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

NN := ideal<ZF | {1, 1, 1}>;

primesArray := [
[2, 2, w^3 - w^2 - 6*w - 2],
[3, 3, -w^3 + 2*w^2 + 4*w - 2],
[4, 2, w^3 - w^2 - 5*w - 2],
[17, 17, -w^3 + w^2 + 6*w - 1],
[17, 17, -w + 2],
[29, 29, -w^3 + 2*w^2 + 4*w - 4],
[29, 29, w^3 - 2*w^2 - 4*w],
[31, 31, -w^2 + 2],
[31, 31, -w^3 + 7*w + 5],
[41, 41, -2*w^3 + 2*w^2 + 13*w - 2],
[41, 41, 3*w^3 - 4*w^2 - 17*w + 5],
[67, 67, 3*w^3 - 4*w^2 - 17*w + 1],
[67, 67, -w^2 + 4*w + 2],
[83, 83, 2*w^2 - 5*w - 2],
[83, 83, -w^3 + 8*w + 6],
[83, 83, w^3 - 2*w^2 - 2*w - 2],
[83, 83, w^3 - 2*w^2 - 6*w],
[97, 97, -3*w^3 + 2*w^2 + 17*w + 7],
[97, 97, w^3 - 4*w^2 + 3*w + 1],
[97, 97, -5*w^3 + 8*w^2 + 24*w - 8],
[97, 97, 2*w^3 - 2*w^2 - 13*w - 4],
[101, 101, -w^3 + 6*w + 6],
[101, 101, -w^3 + 6*w + 2],
[103, 103, -4*w^3 + 3*w^2 + 26*w + 12],
[103, 103, -w^3 + 2*w^2 + 9*w + 3],
[107, 107, 2*w^3 - 2*w^2 - 11*w],
[107, 107, -w^3 + 2*w^2 + 3*w - 3],
[107, 107, w^3 - w^2 - 4*w - 3],
[107, 107, 2*w^3 - 3*w^2 - 10*w],
[121, 11, -2*w^3 + 3*w^2 + 8*w + 2],
[149, 149, -3*w^3 + 5*w^2 + 14*w - 3],
[149, 149, -2*w^3 + 4*w^2 + 7*w - 4],
[157, 157, 2*w^3 - 2*w^2 - 10*w - 1],
[157, 157, 2*w^3 - 3*w^2 - 8*w + 2],
[157, 157, 2*w^3 - 2*w^2 - 10*w - 3],
[157, 157, -3*w^3 + 4*w^2 + 15*w + 1],
[163, 163, -2*w^2 + 4*w + 5],
[163, 163, 2*w^3 - 4*w^2 - 10*w + 5],
[169, 13, 2*w^3 - 3*w^2 - 12*w + 2],
[169, 13, -w^3 + 9*w + 3],
[173, 173, 3*w^3 - 2*w^2 - 17*w - 9],
[173, 173, -2*w^3 + 14*w + 7],
[197, 197, 2*w^3 - 2*w^2 - 12*w + 1],
[197, 197, 3*w^3 - 3*w^2 - 18*w - 1],
[199, 199, 3*w^3 - 4*w^2 - 18*w + 8],
[199, 199, 4*w^3 - 5*w^2 - 24*w + 6],
[223, 223, w^3 - 2*w^2 - 3*w - 3],
[223, 223, 2*w^3 - 3*w^2 - 10*w + 6],
[227, 227, -w^3 + 2*w^2 + 6*w - 8],
[227, 227, 3*w + 4],
[227, 227, 3*w^3 - 3*w^2 - 18*w - 7],
[227, 227, -2*w^3 + 3*w^2 + 12*w - 6],
[229, 229, -w - 4],
[229, 229, w^2 - 2*w + 2],
[229, 229, -w^3 + w^2 + 6*w + 5],
[229, 229, -w^3 + 2*w^2 + 5*w - 7],
[233, 233, -2*w^3 + 3*w^2 + 8*w - 4],
[233, 233, -3*w^3 + 4*w^2 + 15*w + 3],
[281, 281, -2*w^3 + 3*w^2 + 10*w + 2],
[281, 281, -w^3 + 2*w^2 + 3*w - 5],
[289, 17, -2*w^3 + 2*w^2 + 14*w + 5],
[293, 293, w^2 - 6],
[293, 293, w^3 - 7*w - 1],
[331, 331, -6*w^3 + 10*w^2 + 27*w - 6],
[331, 331, 2*w^3 - 2*w^2 - 12*w - 7],
[347, 347, 3*w^3 - 2*w^2 - 18*w - 6],
[347, 347, -4*w^3 + 2*w^2 + 23*w + 10],
[347, 347, 2*w^3 - 6*w^2 - w + 4],
[347, 347, -4*w^3 + 3*w^2 + 24*w + 8],
[359, 359, -3*w^3 + 4*w^2 + 17*w + 1],
[359, 359, w^2 - 4*w - 4],
[359, 359, w^3 - 8*w],
[359, 359, -w^3 + 2*w^2 + 6*w - 6],
[367, 367, 2*w^2 - 3*w - 4],
[367, 367, -w^3 + 3*w^2 + 4*w - 7],
[379, 379, 3*w^3 - 3*w^2 - 16*w - 3],
[379, 379, 2*w^3 - 2*w^2 - 9*w - 2],
[461, 461, -2*w^3 + 5*w^2 + 2*w + 2],
[461, 461, 4*w^3 - 4*w^2 - 26*w - 11],
[463, 463, -2*w^3 + 13*w + 8],
[463, 463, -w^3 - w^2 + 6*w + 7],
[487, 487, 2*w^3 - 17*w - 8],
[487, 487, -w^3 + 10*w + 4],
[491, 491, -5*w^3 + 3*w^2 + 30*w + 13],
[491, 491, -5*w^3 + 4*w^2 + 28*w + 14],
[491, 491, 3*w^3 - 8*w^2 - 4*w + 4],
[491, 491, 7*w^3 - 12*w^2 - 33*w + 15],
[499, 499, -2*w^3 + 2*w^2 + 12*w - 5],
[499, 499, 2*w^2 - 7*w],
[529, 23, -w^3 + 3*w^2 + 4*w - 5],
[529, 23, 2*w^2 - 3*w - 6],
[557, 557, -2*w^3 + 3*w^2 + 12*w + 4],
[557, 557, w^3 - 9*w - 9],
[569, 569, 4*w^3 - 5*w^2 - 22*w + 4],
[569, 569, -3*w^3 + 2*w^2 + 20*w + 4],
[577, 577, 3*w^3 - 3*w^2 - 20*w + 3],
[577, 577, -4*w^3 + 8*w^2 + 16*w - 9],
[577, 577, 7*w^3 - 11*w^2 - 34*w + 9],
[577, 577, -4*w^3 + 6*w^2 + 22*w - 11],
[593, 593, -6*w^3 + 10*w^2 + 29*w - 14],
[593, 593, -w^3 + 4*w^2 - w - 7],
[619, 619, -3*w^3 + 5*w^2 + 12*w - 1],
[619, 619, -4*w^3 + 8*w^2 + 18*w - 15],
[625, 5, -5],
[631, 631, -3*w^3 + 8*w^2 + 6*w - 4],
[631, 631, -2*w^3 + 8*w^2 - 3*w - 8],
[643, 643, 3*w^3 - 4*w^2 - 14*w + 2],
[643, 643, -3*w^3 + 4*w^2 + 14*w + 2],
[661, 661, w^3 - 8*w - 10],
[661, 661, -3*w^3 + 2*w^2 + 21*w + 3],
[661, 661, w^3 - 2*w^2 - 6*w - 4],
[661, 661, 2*w^3 - 3*w^2 - 14*w + 4],
[677, 677, -9*w^3 + 12*w^2 + 50*w - 10],
[677, 677, 3*w^3 - 4*w^2 - 17*w - 7],
[691, 691, 3*w^3 - 2*w^2 - 21*w - 9],
[691, 691, 2*w^3 - 3*w^2 - 14*w - 2],
[701, 701, -9*w^3 + 13*w^2 + 48*w - 15],
[701, 701, 2*w^3 - 6*w^2 + w - 2],
[709, 709, 2*w^2 - 2*w - 5],
[709, 709, 2*w^2 - 2*w - 7],
[709, 709, 4*w^3 - 5*w^2 - 20*w - 4],
[709, 709, -3*w^3 + 6*w^2 + 14*w - 10],
[727, 727, 7*w^3 - 10*w^2 - 36*w + 8],
[727, 727, 3*w^2 - 6*w - 8],
[743, 743, 9*w^3 - 12*w^2 - 51*w + 13],
[743, 743, w^3 - 2*w^2 - w - 5],
[743, 743, -4*w^3 + 4*w^2 + 27*w - 8],
[743, 743, -3*w^3 + 4*w^2 + 21*w + 9],
[751, 751, -w^3 + 10*w + 2],
[751, 751, 3*w^3 - 4*w^2 - 18*w + 2],
[761, 761, 4*w^3 - 4*w^2 - 23*w - 2],
[761, 761, -3*w^3 + 2*w^2 + 19*w + 5],
[809, 809, -3*w^3 + 5*w^2 + 10*w + 5],
[809, 809, -w^3 - w^2 + 10*w + 3],
[821, 821, -w^3 + 3*w^2 + 2*w - 9],
[821, 821, 2*w^2 - 4*w - 9],
[823, 823, w^3 - 6*w^2 + 9*w + 1],
[823, 823, -3*w^3 + 2*w^2 + 15*w + 5],
[827, 827, 3*w^3 - 3*w^2 - 18*w + 1],
[827, 827, 2*w^3 - 16*w - 11],
[827, 827, 3*w^3 - 24*w - 14],
[827, 827, 3*w - 4],
[829, 829, w^2 - 6*w - 2],
[829, 829, 2*w^3 - 2*w^2 - 10*w - 9],
[829, 829, 5*w^3 - 6*w^2 - 29*w - 1],
[829, 829, w^3 + 2*w^2 - 14*w - 6],
[841, 29, -2*w^3 + 2*w^2 + 14*w + 3],
[857, 857, -3*w^3 + 6*w^2 + 11*w - 7],
[857, 857, -4*w^3 + 7*w^2 + 18*w - 4],
[859, 859, -w^3 + w^2 + 6*w - 5],
[859, 859, w - 6],
[883, 883, 5*w^3 - 5*w^2 - 26*w - 11],
[883, 883, -3*w^3 + 3*w^2 + 14*w + 5],
[887, 887, 5*w^3 - 2*w^2 - 34*w - 16],
[887, 887, -w^3 + 4*w^2 - 10],
[887, 887, 4*w^3 - 2*w^2 - 26*w - 11],
[887, 887, 2*w^3 - 8*w^2 + 4*w + 7],
[907, 907, 6*w^3 - 8*w^2 - 34*w + 7],
[907, 907, -w^3 + w^2 + 8*w - 7],
[941, 941, 2*w^3 - 2*w^2 - 8*w - 5],
[941, 941, -3*w^3 + 23*w + 11],
[953, 953, 2*w^3 - 10*w - 3],
[953, 953, -4*w^3 + 2*w^2 + 24*w + 15],
[961, 31, -2*w^3 + 2*w^2 + 14*w - 7],
[991, 991, -3*w^3 + 4*w^2 + 13*w + 3],
[991, 991, 4*w^3 - 5*w^2 - 20*w + 2]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^4 - 11*x^2 + 26;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, -e^2 + 5, -e^2 + 6, -2*e, -2*e, 2*e, 2*e, e^2 - 1, e^2 - 1, 2*e^3 - 12*e, 2*e^3 - 12*e, e^2 - 13, e^2 - 13, -2*e^3 + 14*e, -2*e^3 + 14*e, -2*e^3 + 14*e, -2*e^3 + 14*e, e^2 - 15, e^2 - 15, 3*e^2 - 17, 3*e^2 - 17, -2*e^3 + 12*e, -2*e^3 + 12*e, -4*e^2 + 28, -4*e^2 + 28, 2*e^3 - 14*e, -4*e^3 + 24*e, 2*e^3 - 14*e, -4*e^3 + 24*e, 4*e^2 - 10, -6*e, -6*e, -e^2 + 15, 3*e^2 - 21, -e^2 + 15, 3*e^2 - 21, -4*e^2 + 24, -4*e^2 + 24, 4*e^2 - 26, 4*e^2 - 26, 2*e^3 - 16*e, 2*e^3 - 16*e, -2*e^3 + 12*e, -2*e^3 + 12*e, -4*e^2 + 12, -4*e^2 + 12, -e^2 + 17, -e^2 + 17, 2*e^3 - 6*e, -4*e, -4*e, 2*e^3 - 6*e, -3*e^2 + 25, -5*e^2 + 27, -3*e^2 + 25, -5*e^2 + 27, 2*e^3 - 12*e, 2*e^3 - 12*e, 2*e^3 - 12*e, 2*e^3 - 12*e, 34, -4*e^3 + 30*e, -4*e^3 + 30*e, -5*e^2 + 17, -5*e^2 + 17, -2*e^3 + 6*e, 4*e, 4*e, -2*e^3 + 6*e, -8*e, -8*e, 2*e^3 - 10*e, 2*e^3 - 10*e, 5*e^2 - 37, 5*e^2 - 37, 7*e^2 - 43, 7*e^2 - 43, -14*e, -14*e, -5*e^2 + 21, -5*e^2 + 21, -11*e^2 + 67, -11*e^2 + 67, 2*e^3 - 14*e, 2*e^3 - 14*e, 2*e^3 - 14*e, 2*e^3 - 14*e, 4*e^2, 4*e^2, -9*e^2 + 43, -9*e^2 + 43, -2*e^3 + 20*e, -2*e^3 + 20*e, -10*e, -10*e, 9*e^2 - 55, -7*e^2 + 25, -7*e^2 + 25, 9*e^2 - 55, 2*e^3 - 4*e, 2*e^3 - 4*e, -13*e^2 + 73, -13*e^2 + 73, -3*e^2 + 37, -5*e^2 + 29, -5*e^2 + 29, -3*e^2 - 9, -3*e^2 - 9, 11*e^2 - 69, -9*e^2 + 47, 11*e^2 - 69, -9*e^2 + 47, -4*e^3 + 14*e, -4*e^3 + 14*e, 3*e^2 + 1, 3*e^2 + 1, -2*e^3 + 4*e, -2*e^3 + 4*e, 7*e^2 - 49, 7*e^2 - 49, 17*e^2 - 91, 17*e^2 - 91, 5*e^2 + 3, 5*e^2 + 3, 6*e^3 - 46*e, -2*e^3 + 26*e, 6*e^3 - 46*e, -2*e^3 + 26*e, 19*e^2 - 99, 19*e^2 - 99, 4*e^3 - 14*e, 4*e^3 - 14*e, 2*e^3 - 12*e, 2*e^3 - 12*e, 8*e^3 - 46*e, 8*e^3 - 46*e, e^2 + 7, e^2 + 7, -2*e^3 + 6*e, -4*e^3 + 40*e, -4*e^3 + 40*e, -2*e^3 + 6*e, 17*e^2 - 99, -17*e^2 + 95, 17*e^2 - 99, -17*e^2 + 95, -8*e^2 + 82, -4*e^3 + 42*e, -4*e^3 + 42*e, 15*e^2 - 83, 15*e^2 - 83, -4*e^2 + 24, -4*e^2 + 24, 2*e^3 - 26*e, 2*e^3 - 26*e, 2*e^3 - 26*e, 2*e^3 - 26*e, -20*e^2 + 96, -20*e^2 + 96, -6*e^3 + 40*e, -6*e^3 + 40*e, -8*e^3 + 46*e, -8*e^3 + 46*e, e^2 + 49, 4*e^2 + 12, 4*e^2 + 12];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();

// 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;