/*
  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, -3, -6, 0, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {25, 25, w^3 - 2*w^2 - 4*w + 5}>;

primesArray := [
[3, 3, w],
[5, 5, w - 1],
[7, 7, w^3 - w^2 - 4*w + 1],
[11, 11, -w^2 + w + 2],
[13, 13, -w^3 + w^2 + 3*w - 1],
[16, 2, 2],
[17, 17, -w^3 + 5*w + 2],
[29, 29, -w^2 + w + 1],
[37, 37, -w^3 + 2*w^2 + 4*w - 4],
[41, 41, -w^3 + w^2 + 5*w + 1],
[41, 41, w^2 - 5],
[47, 47, w^3 - w^2 - 5*w - 2],
[47, 47, -w^3 + 2*w^2 + 4*w - 1],
[53, 53, w^3 - 4*w - 4],
[53, 53, 2*w^3 - 2*w^2 - 9*w + 1],
[61, 61, -w^3 + w^2 + 6*w - 2],
[71, 71, w^3 - w^2 - 6*w - 2],
[103, 103, 2*w^3 - 2*w^2 - 8*w + 1],
[103, 103, -3*w^3 + 3*w^2 + 13*w - 5],
[109, 109, 2*w^3 - w^2 - 8*w - 4],
[109, 109, 2*w^3 - w^2 - 9*w - 1],
[113, 113, -2*w^3 + w^2 + 11*w - 1],
[113, 113, 2*w^3 - w^2 - 10*w - 4],
[125, 5, 2*w^3 - 3*w^2 - 10*w + 4],
[131, 131, -2*w^3 + w^2 + 9*w - 1],
[139, 139, 2*w^3 - 4*w^2 - 7*w + 5],
[139, 139, 3*w^3 - w^2 - 16*w - 5],
[149, 149, w^2 + w - 4],
[149, 149, 3*w^3 - 3*w^2 - 13*w + 1],
[151, 151, w^2 - 2*w - 8],
[157, 157, w^3 - w^2 - 7*w - 2],
[157, 157, -2*w^2 + w + 7],
[167, 167, 2*w^3 - 2*w^2 - 7*w + 5],
[173, 173, -w^3 + 6*w - 2],
[173, 173, -w^3 + 2*w^2 + 5*w - 7],
[181, 181, -3*w^3 + 2*w^2 + 14*w + 1],
[191, 191, -w^3 + 2*w^2 + 2*w - 5],
[191, 191, -w^3 + 2*w^2 + 6*w - 5],
[193, 193, -w^3 + 3*w^2 + 4*w - 4],
[193, 193, 2*w^3 + 3*w^2 - 13*w - 20],
[197, 197, w^3 - 2*w - 2],
[199, 199, 2*w^3 - w^2 - 11*w - 2],
[211, 211, -2*w^3 + 9*w + 5],
[223, 223, -w^3 + w^2 + 4*w - 5],
[223, 223, w^2 - 3*w - 2],
[229, 229, 2*w^3 - 2*w^2 - 7*w + 1],
[241, 241, 3*w^2 - 2*w - 13],
[241, 241, -w^3 + w^2 + 5*w + 4],
[257, 257, -3*w^3 + w^2 + 15*w + 8],
[263, 263, -3*w^3 + 5*w^2 + 9*w - 7],
[263, 263, 2*w^3 - w^2 - 9*w + 2],
[263, 263, -w^3 + 4*w^2 + 2*w - 16],
[263, 263, w^3 - w^2 - 2*w - 2],
[271, 271, -3*w - 1],
[281, 281, 2*w^3 - 3*w^2 - 8*w + 2],
[281, 281, 3*w^3 - 4*w^2 - 12*w + 4],
[283, 283, w^3 - 5*w - 7],
[293, 293, -2*w^3 + w^2 + 10*w - 1],
[307, 307, w^3 - 3*w^2 - 3*w + 8],
[307, 307, 3*w^3 - 15*w - 10],
[311, 311, 2*w^3 - 3*w^2 - 7*w + 7],
[313, 313, -w^3 + 7*w + 2],
[337, 337, 3*w^3 - 17*w - 10],
[337, 337, 2*w^3 - 2*w^2 - 9*w - 5],
[343, 7, -2*w^3 + w^2 + 8*w + 2],
[347, 347, -3*w^3 + 17*w + 14],
[347, 347, -3*w^3 - w^2 + 17*w + 13],
[349, 349, 2*w^3 - w^2 - 8*w - 1],
[349, 349, -w^3 + 3*w^2 + 4*w - 8],
[353, 353, -w^3 - w^2 + 7*w + 4],
[359, 359, -2*w^3 + 9*w + 10],
[367, 367, 3*w^3 - 4*w^2 - 9*w + 4],
[367, 367, -w^3 + w^2 + 4*w + 4],
[367, 367, -w^3 + 3*w^2 + 3*w - 7],
[373, 373, -w^2 + w - 2],
[373, 373, -2*w^3 + 2*w^2 + 11*w - 2],
[379, 379, -2*w^3 + 4*w^2 + 7*w - 8],
[379, 379, -w^2 - 2],
[389, 389, -w^3 - 2*w^2 + 5*w + 11],
[397, 397, -2*w^3 + 4*w^2 + 7*w - 7],
[401, 401, w^3 - 3*w^2 - 2*w + 10],
[401, 401, -2*w^3 + 11*w + 4],
[401, 401, 3*w^3 - 2*w^2 - 14*w - 4],
[401, 401, -3*w^3 + 4*w^2 + 15*w - 5],
[421, 421, -w^3 + w^2 + 7*w - 1],
[421, 421, -2*w^3 + 2*w^2 + 9*w - 7],
[433, 433, w^2 - w - 8],
[439, 439, -w^3 + 3*w^2 + 4*w - 7],
[443, 443, -2*w^3 + w^2 + 10*w - 2],
[449, 449, w^2 + 2*w - 4],
[449, 449, 2*w^3 - 2*w^2 - 10*w - 1],
[461, 461, 2*w^3 - 2*w^2 - 6*w + 5],
[461, 461, -2*w^3 + 4*w^2 + 7*w - 13],
[463, 463, -3*w^3 + 3*w^2 + 12*w - 4],
[463, 463, w - 5],
[467, 467, -3*w^3 + w^2 + 16*w + 8],
[467, 467, 2*w^2 - 3*w - 10],
[479, 479, 2*w^2 - 4*w - 1],
[479, 479, -w^3 + 2*w^2 + 2*w - 7],
[487, 487, -3*w^3 + 5*w^2 + 12*w - 10],
[503, 503, w^3 + 2*w^2 - 8*w - 8],
[509, 509, -3*w^3 + 5*w^2 + 11*w - 5],
[521, 521, 2*w^2 - 3*w - 11],
[523, 523, -2*w^3 - w^2 + 13*w + 11],
[523, 523, -2*w^3 + 4*w^2 + 8*w - 7],
[541, 541, 3*w^3 - 2*w^2 - 13*w + 1],
[563, 563, w^3 - 3*w - 5],
[563, 563, -w^3 + 3*w^2 + 2*w - 11],
[569, 569, -2*w^3 + 4*w^2 + 9*w - 4],
[569, 569, 3*w^2 - w - 14],
[571, 571, -w^3 + 8*w - 1],
[577, 577, 3*w^3 - 2*w^2 - 16*w + 1],
[587, 587, -4*w^3 + 5*w^2 + 18*w - 7],
[587, 587, 2*w^3 + w^2 - 10*w - 7],
[599, 599, 2*w^3 + w^2 - 11*w - 8],
[607, 607, w^3 - 3*w + 4],
[613, 613, 3*w^3 - w^2 - 14*w - 5],
[613, 613, 3*w^2 - w - 11],
[619, 619, 2*w^2 - 5*w - 8],
[631, 631, -2*w^3 - 2*w^2 + 12*w + 19],
[647, 647, 2*w^3 - 2*w^2 - 11*w - 2],
[653, 653, 3*w^3 - 5*w^2 - 8*w + 7],
[677, 677, -2*w^3 + 4*w^2 + 5*w - 8],
[683, 683, 2*w^3 + w^2 - 12*w - 8],
[683, 683, -w^3 + 3*w^2 - 8],
[683, 683, -2*w^3 + 3*w^2 + 11*w - 8],
[683, 683, -2*w^3 + 3*w^2 + 10*w - 2],
[691, 691, w^2 - 4*w - 1],
[709, 709, 3*w^3 - 2*w^2 - 13*w - 2],
[709, 709, w^3 - 2*w^2 - 3*w - 2],
[727, 727, -w^3 + 4*w^2 + 2*w - 7],
[727, 727, -w^3 + 8*w + 5],
[733, 733, 3*w^2 - 2*w - 10],
[733, 733, -w^3 + 5*w + 8],
[757, 757, 3*w^3 - 14*w - 8],
[761, 761, -w^3 + 3*w^2 + 5*w - 10],
[773, 773, -2*w^3 + 3*w^2 + 9*w - 1],
[787, 787, -w^3 + 4*w - 4],
[797, 797, w^2 + 2*w - 5],
[797, 797, w^3 + w^2 - 5*w - 1],
[809, 809, w^3 + 3*w^2 - 9*w - 17],
[811, 811, -4*w^2 + 3*w + 17],
[811, 811, w^3 - 2*w^2 - 8*w + 2],
[823, 823, -4*w^3 + 4*w^2 + 17*w - 4],
[857, 857, 3*w^2 - w - 16],
[881, 881, -2*w^3 + 9*w + 1],
[883, 883, w^3 - w^2 - 4*w - 5],
[887, 887, 3*w^2 + w - 10],
[911, 911, w^3 - 3*w - 7],
[919, 919, -w^3 + w^2 + 3*w - 7],
[919, 919, 3*w^3 - 2*w^2 - 12*w + 2],
[929, 929, 3*w^3 - w^2 - 16*w - 2],
[929, 929, w^3 + w^2 - 6*w - 2],
[941, 941, -3*w^3 + 5*w^2 + 14*w - 14],
[941, 941, -w^3 + 2*w^2 + 6*w - 8],
[947, 947, 3*w^3 - 3*w^2 - 13*w - 5],
[947, 947, 3*w^3 - 4*w^2 - 12*w + 2],
[953, 953, -3*w^3 + 5*w^2 + 10*w - 10],
[961, 31, -w^3 + 4*w^2 + 3*w - 13],
[961, 31, -4*w^3 + 3*w^2 + 17*w - 4],
[967, 967, -w^3 - 4*w^2 + 9*w + 19],
[967, 967, w^3 + 5*w^2 - 9*w - 26],
[971, 971, -3*w^3 + 6*w^2 + 13*w - 7],
[971, 971, w^3 + 2*w^2 - 7*w - 7],
[977, 977, 2*w^3 - 3*w^2 - 4*w - 2],
[983, 983, 3*w^3 - 5*w^2 - 7*w + 7],
[991, 991, 2*w^2 - w - 13],
[991, 991, w^3 - 8*w - 4]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^4 - 20*x^2 + 20;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [1/8*e^2 - 7/4, 0, -3/8*e^2 + 9/4, e, 1/8*e^2 - 15/4, 1/4*e^2 - 3/2, -e, 1/2*e^3 - 10*e, 1/8*e^2 - 11/4, -1/4*e^3 + 11/2*e, -3/8*e^3 + 17/4*e, 1/2*e^3 - 8*e, -3/8*e^3 + 33/4*e, 1/8*e^3 + 1/4*e, 1/8*e^3 - 19/4*e, -3/8*e^2 - 7/4, -3/4*e^3 + 27/2*e, 7/8*e^2 - 45/4, -1/4*e^2 - 1/2, -3/8*e^2 + 13/4, 1/8*e^2 - 7/4, -1/8*e^3 + 3/4*e, -3/4*e^3 + 25/2*e, 3/4*e^3 - 35/2*e, 1/2*e^3 - 6*e, 13/8*e^2 - 59/4, -1/8*e^2 - 49/4, -3/8*e^3 + 25/4*e, -1/8*e^3 + 23/4*e, -1/8*e^2 - 81/4, 5/8*e^2 + 45/4, -1/2*e^2 + 15, -9/8*e^3 + 79/4*e, 1/8*e^3 - 3/4*e, -1/8*e^3 + 27/4*e, -3/4*e^2 + 21/2, -5/8*e^3 + 51/4*e, 3/4*e^3 - 25/2*e, -9/8*e^2 - 5/4, 3/8*e^2 - 25/4, -3/8*e^3 + 29/4*e, -1/8*e^2 + 63/4, -23/8*e^2 + 121/4, 7/8*e^2 - 53/4, 13/8*e^2 - 47/4, -1/8*e^2 - 13/4, 19/8*e^2 - 117/4, 3/8*e^2 + 11/4, 5/8*e^3 - 59/4*e, 3/4*e^3 - 41/2*e, -11/8*e^3 + 85/4*e, 1/4*e^3 - 1/2*e, 3/8*e^3 - 53/4*e, -13/8*e^2 + 31/4, -e^3 + 16*e, 3/2*e^3 - 25*e, -3/8*e^2 - 43/4, -9/8*e^3 + 59/4*e, 15/8*e^2 - 49/4, -11/8*e^2 + 45/4, 7/8*e^3 - 57/4*e, 7/8*e^2 - 17/4, -3/4*e^2 - 35/2, -7/8*e^2 + 101/4, -11/8*e^2 + 113/4, 6*e, 1/4*e^3 - 23/2*e, -1/8*e^2 - 13/4, 1/8*e^2 - 3/4, 9/8*e^3 - 59/4*e, 2*e^3 - 34*e, -1/8*e^2 + 39/4, 7/8*e^2 - 85/4, 3/8*e^2 - 49/4, -21/8*e^2 + 87/4, 3/8*e^2 - 77/4, e^2 - 32, -9/8*e^2 - 13/4, -1/2*e^3 + 11*e, 1/8*e^2 - 19/4, -3/8*e^3 + 1/4*e, 13/8*e^3 - 115/4*e, -9/8*e^3 + 71/4*e, e^3 - 14*e, 3/8*e^2 - 21/4, -5/8*e^2 - 81/4, -3/8*e^2 - 123/4, 13/8*e^2 - 47/4, 1/8*e^3 + 21/4*e, -1/4*e^3 - 1/2*e, -1/8*e^3 + 15/4*e, 3/8*e^3 - 17/4*e, 1/8*e^3 + 1/4*e, 17/8*e^2 - 115/4, -3/8*e^2 + 5/4, -1/8*e^3 - 9/4*e, -11/8*e^3 + 109/4*e, 5/8*e^3 - 55/4*e, 5/4*e^3 - 61/2*e, -7/8*e^2 - 19/4, -3/2*e^3 + 30*e, -11/8*e^3 + 73/4*e, 1/2*e^3 - 10*e, -13/8*e^2 + 15/4, 17/8*e^2 - 127/4, -7/8*e^2 - 47/4, -3/4*e^3 + 23/2*e, 11/4*e^3 - 91/2*e, -13/8*e^3 + 99/4*e, -11/8*e^3 + 141/4*e, -13/8*e^2 + 31/4, -13/8*e^2 - 21/4, -19/8*e^3 + 161/4*e, -3/4*e^3 + 19/2*e, 13/4*e^3 - 101/2*e, 21/8*e^2 - 111/4, 3/8*e^2 - 53/4, -31/8*e^2 + 173/4, -15/8*e^2 + 113/4, -7/8*e^2 + 1/4, -e^3 + 14*e, e^3 - 9*e, -21/8*e^3 + 175/4*e, 17/8*e^3 - 171/4*e, 5/4*e^3 - 43/2*e, 5/8*e^3 - 39/4*e, -11/4*e^3 + 89/2*e, -7/8*e^2 - 91/4, -11/4*e^2 + 101/2, -17/8*e^2 + 91/4, 19/4*e^2 - 77/2, 19/8*e^2 - 89/4, -25/8*e^2 + 203/4, -17/8*e^2 - 13/4, 1/8*e^2 + 101/4, 9/8*e^3 - 87/4*e, -7/8*e^3 + 89/4*e, -21/8*e^2 + 171/4, 1/2*e^3 - 11*e, -1/8*e^3 + 51/4*e, e^3 - 27*e, -13/8*e^2 + 87/4, 1/4*e^2 + 1/2, -25/8*e^2 + 23/4, 23/8*e^3 - 225/4*e, -7/4*e^3 + 61/2*e, -3*e^2, 15/8*e^3 - 109/4*e, 1/8*e^3 + 9/4*e, -3/8*e^2 + 117/4, -19/8*e^2 + 61/4, -7/8*e^3 + 97/4*e, -23/8*e^3 + 221/4*e, 11/4*e^3 - 83/2*e, 7/8*e^3 - 29/4*e, -3/4*e^3 + 19/2*e, 1/4*e^3 + 1/2*e, -1/8*e^3 - 1/4*e, 5/8*e^2 + 21/4, 13/8*e^2 - 59/4, 9/4*e^2 - 23/2, -29/8*e^2 + 119/4, 3/8*e^3 - 29/4*e, -9/4*e^3 + 67/2*e, 2*e^3 - 34*e, -21/8*e^3 + 163/4*e, -3/4*e^2 + 49/2, 7/8*e^2 + 31/4];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {5, 5, 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;