/*
  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![3, 4, -5, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {16, 16, w^2 + w - 4}>;

primesArray := [
[2, 2, w - 1],
[3, 3, w],
[8, 2, -w^3 + 5*w + 1],
[19, 19, w^3 - w^2 - 3*w + 2],
[19, 19, -w^3 + 4*w - 2],
[23, 23, w^3 + w^2 - 4*w - 5],
[27, 3, -w^3 + w^2 + 5*w - 4],
[31, 31, w^3 + w^2 - 4*w - 1],
[47, 47, -w^2 - w + 5],
[53, 53, 2*w^3 - 2*w^2 - 9*w + 10],
[59, 59, 2*w - 1],
[59, 59, w^3 - 2*w - 2],
[61, 61, -w^2 - 2*w + 4],
[67, 67, 2*w^2 - 7],
[73, 73, 5*w^3 + w^2 - 23*w - 8],
[79, 79, -2*w^3 - w^2 + 9*w + 7],
[79, 79, -w^3 + w^2 + 4*w - 1],
[79, 79, w^3 - w^2 - w - 2],
[79, 79, w^3 - 2*w^2 - 2*w + 2],
[83, 83, w^3 + w^2 - 3*w - 4],
[83, 83, 3*w^3 + w^2 - 13*w - 4],
[89, 89, -w^3 + 6*w - 2],
[89, 89, w^3 - 3*w^2 - 4*w + 11],
[97, 97, w^2 + 2*w - 2],
[97, 97, 2*w^2 - w - 4],
[103, 103, -w^3 + 3*w^2 + 2*w - 7],
[107, 107, 2*w^3 + 2*w^2 - 8*w - 5],
[113, 113, -2*w^3 + 7*w + 2],
[113, 113, w^3 - 3*w^2 - 3*w + 8],
[127, 127, w^3 - 4*w + 4],
[131, 131, -2*w^2 - w + 8],
[131, 131, w - 4],
[139, 139, w^2 - w - 7],
[157, 157, w^3 + w^2 - 6*w - 5],
[167, 167, w^3 - 4*w - 4],
[169, 13, 2*w^3 + w^2 - 10*w - 2],
[169, 13, 2*w^3 + 2*w^2 - 9*w - 4],
[173, 173, 4*w^3 + 3*w^2 - 18*w - 14],
[173, 173, w^3 - w^2 - 6*w - 1],
[193, 193, w^3 - 2*w - 4],
[193, 193, 3*w^3 - 3*w^2 - 12*w + 11],
[197, 197, -2*w^3 - w^2 + 6*w + 4],
[199, 199, 2*w^3 + 2*w^2 - 10*w - 11],
[199, 199, -2*w^3 + 10*w + 1],
[211, 211, w^2 - 2*w - 4],
[211, 211, -4*w^3 + 17*w + 2],
[223, 223, w^3 - w^2 - 6*w + 5],
[227, 227, -w - 4],
[227, 227, -w^3 - 2*w^2 + 6*w + 2],
[229, 229, 2*w^3 - 2*w^2 - 8*w + 5],
[239, 239, 4*w^3 + w^2 - 19*w - 7],
[251, 251, 2*w^3 - w^2 - 10*w + 2],
[263, 263, w^3 - w^2 - 5*w + 8],
[263, 263, -3*w^3 + 5*w^2 + 12*w - 17],
[263, 263, w^3 - 3*w^2 + w + 4],
[263, 263, 2*w^2 + w - 2],
[269, 269, -w^3 + 2*w^2 + 4*w - 4],
[271, 271, 3*w - 2],
[277, 277, -w^3 - w^2 + 6*w - 1],
[277, 277, w^2 + 3*w - 5],
[281, 281, 2*w^3 - 12*w - 5],
[283, 283, -2*w^3 + 9*w - 2],
[293, 293, -w^2 - 3*w + 7],
[313, 313, 2*w^3 - 2*w^2 - 6*w + 5],
[331, 331, -4*w^2 - w + 14],
[331, 331, -3*w^2 + 3*w + 5],
[337, 337, 4*w^3 - 16*w + 1],
[347, 347, -4*w^3 - w^2 + 16*w + 2],
[359, 359, 2*w^2 - 2*w - 7],
[359, 359, -2*w^3 + w^2 + 10*w - 4],
[361, 19, 3*w^3 - 6*w^2 - 12*w + 22],
[373, 373, -w^3 + w^2 + 2*w - 5],
[373, 373, -2*w^3 + w^2 + 8*w + 2],
[379, 379, 2*w^3 - 2*w^2 - 5*w + 4],
[397, 397, -4*w^3 - w^2 + 18*w + 10],
[397, 397, -2*w^3 - 3*w^2 + 10*w + 14],
[409, 409, -2*w^3 + w^2 + 10*w - 8],
[409, 409, 3*w^2 - 10],
[419, 419, 2*w^3 + w^2 - 9*w - 1],
[419, 419, -2*w^3 + w^2 + 6*w - 2],
[431, 431, 3*w^3 - w^2 - 12*w - 1],
[431, 431, 4*w^3 + w^2 - 18*w - 4],
[433, 433, 2*w^3 + 2*w^2 - 11*w - 8],
[457, 457, 3*w^2 - 8],
[461, 461, -3*w - 4],
[461, 461, w^3 + 3*w^2 - 3*w - 8],
[479, 479, -3*w^3 + w^2 + 12*w - 7],
[487, 487, 2*w^3 - 3*w^2 - 7*w + 7],
[491, 491, -w^3 - 3*w^2 + 6*w + 5],
[557, 557, 4*w^3 - w^2 - 19*w - 1],
[569, 569, 3*w^2 - w - 7],
[571, 571, -6*w^3 - 2*w^2 + 25*w + 10],
[571, 571, w^3 + w^2 - 8*w - 5],
[577, 577, 2*w^3 - w^2 - 9*w - 1],
[577, 577, -2*w^3 - 3*w^2 + 7*w + 7],
[577, 577, w^2 + 2*w - 10],
[577, 577, 4*w^3 - 18*w - 5],
[587, 587, -w^3 - 2*w^2 + 6*w + 10],
[599, 599, 3*w^3 + w^2 - 12*w - 1],
[599, 599, 3*w^3 + 2*w^2 - 12*w - 10],
[601, 601, w^2 - 8],
[613, 613, -3*w^2 - w + 5],
[617, 617, 3*w^3 - 3*w^2 - 11*w + 10],
[619, 619, 2*w^3 - 4*w^2 - 7*w + 16],
[625, 5, -5],
[631, 631, -w^3 + 2*w^2 + 4*w - 2],
[641, 641, -3*w^3 - w^2 + 13*w + 2],
[641, 641, -2*w^3 + 3*w^2 + 8*w - 8],
[643, 643, -2*w^3 + 9*w - 4],
[653, 653, -3*w^3 - w^2 + 8*w + 1],
[673, 673, -2*w^3 + 11*w - 2],
[673, 673, -w^3 + w^2 + w - 4],
[677, 677, 4*w^3 + 2*w^2 - 19*w - 14],
[691, 691, 2*w^3 + 3*w^2 - 8*w - 10],
[691, 691, -w^3 + 3*w^2 + 5*w - 16],
[701, 701, 2*w^3 + 2*w^2 - 8*w - 1],
[709, 709, 7*w^3 + w^2 - 30*w - 5],
[733, 733, -w^3 + 2*w^2 + 2*w - 8],
[733, 733, -10*w^3 - 3*w^2 + 44*w + 16],
[739, 739, -3*w^3 - w^2 + 11*w + 10],
[739, 739, 2*w^3 - 2*w^2 - 11*w + 4],
[757, 757, w^3 - 3*w^2 - 7*w + 14],
[769, 769, w^3 + w^2 - 2*w - 7],
[773, 773, 2*w^3 - 2*w^2 - 9*w + 4],
[787, 787, 3*w^3 - 14*w + 2],
[787, 787, 2*w^3 + 2*w^2 - 6*w - 7],
[797, 797, 2*w^3 + 3*w^2 - 9*w - 11],
[797, 797, -4*w^3 - 4*w^2 + 19*w + 20],
[809, 809, w^3 + 3*w^2 - 4*w - 7],
[821, 821, -w^3 - 3*w^2 + 5*w + 10],
[839, 839, 2*w^3 + w^2 - 5*w - 5],
[839, 839, 3*w^3 + 3*w^2 - 14*w - 11],
[853, 853, -3*w^3 + 3*w^2 + 10*w - 13],
[859, 859, -4*w^3 + 19*w + 2],
[859, 859, -2*w^3 + 4*w^2 + 7*w - 10],
[863, 863, w^2 - 3*w - 5],
[863, 863, -2*w^3 + 3*w^2 + 12*w - 16],
[877, 877, -w^3 + 5*w^2 + 4*w - 19],
[877, 877, -3*w^3 - 3*w^2 + 11*w + 4],
[881, 881, 2*w^3 + 2*w^2 - 9*w - 2],
[883, 883, -3*w^3 + 7*w^2 + 13*w - 26],
[887, 887, -w^3 + 2*w^2 + 6*w - 2],
[907, 907, 8*w^3 + 3*w^2 - 33*w - 11],
[907, 907, -w^3 - w^2 + w - 4],
[907, 907, -2*w^3 + 6*w^2 + 8*w - 23],
[907, 907, -3*w^3 + w^2 + 11*w - 4],
[911, 911, -3*w^3 - 3*w^2 + 12*w + 13],
[911, 911, w^3 + 3*w^2 - 6*w - 7],
[929, 929, 2*w^3 + 3*w^2 - 9*w - 7],
[941, 941, 4*w - 1],
[947, 947, w^3 + w^2 - 7*w - 8],
[953, 953, -4*w^3 + w^2 + 14*w - 8],
[953, 953, 2*w^3 + w^2 - 9*w + 1],
[967, 967, -w^3 + w^2 + 7*w - 2],
[967, 967, 4*w^3 + 3*w^2 - 18*w - 8],
[971, 971, 2*w^3 + 3*w^2 - 8*w - 8],
[977, 977, 4*w^3 - 16*w - 1],
[977, 977, 7*w^3 + w^2 - 29*w - 4],
[977, 977, 3*w^3 - w^2 - 14*w - 1],
[977, 977, -3*w^3 + 10*w + 2],
[983, 983, -2*w^3 - 2*w^2 + 7*w - 2],
[991, 991, 7*w^3 + w^2 - 30*w - 7],
[991, 991, 4*w^3 - 2*w^2 - 17*w + 4],
[997, 997, 4*w^3 + 2*w^2 - 16*w - 11],
[997, 997, 6*w^3 + 4*w^2 - 25*w - 14]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^3 - 9*x - 4;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [0, e, -e^2 + e + 5, -e, 4, -e^2 + 8, 4, -e - 4, -e^2 + 4*e + 8, -2*e^2 + e + 10, 2*e + 4, -2*e^2 + 12, 2*e^2 - 10, 2*e^2 - 4*e - 12, 2*e^2 + 2*e - 14, -4*e, 2*e - 8, e^2 + 4*e - 8, e + 4, 2*e^2 - 4*e - 12, -4*e + 4, 2*e^2 - 3*e - 18, 4*e + 2, -2*e^2 + 18, -2*e^2 + 10, 2*e^2 - 4*e - 8, -3*e^2 + 4*e + 20, 2*e^2 - e - 18, e^2 + 4*e - 14, -e^2 + 4*e + 16, -e^2 + 12, 3*e^2 - 4*e - 12, 12, e^2 - 2, -16, 2*e^2 - 6*e - 14, 2*e^2 - 2*e - 22, -4*e^2 + 4*e + 22, -2*e^2 + e + 18, -2*e^2 + 8*e + 18, -2*e^2 - 5*e + 14, -2*e^2 - 4*e + 6, 3*e^2 - 16, 4*e^2 - 8*e - 32, 12, 4*e - 12, 3*e + 4, 3*e^2 + 4*e - 20, 4*e^2 - e - 24, -2*e^2 + 4*e - 2, e - 4, -4*e - 4, -e^2 - 4*e + 8, 4*e^2 - 3*e - 28, 4*e + 8, -24, 2*e^2 - e - 6, -8*e, 2*e^2 - 6*e - 26, 2*e^2 - e - 14, -2*e^2 - 6*e + 26, -2*e + 4, -3*e^2 + 8*e + 22, -2*e^2 - 2*e + 26, 4*e^2 - 4*e - 20, 4*e^2 + 4*e - 20, -2*e^2 + 3*e + 14, 6*e + 4, -2*e^2 + 8, 4*e^2 - 3*e - 4, 2*e^2 - 4*e + 2, 2*e^2 - 2, -2*e^2 - 2*e - 2, -9*e - 8, -18, -6*e^2 - 4*e + 38, -2*e^2 - 10*e + 18, 3*e^2 - 4*e - 22, -4*e^2 - 2*e + 12, -e - 8, -4*e^2 + 2*e + 8, 4*e^2 - 8*e - 40, -6*e^2 + 26, -e^2 + 10, 6*e^2 - 8*e - 34, -2*e^2 - 4*e + 30, -2*e + 8, 9*e + 12, 2*e^2 - 4*e - 20, -2*e^2 + 10*e + 6, -8*e + 2, -8*e - 4, e^2 - 4*e - 28, -6*e^2 + 8*e + 26, -3*e^2 - 4*e + 34, -4*e^2 + 8*e + 18, -6*e^2 - 6*e + 42, e^2 + 12, -4*e^2 + 40, -8*e + 16, -4*e^2 + 26, 6*e^2 - 6*e - 18, 2*e^2 + 6*e - 22, 8*e - 4, 4*e^2 - 8*e - 30, -7*e + 12, 2*e^2 - 2*e - 30, -10*e^2 + 9*e + 54, 4*e^2 - 5*e - 48, -2*e^2 - e + 10, -6*e^2 + 6*e + 10, 2*e^2 - 4*e - 22, -8*e + 6, 6*e^2 - 8*e - 28, 2*e^2 + 12*e - 12, 2*e^2 + 4*e - 2, 10*e^2 - 8*e - 58, -e^2 - 12*e + 14, 2*e^2 + 3*e - 22, 7*e^2 - 20, -12*e + 12, 4*e^2 - 26, -2*e^2 + 10, 2*e^2 - 10*e - 18, -4*e^2 - 2*e + 28, e^2 + 8*e - 4, -4*e - 26, 2*e^2 - 8*e - 18, e^2 + 8*e + 10, -8*e^2 + 8*e + 38, -2*e^2 + 4*e - 8, -6*e^2 - 4*e + 56, 9*e^2 - 4*e - 42, 4*e^2 - 8*e - 4, 8*e^2 - 10*e - 60, -7*e^2 + 32, 8*e^2 - 4*e - 48, 6*e^2 - 12*e - 26, -2*e^2 - 2*e + 6, 2*e^2 + 2*e + 26, 8*e - 20, 4*e^2 - 15*e - 44, -8*e^2 - 5*e + 40, -4*e^2 + 4*e - 4, 5*e^2 - 44, -4*e^2 + 10*e + 36, -4*e^2 + 20*e + 32, 7*e^2 - 32, 6*e^2 - 12*e - 46, -2*e^2 + 6*e + 38, -6*e^2 - 4*e + 36, 6*e^2 - 2*e - 46, 6*e^2 - e - 10, 8*e^2 - e - 20, -8*e^2 + 11*e + 60, -4*e^2 - 4*e + 44, -2*e^2 + 9*e - 10, 2*e^2 + 8*e + 2, -6*e^2 + 12*e + 50, 2*e^2 - 6*e + 26, -8*e^2 + 4*e + 40, -2*e + 40, 8*e^2 - e - 44, -7*e^2 + 8*e + 22, -7*e^2 + 12*e + 38];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {2, 2, w - 1}>] := 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;