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

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

primesArray := [
[5, 5, w + 1],
[5, 5, -w + 2],
[7, 7, -w^2 + 2*w + 1],
[7, 7, -w^2 + 2],
[9, 3, w^3 - w^2 - 4*w],
[16, 2, 2],
[17, 17, -w^3 + 2*w^2 + 3*w - 2],
[17, 17, -w^3 + w^2 + 4*w - 2],
[25, 5, w^2 - w - 3],
[37, 37, 2*w - 1],
[43, 43, -w^3 + 2*w^2 + 2*w - 2],
[43, 43, w^3 - w^2 - 3*w + 1],
[59, 59, 2*w - 5],
[59, 59, -2*w - 3],
[79, 79, -w^3 + 2*w^2 + 5*w - 4],
[79, 79, -w^3 + w^2 + 6*w - 2],
[83, 83, -w^3 + 7*w],
[83, 83, -w^3 + 2*w^2 + 4*w - 2],
[83, 83, -w^3 + w^2 + 5*w - 3],
[83, 83, w^2 - 2*w - 6],
[89, 89, w^2 - 2*w - 4],
[89, 89, w^2 - 5],
[101, 101, -2*w^3 + 5*w^2 + 5*w - 11],
[101, 101, w^2 + w - 4],
[101, 101, w^2 - 3*w - 2],
[101, 101, 2*w^3 - w^2 - 9*w - 3],
[109, 109, w^3 - w^2 - 6*w - 3],
[109, 109, -w^3 + 2*w^2 + 5*w - 9],
[121, 11, w^3 - 8*w],
[121, 11, w^3 - 3*w^2 - 5*w + 7],
[131, 131, w^3 - 5*w - 6],
[131, 131, -w^3 + 3*w^2 + 2*w - 10],
[163, 163, 2*w^2 - 3*w - 6],
[163, 163, 2*w^2 - w - 7],
[167, 167, w^3 - w^2 - 3*w - 4],
[167, 167, -w^3 + 2*w^2 + 2*w - 7],
[193, 193, 2*w^3 - 3*w^2 - 7*w + 2],
[193, 193, 2*w^3 - 3*w^2 - 7*w + 6],
[227, 227, 2*w^3 - 3*w^2 - 8*w + 1],
[227, 227, -2*w^3 + 3*w^2 + 8*w - 8],
[251, 251, 2*w^2 - w - 9],
[251, 251, 2*w^2 - 3*w - 8],
[257, 257, w^3 + w^2 - 8*w - 7],
[257, 257, -4*w^3 + 5*w^2 + 17*w - 1],
[269, 269, 2*w^3 - 2*w^2 - 10*w + 3],
[269, 269, 2*w^3 - 3*w^2 - 9*w + 2],
[269, 269, 2*w^3 - 3*w^2 - 9*w + 8],
[269, 269, -2*w^3 + 4*w^2 + 8*w - 7],
[277, 277, w^3 - w^2 - 4*w - 4],
[277, 277, -w^3 + 2*w^2 + 3*w - 8],
[289, 17, 2*w^2 - 2*w - 3],
[293, 293, -w^2 - 2*w + 5],
[293, 293, -w^3 + 3*w^2 + 3*w - 4],
[293, 293, w^3 - 6*w + 1],
[293, 293, -2*w^3 + 4*w^2 + 7*w - 6],
[311, 311, -w^3 + 3*w^2 + w - 7],
[311, 311, w^3 - 4*w - 4],
[331, 331, -2*w^3 + 4*w^2 + 9*w - 10],
[331, 331, 2*w^3 - 2*w^2 - 11*w + 1],
[337, 337, 2*w^2 - 3*w - 4],
[337, 337, -w^3 + 2*w^2 + 6*w - 3],
[337, 337, w^3 - w^2 - 7*w + 4],
[337, 337, 2*w^2 - w - 5],
[353, 353, -3*w^3 + 4*w^2 + 13*w - 2],
[353, 353, 3*w^3 - 5*w^2 - 12*w + 12],
[373, 373, -2*w^3 + 6*w^2 + 3*w - 11],
[373, 373, -w^3 - w^2 + 6*w + 7],
[373, 373, -w^3 + 4*w^2 + w - 11],
[373, 373, 2*w^3 - 9*w - 4],
[379, 379, w^3 - w^2 - 7*w + 1],
[379, 379, 3*w^3 - 7*w^2 - 7*w + 11],
[379, 379, 3*w^3 - 2*w^2 - 12*w],
[379, 379, -w^3 + 2*w^2 + 6*w - 6],
[383, 383, -3*w^3 + 4*w^2 + 12*w - 8],
[383, 383, 3*w^3 - 5*w^2 - 11*w + 5],
[421, 421, 2*w^3 - 2*w^2 - 9*w - 4],
[421, 421, -2*w^3 + 4*w^2 + 7*w - 13],
[457, 457, -w^3 + 2*w^2 + 6*w - 5],
[457, 457, w^3 - w^2 - 7*w + 2],
[461, 461, -w^3 + 3*w^2 + 4*w - 4],
[461, 461, -w^3 + 7*w - 2],
[463, 463, -w^3 + 4*w^2 + 3*w - 10],
[463, 463, w^3 + w^2 - 8*w - 4],
[467, 467, -w^3 + 6*w - 2],
[467, 467, -w^3 + 3*w^2 + 3*w - 3],
[479, 479, -w^3 + w^2 + w - 4],
[479, 479, w^3 - 2*w^2 - 3],
[487, 487, -3*w^3 + 4*w^2 + 12*w - 3],
[487, 487, -3*w^3 + 5*w^2 + 11*w - 10],
[499, 499, 3*w^3 - 4*w^2 - 11*w + 8],
[499, 499, 3*w^3 - 5*w^2 - 10*w + 4],
[503, 503, -w^3 + 2*w^2 + w - 5],
[503, 503, w^3 - w^2 - 2*w - 3],
[521, 521, 2*w^3 - 4*w^2 - 6*w + 3],
[521, 521, -w^3 + 10*w - 8],
[521, 521, w^3 - 3*w^2 - 7*w + 1],
[521, 521, -2*w^3 + 2*w^2 + 8*w - 5],
[541, 541, 2*w - 7],
[541, 541, 2*w + 5],
[547, 547, w^3 - 4*w^2 + 7],
[547, 547, w^3 + w^2 - 5*w - 4],
[563, 563, -2*w^3 + 4*w^2 + 7*w - 5],
[563, 563, -2*w^3 + 2*w^2 + 9*w - 4],
[587, 587, -w^3 - w^2 + 4*w + 6],
[587, 587, w^3 - 4*w^2 + w + 8],
[593, 593, w^2 + w - 7],
[593, 593, 2*w^3 - 5*w^2 - 4*w + 5],
[593, 593, -2*w^3 + w^2 + 8*w - 2],
[593, 593, w^2 - 3*w - 5],
[613, 613, 2*w^3 - 3*w^2 - 11*w + 2],
[613, 613, w^3 + w^2 - 6*w - 3],
[613, 613, 3*w^3 - 4*w^2 - 12*w + 4],
[613, 613, 2*w^3 - 3*w^2 - 11*w + 10],
[631, 631, -w^3 - w^2 + 7*w + 6],
[631, 631, -w^3 + 4*w^2 + 2*w - 11],
[647, 647, w^2 - 4*w - 3],
[647, 647, w^2 + 2*w - 6],
[677, 677, w^2 + w - 8],
[677, 677, -w^3 + 7*w - 3],
[677, 677, -w^3 + 3*w^2 + 4*w - 3],
[677, 677, w^2 - 3*w - 6],
[709, 709, w^3 - 7*w - 8],
[709, 709, w^3 - 3*w^2 - 4*w + 14],
[739, 739, w^3 - w^2 - 4*w - 5],
[739, 739, -3*w^3 + 3*w^2 + 16*w - 3],
[739, 739, -3*w^3 + 6*w^2 + 13*w - 13],
[739, 739, -w^3 + 2*w^2 + 3*w - 9],
[751, 751, -w - 5],
[751, 751, w^3 + w^2 - 7*w - 4],
[751, 751, -w^3 + 4*w^2 + 2*w - 9],
[751, 751, w - 6],
[757, 757, -w^3 + 4*w^2 + 2*w - 10],
[757, 757, w^3 + w^2 - 7*w - 5],
[761, 761, -3*w^3 + 2*w^2 + 17*w - 1],
[761, 761, -3*w^3 + 6*w^2 + 9*w - 17],
[761, 761, 2*w^3 - 5*w^2 - 9*w + 9],
[761, 761, 2*w^3 - 2*w^2 - 7*w - 6],
[773, 773, -3*w^3 + 8*w^2 + 6*w - 12],
[773, 773, w^3 - 4*w - 6],
[773, 773, -w^3 + 3*w^2 + w - 9],
[773, 773, -3*w^3 + w^2 + 13*w + 1],
[797, 797, 3*w^3 - 2*w^2 - 13*w + 1],
[797, 797, 3*w^3 - 7*w^2 - 8*w + 11],
[823, 823, 2*w^3 - 3*w^2 - 6*w - 4],
[823, 823, -2*w^3 + 3*w^2 + 7*w + 3],
[823, 823, w^3 - 2*w^2 - 7*w - 3],
[823, 823, -3*w^3 + 5*w^2 + 9*w - 2],
[857, 857, 2*w^3 - w^2 - 10*w + 1],
[857, 857, -2*w^3 + 5*w^2 + 6*w - 8],
[883, 883, 3*w^3 - 3*w^2 - 12*w + 1],
[883, 883, 3*w^3 - 6*w^2 - 9*w + 11],
[887, 887, -2*w^3 + 5*w^2 + 7*w - 9],
[887, 887, -4*w^3 + 6*w^2 + 17*w - 7],
[887, 887, -2*w^3 + 5*w^2 + 9*w - 16],
[887, 887, 2*w^3 - w^2 - 11*w + 1],
[907, 907, -3*w^3 + 3*w^2 + 16*w - 2],
[907, 907, -3*w^3 + 6*w^2 + 13*w - 14],
[919, 919, 2*w^3 - 2*w^2 - 7*w + 9],
[919, 919, -2*w^3 + 2*w^2 + 12*w - 5],
[929, 929, -2*w^3 + 4*w^2 + 8*w - 5],
[929, 929, 3*w^3 - 4*w^2 - 11*w + 12],
[929, 929, 3*w^3 - 5*w^2 - 10*w],
[929, 929, -2*w^3 + 2*w^2 + 10*w - 5],
[967, 967, 3*w^3 - 5*w^2 - 9*w + 11],
[967, 967, -3*w^3 + 7*w^2 + 6*w - 10],
[971, 971, 2*w^3 - 2*w^2 - 7*w - 5],
[971, 971, -2*w^3 + w^2 + 8*w - 3],
[971, 971, 2*w^3 - 5*w^2 - 4*w + 4],
[971, 971, -2*w^3 + 4*w^2 + 5*w - 12],
[983, 983, 4*w - 7],
[983, 983, 2*w^2 - 4*w - 7],
[983, 983, 2*w^2 - 9],
[983, 983, 4*w + 3],
[991, 991, 2*w^3 - 6*w^2 - 5*w + 14],
[991, 991, -2*w^3 + 11*w + 5]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^4 - 16*x^2 + 4;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, 1/2*e^3 - 8*e, 1/2*e^2 - 3, -1/2*e^2 + 5, -1, 1, -1/2*e^3 + 9*e, 1/2*e^3 - 9*e, 4, 6, 1/2*e^2 - 3, -1/2*e^2 + 5, 1/2*e^3 - 11*e, -3/2*e^3 + 25*e, e^2 - 6, -e^2 + 10, -3/2*e^3 + 23*e, -1/2*e^3 + 9*e, 1/2*e^3 - 9*e, -1/2*e^3 + 5*e, -1/2*e^3 + 7*e, -1/2*e^3 + 7*e, 3/2*e^3 - 20*e, -1/2*e^3 + 9*e, 1/2*e^3 - 9*e, 2*e^3 - 29*e, e^2 - 14, -e^2 + 2, e^2, -e^2 + 16, e^3 - 12*e, 2*e^3 - 30*e, -e^2 + 2, e^2 - 14, -e^3 + 10*e, -3*e^3 + 46*e, -e^2 + 18, e^2 + 2, -2*e^3 + 32*e, -4*e, -7/2*e^3 + 55*e, -1/2*e^3 + e, -3*e^3 + 53*e, 5/2*e^3 - 46*e, -e^3 + 15*e, -1/2*e^3 + 7*e, -1/2*e^3 + 7*e, -1/2*e^3 + 6*e, -2*e^2 + 32, 2*e^2, -10, -3/2*e^3 + 22*e, 7/2*e^3 - 53*e, 3/2*e^3 - 17*e, -e^3 + 13*e, -3*e^3 + 46*e, -e^3 + 10*e, 1/2*e^2 - 17, -1/2*e^2 - 9, -2, 2*e^2 + 4, -2*e^2 + 36, -2, 9/2*e^3 - 65*e, 7/2*e^3 - 47*e, -26, e^2 + 2, -e^2 + 18, -26, 3/2*e^2 - 11, 1/2*e^2 - 19, -1/2*e^2 - 11, -3/2*e^2 + 13, -2*e^3 + 26*e, -3*e^3 + 44*e, 14, 14, -2*e^2 + 6, 2*e^2 - 26, -3/2*e^3 + 26*e, e^3 - 19*e, -2*e^2 + 24, 2*e^2 - 8, 5/2*e^3 - 41*e, -1/2*e^3 + 13*e, -3/2*e^3 + 17*e, -7/2*e^3 + 53*e, -9/2*e^2 + 41, 9/2*e^2 - 31, 1/2*e^2 - 35, -1/2*e^2 - 27, -1/2*e^3 + 3*e, -5/2*e^3 + 39*e, -9/2*e^3 + 69*e, -e^3 + 11*e, -5/2*e^3 + 38*e, -3/2*e^3 + 15*e, -2*e^2 + 16, 2*e^2 - 16, -8, -8, -11/2*e^3 + 87*e, -1/2*e^3 - 3*e, 9/2*e^3 - 61*e, 11/2*e^3 - 79*e, 7/2*e^3 - 61*e, -1/2*e^3 + 16*e, 4*e^3 - 65*e, -5/2*e^3 + 47*e, -6, -3*e^2 + 30, 3*e^2 - 18, -6, 2*e^2 - 4, -2*e^2 + 28, -6*e^3 + 96*e, -12*e, -9/2*e^3 + 82*e, 9/2*e^3 - 61*e, 11/2*e^3 - 79*e, 5*e^3 - 89*e, -3*e^2 + 50, 3*e^2 + 2, -2*e^2 + 52, -5*e^2 + 34, 5*e^2 - 46, 2*e^2 + 20, 1/2*e^2 - 5, -1/2*e^2 - 7, 1/2*e^2 - 15, -1/2*e^2 + 3, e^2 + 16, -e^2 + 32, -3/2*e^3 + 35*e, e^3 - 9*e, 7/2*e^3 - 54*e, 11/2*e^3 - 91*e, -1/2*e^3 + 13*e, e^3 - 23*e, -7/2*e^3 + 58*e, 5/2*e^3 - 41*e, -3/2*e^3 + 13*e, -11/2*e^3 + 85*e, 2*e^2 - 12, -e^2 - 6, e^2 - 22, -2*e^2 + 20, -3/2*e^3 + 11*e, -13/2*e^3 + 101*e, 3/2*e^2 - 3, -3/2*e^2 + 21, -1/2*e^3 + 17*e, 11/2*e^3 - 93*e, -5/2*e^3 + 51*e, 9/2*e^3 - 73*e, -7/2*e^2 + 5, 7/2*e^2 - 51, 3/2*e^2 - 35, -3/2*e^2 - 11, 15/2*e^3 - 118*e, -3/2*e^3 + 13*e, -11/2*e^3 + 85*e, e^3 - e, 4*e^2 - 16, -4*e^2 + 48, -5/2*e^3 + 27*e, 12*e, 6*e^3 - 96*e, -13/2*e^3 + 99*e, 5*e^3 - 90*e, -5*e^3 + 92*e, 6*e^3 - 106*e, -5*e^3 + 90*e, 1/2*e^2 - 49, -1/2*e^2 - 41];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {9, 3, w^3 - w^2 - 4*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;