/*
  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, -5, 0, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {16, 2, 2}>;

primesArray := [
[2, 2, w + 1],
[5, 5, w^3 - 4*w],
[8, 2, w^3 - w^2 - 4*w + 3],
[11, 11, w^3 - 5*w + 1],
[17, 17, -w^3 - w^2 + 5*w + 4],
[23, 23, -w^2 + 2],
[29, 29, -w^2 - w + 3],
[31, 31, -w^3 + 6*w + 2],
[43, 43, w^3 - 6*w],
[47, 47, 2*w^3 - 9*w],
[47, 47, -2*w^3 + w^2 + 8*w - 2],
[53, 53, -4*w^3 + 2*w^2 + 18*w - 5],
[59, 59, w^2 - w - 5],
[59, 59, 3*w^3 - 15*w - 5],
[71, 71, w^3 - 3*w - 3],
[81, 3, -3],
[83, 83, -2*w^3 + 8*w + 3],
[89, 89, -2*w^3 + 11*w - 2],
[101, 101, 2*w^3 - w^2 - 10*w],
[101, 101, 2*w^3 - 2*w^2 - 10*w + 3],
[107, 107, 3*w^3 - 15*w - 1],
[107, 107, 4*w^3 - 2*w^2 - 18*w + 3],
[109, 109, 2*w^3 - 10*w + 3],
[109, 109, 3*w^3 - 2*w^2 - 13*w + 3],
[113, 113, 3*w^3 - w^2 - 15*w + 2],
[113, 113, 3*w^3 - 13*w - 5],
[121, 11, 3*w^3 - 13*w - 1],
[125, 5, -2*w^3 + w^2 + 7*w + 1],
[127, 127, 2*w^3 - 8*w - 1],
[127, 127, -w^3 + 2*w^2 + 3*w - 3],
[137, 137, 2*w^3 - 10*w + 1],
[137, 137, 2*w^3 - 11*w],
[139, 139, 2*w^3 - 9*w - 6],
[157, 157, -4*w^3 - w^2 + 19*w + 7],
[163, 163, 5*w^3 - 2*w^2 - 25*w + 7],
[167, 167, w^2 + w - 5],
[167, 167, -w^3 + 2*w^2 + 4*w - 6],
[169, 13, -w^3 + w^2 + 3*w - 4],
[169, 13, w^2 - 2*w - 4],
[173, 173, -2*w^3 + w^2 + 11*w - 7],
[173, 173, w - 4],
[181, 181, w + 4],
[181, 181, -2*w - 5],
[191, 191, -w^3 + 2*w^2 + 6*w - 6],
[191, 191, 2*w^3 - w^2 - 11*w + 1],
[193, 193, -3*w^3 + 16*w],
[227, 227, 6*w^3 - 3*w^2 - 27*w + 5],
[227, 227, 2*w^3 - 11*w - 2],
[229, 229, -2*w^3 + w^2 + 10*w - 6],
[233, 233, 4*w^3 - 3*w^2 - 17*w + 5],
[233, 233, 2*w^3 + w^2 - 7*w - 5],
[233, 233, w^3 - 7*w - 1],
[233, 233, 2*w^3 - 7*w],
[239, 239, 2*w^2 - 2*w - 7],
[239, 239, -4*w^3 + w^2 + 18*w],
[241, 241, w^3 - 5*w - 5],
[263, 263, -4*w^3 + w^2 + 20*w - 4],
[269, 269, w^3 + 2*w^2 - 6*w - 6],
[271, 271, 2*w^3 + w^2 - 12*w - 8],
[277, 277, w^3 + w^2 - 7*w - 6],
[283, 283, 6*w^3 - 3*w^2 - 27*w + 7],
[283, 283, -3*w^3 + 3*w^2 + 13*w - 8],
[293, 293, -2*w^3 + w^2 + 9*w + 3],
[293, 293, 4*w^3 - w^2 - 19*w + 3],
[313, 313, 2*w^2 - w - 4],
[313, 313, -2*w^3 + 2*w^2 + 8*w - 1],
[317, 317, 3*w^3 + w^2 - 13*w - 10],
[331, 331, w^3 + 2*w^2 - 4*w - 6],
[349, 349, -w^3 + 3*w^2 + w - 4],
[353, 353, 3*w^3 - 14*w],
[353, 353, -2*w^3 + w^2 + 7*w + 5],
[361, 19, -5*w^3 + w^2 + 23*w - 2],
[361, 19, -2*w^3 + 7*w + 6],
[373, 373, -w^3 + 4*w - 4],
[379, 379, -w^3 - 2*w^2 + 5*w + 7],
[383, 383, w^2 - 2*w - 6],
[389, 389, -4*w^3 + 2*w^2 + 16*w - 5],
[397, 397, 5*w^3 - 2*w^2 - 24*w + 2],
[397, 397, 3*w^3 + 2*w^2 - 16*w - 14],
[401, 401, 2*w^2 - 5],
[419, 419, -w^3 - w^2 + 9*w + 6],
[439, 439, 3*w^3 - w^2 - 15*w - 2],
[449, 449, 4*w^3 + w^2 - 17*w - 5],
[449, 449, 4*w^3 - w^2 - 17*w - 5],
[467, 467, -3*w^3 + 16*w - 2],
[467, 467, 2*w^3 + w^2 - 10*w - 2],
[479, 479, 4*w^3 - 19*w - 8],
[491, 491, 3*w - 4],
[491, 491, -w^3 + 2*w^2 + 5*w - 11],
[509, 509, -7*w^3 + 3*w^2 + 33*w - 10],
[509, 509, 4*w^3 - w^2 - 21*w - 3],
[523, 523, -w^3 - 2*w^2 + 6*w + 2],
[541, 541, -5*w^3 + 2*w^2 + 26*w - 8],
[541, 541, 2*w^3 + w^2 - 9*w - 1],
[571, 571, -4*w^3 + 3*w^2 + 20*w - 14],
[571, 571, -3*w^3 - w^2 + 15*w + 4],
[593, 593, w^3 - 4*w - 6],
[613, 613, 4*w^3 - 2*w^2 - 19*w + 2],
[617, 617, -w^3 + 3*w - 5],
[617, 617, -w^3 + 2*w^2 + 3*w - 9],
[617, 617, 8*w^3 - 3*w^2 - 38*w + 8],
[617, 617, w^3 - 8*w],
[619, 619, 6*w^3 - 3*w^2 - 26*w + 4],
[631, 631, 3*w^3 + w^2 - 17*w - 8],
[631, 631, w^2 - 3*w - 5],
[641, 641, -3*w^3 + 2*w^2 + 16*w - 10],
[643, 643, -2*w^3 + 3*w^2 + 7*w - 9],
[647, 647, w^3 - w^2 - 3*w - 4],
[653, 653, 2*w^3 + w^2 - 11*w - 1],
[653, 653, w^3 + 2*w^2 - 7*w - 9],
[653, 653, -4*w^3 - w^2 + 17*w + 3],
[653, 653, 3*w^3 - 2*w^2 - 13*w + 1],
[661, 661, 2*w^3 - 3*w^2 - 8*w + 8],
[673, 673, 3*w^3 - 12*w - 2],
[673, 673, 2*w^3 - w^2 - 12*w],
[683, 683, -w^3 + 2*w^2 + 2*w - 6],
[691, 691, -5*w^3 + 22*w + 8],
[691, 691, 2*w^3 - 12*w - 3],
[719, 719, 6*w^3 + w^2 - 28*w - 8],
[733, 733, -3*w^2 + 3*w + 7],
[733, 733, 3*w^3 - 14*w + 2],
[733, 733, -w^3 + w^2 + 7*w - 8],
[733, 733, 2*w^3 - 6*w - 5],
[751, 751, 3*w^3 + 3*w^2 - 13*w - 12],
[751, 751, 5*w^3 - 2*w^2 - 25*w + 5],
[751, 751, -4*w^3 + 3*w^2 + 20*w - 12],
[751, 751, 2*w^3 + w^2 - 9*w + 1],
[757, 757, -3*w^3 + w^2 + 11*w + 2],
[773, 773, w^3 - w^2 - 7*w + 6],
[787, 787, -9*w^3 + 4*w^2 + 43*w - 13],
[787, 787, w^2 + 2*w - 6],
[797, 797, -w^3 + 4*w^2 - 2*w - 6],
[797, 797, 2*w^3 - w^2 - 9*w + 7],
[821, 821, -6*w^3 + 27*w + 2],
[823, 823, 8*w^3 - 4*w^2 - 37*w + 8],
[827, 827, -3*w^3 + 3*w^2 + 15*w - 4],
[829, 829, w^3 + w^2 - 3*w - 8],
[839, 839, 2*w^3 - 3*w^2 - 8*w + 6],
[839, 839, 8*w^3 - 2*w^2 - 38*w + 3],
[859, 859, 4*w^3 - 2*w^2 - 18*w + 1],
[859, 859, -4*w^3 + 4*w^2 + 17*w - 10],
[863, 863, -4*w^3 + w^2 + 19*w + 3],
[877, 877, -4*w^3 + w^2 + 18*w + 4],
[883, 883, -6*w^3 + 3*w^2 + 31*w - 13],
[883, 883, 4*w^3 - w^2 - 17*w - 1],
[887, 887, 3*w^3 - 4*w^2 - 11*w + 11],
[907, 907, 5*w^3 - 4*w^2 - 21*w + 11],
[907, 907, -2*w^3 + 2*w^2 + 8*w - 11],
[911, 911, 2*w^3 - 2*w^2 - 11*w],
[919, 919, 4*w^3 - 17*w - 2],
[937, 937, 3*w^3 - 2*w^2 - 15*w + 1],
[937, 937, 3*w^3 - 17*w + 1],
[941, 941, 4*w^3 - w^2 - 16*w - 4],
[953, 953, -2*w^3 + 2*w^2 + 13*w - 6],
[971, 971, -3*w^3 + 11*w + 9],
[977, 977, 8*w^3 - 4*w^2 - 39*w + 16],
[983, 983, 4*w^3 - w^2 - 21*w + 3],
[983, 983, 2*w^3 - 3*w^2 - 9*w + 7],
[997, 997, w^3 - w^2 - w - 6],
[997, 997, w^3 - 9*w - 3]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^3 - x^2 - 14*x + 16;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [1, e, 1, -1/2*e^2 - 1/2*e + 5, -1/2*e^2 + 1/2*e + 5, -e^2 - e + 10, e^2 + e - 12, e^2 - 4, 1/2*e^2 - 1/2*e + 1, -e - 2, -e - 2, -2, -1/2*e^2 - 3/2*e + 3, -1/2*e^2 - 3/2*e + 3, e^2 - e - 14, 4*e + 2, -1/2*e^2 + 5/2*e + 3, 1/2*e^2 - 3/2*e - 3, -e - 4, -2*e + 2, -1/2*e^2 - 5/2*e + 1, 1/2*e^2 + 5/2*e - 9, 2*e^2 - 18, -e + 4, -1/2*e^2 - 3/2*e + 1, 1/2*e^2 + 7/2*e - 9, -1/2*e^2 - 3/2*e + 9, 2*e^2 + 4*e - 26, 2*e^2 + 3*e - 18, e^2 - 2*e - 8, 3/2*e^2 + 5/2*e - 15, 1/2*e^2 - 5/2*e - 5, 1/2*e^2 - 5/2*e - 3, e^2 + 3*e - 8, -1/2*e^2 - 7/2*e + 7, -2*e^2 + 16, 2*e^2 + 2*e - 28, -2*e + 14, 3/2*e^2 - 3/2*e - 15, e, -2*e^2 + e + 16, e + 8, 2*e + 2, -4*e - 8, -e^2 + 4, -3/2*e^2 - 9/2*e + 23, e^2 - 4*e - 8, 4, e^2 + 5*e - 4, e^2 - 4*e - 2, -1/2*e^2 - 7/2*e + 21, -e^2 - 4*e + 14, -3/2*e^2 - 11/2*e + 13, 16, -3*e^2 - 2*e + 24, -7/2*e^2 - 13/2*e + 43, -3*e^2 + 2*e + 24, 3*e^2 + 5*e - 36, -2*e^2 - 2*e + 20, -e^2 + 3*e + 16, 1/2*e^2 + 7/2*e - 7, 3/2*e^2 - 1/2*e - 19, e^2 + 8*e - 14, 14, 1/2*e^2 - 1/2*e - 9, 3*e^2 - 18, -e^2 - 7*e + 4, 5/2*e^2 - 11/2*e - 33, -3*e^2 - 4*e + 34, 1/2*e^2 - 5/2*e - 13, -1/2*e^2 + 5/2*e - 7, -7/2*e^2 - 7/2*e + 33, -7/2*e^2 - 9/2*e + 39, 2*e + 26, -3/2*e^2 + 1/2*e + 11, e^2 + 6*e - 24, -e^2 - 4*e + 2, 2*e^2 + 2*e - 22, 2*e - 6, 2*e^2 - 6*e - 26, -1/2*e^2 + 9/2*e - 9, -8, 9/2*e^2 + 5/2*e - 35, -1/2*e^2 + 9/2*e + 21, e^2 - 16, 4*e - 4, e^2 - 5*e - 30, -e^2 - 6*e + 4, -3/2*e^2 - 21/2*e + 21, -4*e^2 + e + 32, e^2 - e - 24, 5/2*e^2 - 9/2*e - 15, -4*e^2 - 7*e + 56, 2*e^2 - 5*e - 4, -1/2*e^2 + 13/2*e + 19, 5/2*e^2 - 3/2*e - 41, 5/2*e^2 + 15/2*e - 17, e^2 + 2*e + 6, -2*e + 14, 3/2*e^2 + 15/2*e - 13, 9/2*e^2 - 9/2*e - 49, -e^2 + 2*e + 26, -3/2*e^2 + 9/2*e + 3, 3*e^2 + 7*e - 22, -4*e - 16, 1/2*e^2 + 11/2*e + 3, 3/2*e^2 - 7/2*e - 33, e^2 + 10*e - 8, -6*e^2 - 2*e + 42, -5*e^2 - e + 56, -3*e^2 + 3*e + 16, -2*e^2 + 4*e + 30, 3*e^2 + 7*e - 48, -3/2*e^2 + 1/2*e + 25, -8*e - 6, 5/2*e^2 + 3/2*e - 3, 3/2*e^2 + 19/2*e - 7, -3/2*e^2 - 3/2*e - 1, 4*e^2 + e - 46, 4*e^2 - 2*e - 14, 2*e^2 - 3*e + 8, -2*e^2 - e + 28, -3*e^2 - 2*e + 14, -5*e^2 - 4*e + 60, -6*e^2 + 4*e + 48, -e^2 + 7*e + 18, -e^2 - 11*e + 14, -3*e^2 + 2*e + 14, -e^2 + 11*e + 16, 7/2*e^2 + 25/2*e - 57, -9/2*e^2 - 9/2*e + 53, -3*e^2 - 3*e + 28, -4*e^2 - 4*e + 14, 5*e^2 - 5*e - 56, e^2 - 3*e - 34, -7/2*e^2 - 1/2*e + 33, 2*e^2 + 4*e + 6, 4*e^2 + 3*e - 50, -4*e - 32, -4*e^2 - 10*e + 56, -1/2*e^2 - 7/2*e + 31, 5*e^2 + 3*e - 62, 2*e^2 + 10*e - 38, -2*e^2 + 4*e + 36, -12, -3*e^2 - 10*e + 48, -7/2*e^2 + 23/2*e + 41, 3/2*e^2 + 9/2*e - 41, -3*e^2 + 3*e + 50, 24, 6*e^2 + 8*e - 54, 2*e^2 - 4*e - 6, -4*e^2 - 12*e + 38, -3*e^2 - 10*e + 58, 7/2*e^2 + 9/2*e - 49, 1/2*e^2 + 7/2*e + 15, -e^2 - 7*e - 2, e^2 + 3*e - 22, -4*e^2 - 3*e + 40, 2*e^2 - 26];
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;
ALEigenvalues[ideal<ZF | {8, 2, w^3 - w^2 - 4*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;