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

NN := ideal<ZF | {10, 10, -w^2 + w + 2}>;

primesArray := [
[2, 2, w - 1],
[3, 3, w],
[5, 5, -w + 2],
[11, 11, w + 2],
[27, 3, -w^3 + 2*w^2 + 4*w - 4],
[29, 29, -w^2 + 3*w + 1],
[31, 31, w^3 - 2*w^2 - 2*w + 2],
[31, 31, -w^2 + 2*w + 4],
[47, 47, w^3 - w^2 - 4*w + 1],
[49, 7, 2*w^3 - 2*w^2 - 8*w - 1],
[49, 7, w^3 - w^2 - 3*w - 2],
[53, 53, w^3 - 3*w^2 - w + 2],
[53, 53, w^3 - 5*w - 5],
[61, 61, 2*w - 1],
[61, 61, -w^3 + 4*w^2 - 4],
[67, 67, 3*w^3 - 3*w^2 - 13*w - 4],
[73, 73, -w^3 + 4*w^2 - 10],
[73, 73, w^2 - w + 1],
[83, 83, -4*w^3 + 5*w^2 + 17*w + 1],
[83, 83, 3*w^3 - 3*w^2 - 14*w - 5],
[89, 89, w^3 - w^2 - 4*w + 5],
[89, 89, w^3 - 3*w^2 - w + 8],
[97, 97, -w^3 + 7*w + 5],
[97, 97, 2*w^3 - 3*w^2 - 7*w + 1],
[103, 103, -3*w^3 + 8*w^2 + 7*w - 19],
[113, 113, -w^2 + 2*w - 2],
[125, 5, 2*w^3 - 5*w^2 - 3*w + 7],
[127, 127, 2*w^2 - 3*w - 4],
[137, 137, -w^3 + 3*w^2 + 3*w - 4],
[137, 137, w^3 - w^2 - 6*w - 1],
[149, 149, 2*w^3 - 3*w^2 - 5*w - 1],
[151, 151, -w^3 + 3*w^2 - 5],
[157, 157, -w^3 + 2*w^2 + w - 5],
[163, 163, 4*w^3 - 4*w^2 - 18*w - 7],
[167, 167, -3*w^3 + 4*w^2 + 13*w - 1],
[167, 167, 2*w^2 - 4*w - 1],
[173, 173, w^3 - 5*w - 1],
[173, 173, 2*w^3 - 3*w^2 - 9*w + 1],
[179, 179, -2*w^3 + 3*w^2 + 8*w - 4],
[181, 181, 2*w^2 - 2*w - 5],
[193, 193, 4*w^3 - 9*w^2 - 11*w + 17],
[197, 197, w^3 - w^2 - 6*w + 1],
[197, 197, -w^3 + 5*w^2 - w - 4],
[199, 199, -3*w^3 + 4*w^2 + 11*w + 5],
[211, 211, -2*w^3 + 5*w^2 + 6*w - 8],
[227, 227, w^2 - 3*w - 5],
[233, 233, 2*w^3 - 6*w^2 - 5*w + 16],
[233, 233, -5*w^3 + 13*w^2 + 13*w - 26],
[251, 251, -2*w^3 + 4*w^2 + 5*w - 4],
[251, 251, -w^3 + 5*w^2 + w - 8],
[251, 251, w^2 + w - 5],
[251, 251, -w^3 + 2*w^2 + 3*w - 7],
[257, 257, w^3 - 5*w^2 + 2*w + 11],
[263, 263, -3*w^3 + 8*w^2 + 9*w - 17],
[269, 269, -3*w^3 + 2*w^2 + 13*w + 7],
[269, 269, 3*w^3 - 4*w^2 - 13*w - 1],
[271, 271, 2*w^3 - 5*w^2 - 5*w + 7],
[277, 277, -w^3 + w^2 + 5*w - 4],
[283, 283, 3*w - 2],
[293, 293, 3*w^3 - 2*w^2 - 15*w - 5],
[307, 307, -w - 4],
[307, 307, -2*w^3 + 11*w + 8],
[313, 313, -2*w^3 + 2*w^2 + 7*w - 2],
[347, 347, -w^3 + 3*w^2 + w - 10],
[347, 347, 3*w^3 - 3*w^2 - 11*w - 4],
[349, 349, -w^3 + w^2 + 4*w + 5],
[349, 349, 2*w^2 - 2*w - 11],
[353, 353, 2*w^3 - 7*w^2 - 3*w + 17],
[353, 353, w^3 - w^2 - 6*w - 5],
[359, 359, 2*w^2 - 4*w - 7],
[359, 359, 3*w^2 - 3*w - 13],
[361, 19, 6*w^3 - 6*w^2 - 28*w - 7],
[361, 19, 2*w^2 - w - 8],
[367, 367, -2*w^3 + w^2 + 9*w + 7],
[367, 367, -4*w^3 + 5*w^2 + 16*w + 4],
[373, 373, -2*w^3 + 3*w^2 + 6*w - 2],
[373, 373, -w^3 + 3*w^2 + 3*w - 2],
[373, 373, -2*w^3 + 7*w^2 + w - 13],
[373, 373, 3*w^3 - 9*w^2 - 7*w + 22],
[379, 379, -2*w^3 + 6*w^2 + 2*w - 11],
[379, 379, 2*w^3 - 5*w^2 - 7*w + 7],
[409, 409, -w^3 + 4*w^2 + 2*w - 14],
[433, 433, -2*w^3 + 4*w^2 + 4*w - 5],
[433, 433, 3*w^2 - 4*w - 10],
[443, 443, -w^3 + 3*w^2 - 7],
[443, 443, -w^3 + 3*w^2 + 4*w - 11],
[457, 457, -3*w^3 + 5*w^2 + 10*w - 5],
[461, 461, -2*w^3 + 3*w^2 + 8*w + 4],
[463, 463, -5*w^3 + 3*w^2 + 25*w + 16],
[479, 479, 3*w^2 - 6*w - 8],
[487, 487, w^3 - 3*w^2 - 5*w + 8],
[491, 491, 2*w^3 - 4*w^2 - 3*w + 2],
[499, 499, w^3 - w^2 - 7*w + 4],
[499, 499, w^3 - 6*w - 8],
[521, 521, 2*w^3 - 4*w^2 - 8*w + 7],
[521, 521, 3*w^3 - 5*w^2 - 8*w - 1],
[521, 521, w^3 - 5*w + 1],
[521, 521, w^3 - 8*w + 2],
[523, 523, -2*w^3 + 5*w^2 + 4*w - 4],
[547, 547, -2*w^3 + 4*w^2 + 6*w - 1],
[547, 547, -w^3 - w^2 + 6*w + 11],
[563, 563, -2*w^3 + 8*w^2 - 3*w - 8],
[569, 569, 3*w^3 - 7*w^2 - 10*w + 13],
[571, 571, -3*w^3 + 4*w^2 + 11*w - 1],
[577, 577, -w^3 + 5*w^2 - 3*w - 10],
[601, 601, -2*w^3 + 6*w^2 + 4*w - 17],
[607, 607, 2*w^3 - 12*w - 5],
[613, 613, 6*w^3 - 5*w^2 - 27*w - 11],
[613, 613, 2*w + 5],
[631, 631, 5*w^3 - 14*w^2 - 11*w + 29],
[641, 641, w^3 + w^2 - 5*w - 8],
[641, 641, -w^3 + 5*w^2 - 13],
[643, 643, w^2 + w - 7],
[643, 643, -3*w^3 + 3*w^2 + 15*w + 8],
[661, 661, 3*w^3 - 2*w^2 - 14*w - 4],
[691, 691, -2*w^3 + 3*w^2 + 9*w - 7],
[691, 691, 2*w^3 - 5*w^2 - 3*w + 11],
[709, 709, 7*w^3 - 16*w^2 - 20*w + 32],
[709, 709, -w^3 + 4*w^2 + 2*w - 4],
[709, 709, -2*w^3 + 3*w^2 + 4*w - 2],
[719, 719, -2*w^3 + 3*w^2 + 7*w - 7],
[719, 719, -w^3 + 4*w^2 - 14],
[739, 739, 5*w^3 - 3*w^2 - 24*w - 17],
[739, 739, -4*w^3 + 10*w^2 + 10*w - 17],
[743, 743, 2*w^3 - 2*w^2 - 11*w - 2],
[757, 757, -3*w^3 + 5*w^2 + 11*w - 2],
[761, 761, w^3 - 3*w - 5],
[761, 761, w^2 - 5*w - 1],
[769, 769, -w^3 - w^2 + 9*w + 8],
[773, 773, -2*w^3 + 4*w^2 + 7*w - 2],
[797, 797, -2*w^3 + w^2 + 12*w + 10],
[797, 797, -2*w^3 + 5*w^2 + 3*w + 1],
[809, 809, 3*w^2 - 4*w - 8],
[821, 821, -3*w^3 + 6*w^2 + 9*w - 5],
[823, 823, -2*w^3 + 5*w^2 + w - 7],
[823, 823, -3*w^3 + 6*w^2 + 6*w - 10],
[827, 827, w^3 + 2*w^2 - 5*w - 7],
[827, 827, w^3 - 2*w^2 - 6*w - 4],
[829, 829, 3*w^3 - w^2 - 15*w - 14],
[829, 829, w^3 + 2*w^2 - 7*w - 13],
[839, 839, 4*w^3 - 11*w^2 - 8*w + 26],
[853, 853, w^3 - 4*w - 8],
[863, 863, 5*w^3 - 2*w^2 - 26*w - 14],
[863, 863, 2*w^3 - 5*w^2 - 5*w + 5],
[881, 881, 2*w^3 + w^2 - 11*w - 11],
[883, 883, -6*w^3 + 15*w^2 + 14*w - 32],
[883, 883, -w^3 + 2*w^2 + 2*w - 8],
[887, 887, w^3 - w^2 - 7*w + 2],
[907, 907, 3*w^3 - 3*w^2 - 10*w - 5],
[907, 907, w^3 - 2*w^2 - 2*w - 4],
[919, 919, 3*w^3 - 3*w^2 - 11*w - 2],
[919, 919, -4*w^3 + 10*w^2 + 7*w - 16],
[929, 929, 3*w^3 - 3*w^2 - 10*w + 1],
[937, 937, 6*w^3 - 8*w^2 - 24*w + 1],
[947, 947, 3*w^3 - 6*w^2 - 8*w + 8],
[953, 953, 4*w - 1],
[953, 953, 4*w^3 - 9*w^2 - 10*w + 16],
[961, 31, -w^3 + 6*w - 2],
[967, 967, 2*w^3 - 8*w - 7],
[967, 967, 4*w^3 - 9*w^2 - 12*w + 14],
[971, 971, 5*w^3 - 6*w^2 - 20*w - 4]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^3 - 10*x - 5;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [1, e, 1, -e^2 + e + 5, -e^2 + 6, -8, e^2 - 3, -2*e + 2, -3*e + 3, -e - 3, e^2 - 3*e - 11, e^2 - 2*e - 11, 4*e, -e^2 + 4*e + 5, -e^2 + 2*e + 13, -2*e^2 + 2*e + 16, e^2 + e - 3, -e^2 + 14, -e^2 - e + 9, -e^2 - e + 9, e^2 - 3*e - 9, e^2 - 3*e - 9, -e^2 - 2*e + 8, 3*e^2 - 2*e - 18, 2*e^2 - 3*e - 11, -3*e^2 + 3*e + 17, -2*e^2 + 3*e + 13, e^2 - 4*e - 5, -e^2 - 2*e + 10, -3*e^2 + e + 11, -2*e + 6, -2*e^2 + 2*e + 24, 4*e - 10, 3*e^2 + e - 21, 2*e^2 - 5*e - 15, 2*e^2 + e - 1, -e^2 + 2*e + 11, -3*e^2 + 13, e^2 - 2*e - 18, 8*e - 2, -3*e^2 + 2*e + 6, 2*e^2 - 4*e - 8, -2*e^2 - 4*e + 18, 2*e^2 - 22, 5*e + 2, e^2 - 5*e - 15, -e + 10, e^2 - 7*e - 1, 4*e^2 - 3*e - 28, 3*e^2 - 4*e - 27, 2*e^2 - 2*e, -2*e^2 + 7*e + 28, -2*e^2 + e + 20, -2*e^2 + 2*e + 22, e^2 + 7, -2*e^2 - 4*e + 14, 3*e^2 - 2*e - 15, 6*e - 12, 5*e - 2, 2*e^2 - 5*e - 3, -e^2 - 3*e + 13, 4, e^2 - e - 21, -e^2 + 3*e + 29, -3*e^2 + 5*e + 15, 4*e^2 - 18, 2*e^2 - 4*e - 8, -e^2 - 2*e + 17, -e^2 - 5*e + 29, 2*e^2 + 5*e - 15, 3*e - 13, 2*e + 12, 3*e^2 + 4*e - 25, 2*e^2 - 6*e - 20, -7*e - 3, 4*e^2 - 8*e - 38, 2*e^2 + 2*e - 8, 8*e, 4*e^2 - 13*e - 37, -10*e + 2, 3*e^2 - 3*e - 17, -2*e^2 + 3*e + 12, e^2 - 9*e - 21, -3*e^2 + 8*e + 32, 3*e^2 + e - 7, -3*e^2 - e + 21, -2*e^2 - 4*e + 12, -3*e^2 + 35, -4*e - 14, 2*e^2 - e + 15, 4*e^2 - 13*e - 37, e^2 - e - 33, e^2 - 9*e + 7, -e + 10, 7*e^2 - 5*e - 45, -4*e^2 - 2*e + 24, -5*e^2 + 7*e + 23, -7*e^2 + 3*e + 33, 5*e^2 + 2*e - 39, -4*e^2 - e + 27, -3*e^2 + 6*e + 21, 6*e^2 - 12*e - 54, 4*e^2 - 26, -5*e^2 - 3*e + 37, -3*e^2 - 5*e + 19, e^2 - 6*e - 17, 7*e^2 - 2*e - 47, -3*e^2 + 4*e + 19, 4*e^2 - 20, 2*e^2 + 8*e - 40, 3*e^2 - 4*e - 17, 2*e^2 - 2*e + 10, 2*e^2 + 7*e - 24, -2*e^2 + 8*e - 2, -7*e^2 + 14*e + 53, e^2 - e - 23, 2*e^2 + 2*e - 32, 2*e^2 + 8*e - 38, 9*e + 9, e^2 + 8*e - 3, 2*e^2 + 6*e - 20, 4*e + 20, 8*e^2 - 8*e - 40, -7*e - 11, 4*e^2 + 8*e - 36, 10*e - 4, 4*e^2 - 4*e - 46, -6*e^2 + 11*e + 35, 2*e^2 + 2*e - 30, 7*e^2 + 2*e - 49, -e + 23, -4*e^2 - 4*e + 42, -4*e^2 + 7*e + 10, -4*e^2 - 4*e + 28, -4*e^2 + 4*e + 36, 3*e^2 - 4*e - 25, -4*e^2 + 11*e + 12, -e^2 - 8*e + 21, -6*e^2 + 4*e + 36, 6*e^2 + 9*e - 43, -7*e^2 + 2*e + 51, e^2 - 8*e - 23, -2*e^2 - 13*e + 17, -2*e^2 - 2*e - 8, 6*e^2 - 11*e - 63, -4*e^2 + 4*e + 20, -e^2 - 11*e + 13, 4*e^2 - 2*e - 42, -5*e^2 - e + 23, 6*e^2 - 17*e - 51, -6*e^2 - 7*e + 37, -2*e^2 + 6*e + 16, -e^2 + 2*e + 45, -6*e^2 - e + 50, -6*e^2 + 9*e + 39, 5*e^2 - 9*e - 59, 3*e^2 - 44, e^2 + 11*e - 29, -8*e^2 + 32, 10*e + 16, -e^2 - 15*e + 3];
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 | {5, 5, -w + 2}>] := -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;