3*w^2 - 6*w + 1],
[199, 199, -w^3 - 2*w^2 + 6*w + 10],
[199, 199, -w^3 + 5*w^2 - w - 13],
[223, 223, -w^3 + 2*w^2 + 3*w - 8],
[223, 223, w^3 - w^2 - 4*w - 4],
[233, 233, -w^2 - 1],
[233, 233, w^2 - 2*w + 2],
[239, 239, 3*w^2 - 2*w - 13],
[239, 239, 3*w^2 - 4*w - 12],
[241, 241, w^3 - w^2 - 6*w + 3],
[241, 241, w^3 - 2*w^2 - 5*w + 3],
[257, 257, -w^3 + 3*w^2 + 4*w - 9],
[257, 257, -2*w^3 + 4*w^2 + 8*w - 11],
[257, 257, -w^3 + 2*w^2 + 4],
[257, 257, w^3 - 7*w - 3],
[263, 263, -w^3 + w^2 + 3*w + 5],
[263, 263, w^3 - 5*w^2 + 12],
[263, 263, -w^3 - 2*w^2 + 7*w + 8],
[263, 263, -w^3 + 7*w + 2],
[271, 271, -w^3 + 3*w^2 + 2*w - 3],
[271, 271, w^3 - 5*w + 1],
[289, 17, 3*w^2 - 3*w - 8],
[289, 17, 3*w^2 - 3*w - 10],
[311, 311, -2*w^3 + 2*w^2 + 7*w - 1],
[311, 311, -w^3 + 5*w^2 - w - 11],
[311, 311, w^3 + 2*w^2 - 6*w - 8],
[311, 311, -2*w^3 + 4*w^2 + 5*w - 6],
[337, 337, w^3 - w^2 - 2*w - 4],
[337, 337, 2*w^3 - 9*w - 4],
[337, 337, 2*w^3 - 6*w^2 - 3*w + 11],
[337, 337, -w^3 + 2*w^2 + w - 6],
[353, 353, -w^3 + 4*w^2 - 12],
[353, 353, w^3 + w^2 - 5*w - 9],
[359, 359, 2*w^3 - 4*w^2 - 6*w + 5],
[359, 359, 2*w^3 - 2*w^2 - 8*w + 3],
[361, 19, -w^3 + 5*w^2 - 13],
[361, 19, w^3 + 2*w^2 - 7*w - 9],
[367, 367, 2*w^3 - 10*w - 5],
[367, 367, w^3 - w^2 - 6*w + 2],
[367, 367, -w^3 + 2*w^2 + 5*w - 4],
[367, 367, 2*w^3 - 6*w^2 - 4*w + 13],
[401, 401, -w^3 + 2*w^2 + 5*w - 5],
[401, 401, -w^3 + w^2 + 6*w - 1],
[409, 409, 2*w^3 - 3*w^2 - 6*w + 4],
[409, 409, 3*w^3 - 3*w^2 - 12*w + 1],
[409, 409, -3*w^3 + 6*w^2 + 9*w - 11],
[409, 409, -2*w^3 + 3*w^2 + 6*w - 3],
[431, 431, 3*w^2 - 5*w - 10],
[431, 431, w^3 + w^2 - 8*w - 3],
[457, 457, -2*w^3 + 4*w^2 + 4*w - 5],
[457, 457, -2*w^3 + 2*w^2 + 6*w - 1],
[463, 463, -2*w^3 + 7*w^2 + w - 16],
[463, 463, 3*w^3 - 6*w^2 - 8*w + 8],
[503, 503, -w^3 + 3*w^2 + 2*w - 12],
[503, 503, -w^3 + 4*w^2 + 4*w - 10],
[521, 521, w^3 + w^2 - 6*w - 2],
[521, 521, -w^3 + 4*w^2 + w - 6],
[529, 23, w^2 - w - 8],
[569, 569, w^3 - 3*w - 6],
[569, 569, -w^3 + 3*w^2 - 8],
[577, 577, -w^3 + 4*w^2 - w - 10],
[577, 577, w^3 + w^2 - 4*w - 8],
[593, 593, -w^3 + 3*w^2 - 12],
[593, 593, w^3 - 3*w - 10],
[599, 599, w^3 + 3*w^2 - 7*w - 9],
[599, 599, -w^3 + 3*w^2 + 3*w - 13],
[601, 601, w^2 + 2*w - 4],
[601, 601, w^2 - 4*w - 1],
[607, 607, 2*w^2 - 13],
[607, 607, w^3 + 2*w^2 - 7*w - 5],
[607, 607, 2*w^3 - 3*w^2 - 8*w + 8],
[607, 607, 2*w^2 - 4*w - 11],
[617, 617, w^3 + w^2 - 9*w - 1],
[617, 617, w^3 - w^2 - 4*w - 5],
[617, 617, -w^3 + 2*w^2 + 3*w - 9],
[617, 617, -w^3 + 4*w^2 + 4*w - 8],
[625, 5, -5],
[631, 631, -w^3 + 5*w^2 - 2*w - 13],
[631, 631, w^3 + 2*w^2 - 5*w - 11],
[641, 641, w^2 - 2*w + 3],
[641, 641, -w^3 + 6*w - 2],
[641, 641, -w^3 + 3*w^2 + 3*w - 3],
[641, 641, -w^2 - 2],
[647, 647, -2*w^3 + 5*w^2 + 4*w - 6],
[647, 647, 2*w^3 - 4*w^2 - 5*w - 1],
[647, 647, 3*w^3 - 7*w^2 - 7*w + 13],
[647, 647, -2*w^3 + w^2 + 8*w - 1],
[673, 673, -w^3 + 7*w - 2],
[673, 673, -w^3 + 3*w^2 + 4*w - 4],
[719, 719, -2*w^3 + 5*w^2 + 3*w - 12],
[719, 719, 4*w^2 - 5*w - 10],
[719, 719, -4*w^2 + 3*w + 11],
[719, 719, 2*w^3 - w^2 - 7*w - 6],
[727, 727, -w - 5],
[727, 727, w - 6],
[743, 743, w^2 + 2*w - 11],
[743, 743, w^3 + 4*w^2 - 11*w - 11],
[751, 751, -2*w^2 + w + 13],
[751, 751, 2*w^2 - 3*w - 12],
[769, 769, 2*w^3 - 6*w^2 - 3*w + 17],
[769, 769, 3*w^3 - 2*w^2 - 11*w + 2],
[809, 809, w^3 + 3*w^2 - 6*w - 11],
[809, 809, -3*w^3 + 5*w^2 + 8*w - 11],
[823, 823, -w^3 + w^2 + 2*w - 6],
[823, 823, 4*w^2 - 5*w - 11],
[823, 823, -4*w^2 + 3*w + 12],
[823, 823, -w^3 + 4*w^2 + 4*w - 12],
[839, 839, -2*w^3 + w^2 + 6*w + 5],
[839, 839, -3*w^3 + 8*w^2 + 7*w - 17],
[839, 839, 3*w^3 - w^2 - 14*w - 5],
[839, 839, -2*w^3 + 5*w^2 + 2*w - 10],
[857, 857, w^3 + w^2 - 8*w - 2],
[857, 857, -w^3 + 4*w^2 + 3*w - 8],
[863, 863, -w^3 + w^2 - w - 1],
[863, 863, w^3 - 2*w^2 + 2*w - 2],
[911, 911, -w^3 + 3*w^2 - 11],
[911, 911, w^3 - 3*w - 9],
[919, 919, -2*w^3 + w^2 + 8*w - 2],
[919, 919, -2*w^3 + 5*w^2 + 4*w - 5],
[929, 929, -2*w^3 + 2*w^2 + 9*w - 4],
[929, 929, 2*w^3 - 9*w - 12],
[929, 929, 2*w^3 - 6*w^2 - 3*w + 19],
[929, 929, -2*w^3 + 4*w^2 + 7*w - 5],
[937, 937, -w^3 + 2*w^2 - 6],
[937, 937, -w^3 + 3*w^2 - 10],
[937, 937, w^3 - 3*w - 8],
[937, 937, w^3 - w^2 - w - 5],
[953, 953, 2*w^3 - 6*w^2 - 3*w + 18],
[953, 953, -3*w^3 + 3*w^2 + 11*w - 5],
[961, 31, 4*w^2 - 4*w - 11],
[967, 967, -w^3 + 7*w - 3],
[967, 967, -2*w^3 + 2*w^2 + 11*w + 2],
[967, 967, -w^3 - 4*w^2 + 9*w + 9],
[967, 967, -w^3 + 3*w^2 + 4*w - 3],
[977, 977, 2*w^2 - 5*w - 8],
[977, 977, -w^3 + 2*w^2 - w + 6],
[977, 977, -w^3 + w^2 - 6],
[977, 977, 2*w^2 + w - 11],
[983, 983, -w^3 + 6*w^2 - 2*w - 16],
[983, 983, w^3 + 3*w^2 - 7*w - 13]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^3 + 3*x^2 - 4*x - 7;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [-1, e, -e^2 - e + 5, -e^2 - e + 5, e, e^2 + 4*e - 2, e^2 + 4*e - 2, 2*e + 6, 2*e + 6, 2*e - 4, 2*e - 4, -2*e^2 - 6*e + 4, -2*e^2 - 6*e + 4, -e^2 - 2*e + 2, -e^2 - 2*e + 2, -2*e^2 - 4*e + 10, -2*e^2 - 4*e + 10, 3*e^2 + 5*e - 5, -2*e^2 - 4*e, -2*e^2 - 4*e, 2*e^2 + 3*e - 14, 2*e^2 + 3*e - 14, -e^2 - 2, -e^2 - 2, 2*e - 4, 2*e - 4, 4*e^2 + 4*e - 20, 4*e^2 + 4*e - 20, 4*e^2 - 28, e^2 - e + 7, -3*e^2 - 10*e + 10, -3*e^2 - 10*e + 10, e^2 - e + 7, 6*e + 4, 6*e + 4, 3*e^2 - 2*e - 22, -2*e^2 - 5*e + 2, -2*e^2 - 5*e + 2, 3*e^2 - 2*e - 22, -e - 2, -e - 2, -2*e^2 + 22, -2*e^2 + 22, -2*e^2 + 2, -2*e^2 + 2, 2*e^2 - 18, 2*e^2 - 18, e^2 - 3*e - 3, e^2 - 3*e - 3, e^2 + 3*e - 15, -e^2 - 3*e - 9, -e^2 - 3*e - 9, e^2 + 3*e - 15, 5*e^2 + 8*e - 10, -4*e^2 - 7*e + 26, -4*e^2 - 7*e + 26, 5*e^2 + 8*e - 10, 4*e^2 + 6*e - 22, 4*e^2 + 6*e - 22, 30, -4*e^2 - 8*e + 30, -4*e^2 - 8*e - 8, e^2 + 7*e + 7, e^2 + 7*e + 7, -4*e^2 - 8*e - 8, -3*e^2 - 9*e - 3, -e^2 + 8*e + 18, -e^2 + 8*e + 18, -3*e^2 - 9*e - 3, 6*e^2 + 17*e - 16, 6*e^2 + 17*e - 16, 3*e^2 + 13*e - 11, 3*e^2 + 13*e - 11, 6*e^2 + 16*e - 20, 6*e^2 + 16*e - 20, -2*e^2 + 16, 4*e^2 + 4*e - 10, 4*e^2 + 4*e - 10, -2*e^2 + 16, -3*e^2 - 11*e - 3, -3*e^2 - 11*e - 3, -4*e^2 - 4*e + 28, -6*e^2 - 16*e + 22, -6*e^2 - 16*e + 22, -4*e^2 - 4*e + 28, -2*e^2 + 6*e + 12, -2*e^2 + 6*e + 12, 2*e^2 + 8*e - 4, 2*e^2 + 8*e - 4, -6*e^2 - 8*e + 22, -6*e^2 - 8*e + 22, -e + 22, -e + 22, -2*e^2 - 6*e + 18, -2*e^2 - 6*e + 18, -4*e^2 + 46, 4*e^2 + 16*e - 24, 4*e^2 + 16*e - 24, 2*e^2 - 5*e - 20, 2*e^2 - 5*e - 20, -2*e^2 + 11*e + 24, -2*e^2 + 11*e + 24, 4*e^2 + 16*e - 24, 4*e^2 + 16*e - 24, -2*e^2 - 8*e - 6, -2*e^2 - 8*e - 6, -6*e^2 - 14*e + 14, 2*e^2 + 4*e - 12, 2*e^2 + 4*e - 12, -6*e^2 - 14*e + 14, -8*e^2 - 18*e + 34, -4*e^2 - 10*e + 4, -4*e^2 - 10*e + 4, -8*e^2 - 18*e + 34, -e^2 + 50, -3*e^2 + 10*e + 34, -3*e^2 + 10*e + 34, 7*e^2 + 15*e - 21, e^2 + e + 5, e^2 + e + 5, 7*e^2 + 15*e - 21, 3*e + 4, 8*e - 16, 8*e - 16, 3*e + 4, -e^2 - 13*e - 13, -e^2 - 13*e - 13, -10*e - 10, 2*e^2 + 14*e, 2*e^2 + 14*e, -10*e - 10, 8*e - 16, 8*e - 16, -17*e - 10, -17*e - 10, 6*e^2 + 10*e - 22, 6*e^2 + 10*e - 22, -e^2 - 14*e - 14, -e^2 - 14*e - 14, -6*e^2 - 6*e + 32, -6*e^2 - 6*e + 32, 3*e^2 - 3*e - 19, 3*e^2 + 15*e - 23, 3*e^2 + 15*e - 23, 3*e^2 - 3*e - 19, e^2 - 3*e + 15, 4*e^2 + 8*e - 20, 4*e^2 + 8*e - 20, e^2 - 3*e + 15, 4*e^2 + 10*e - 8, 4*e^2 + 10*e - 8, -4*e^2 - 10*e + 20, -4*e^2 - 10*e + 20, 2*e^2 + 4*e + 22, 2*e^2 + 4*e + 22, -e^2 + 4*e + 22, -e^2 + 4*e + 22, 11*e^2 + 18*e - 38, 9*e^2 + 14*e - 38, 9*e^2 + 14*e - 38, 11*e^2 + 18*e - 38, -10*e^2 - 14*e + 40, -2*e^2 - 16*e + 24, -2*e^2 - 16*e + 24, -10*e^2 - 14*e + 40, -2*e^2 + 12*e + 26, -2*e^2 + 12*e + 26, 6*e^2 + 11*e + 20, 12*e^2 + 23*e - 34, -12*e^2 - 24*e + 28, -12*e^2 - 24*e + 28, 12*e^2 + 23*e - 34, 5*e^2 + 10*e - 2, -14*e^2 - 23*e + 38, -14*e^2 - 23*e + 38, 5*e^2 + 10*e - 2, 7*e^2 + 8*e - 38, 7*e^2 + 8*e - 38];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {4, 2, -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;