/*
  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![2, 1, -7, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {16, 4, w^2 - 2*w - 3}>;

primesArray := [
[2, 2, w],
[2, 2, -1/2*w^3 + w^2 + 3/2*w],
[11, 11, 1/2*w^3 - w^2 - 5/2*w + 4],
[11, 11, -w^3 + w^2 + 6*w + 1],
[13, 13, -w^3 + w^2 + 6*w - 3],
[19, 19, -1/2*w^3 + w^2 + 5/2*w],
[23, 23, -1/2*w^3 + 2*w^2 - 1/2*w - 2],
[31, 31, -w^3 + w^2 + 6*w - 1],
[31, 31, 1/2*w^3 - 7/2*w],
[43, 43, 1/2*w^3 - w^2 - 9/2*w + 2],
[67, 67, w^2 - w - 5],
[67, 67, -1/2*w^3 + 11/2*w + 2],
[73, 73, w^2 + w + 1],
[79, 79, 1/2*w^3 + w^2 - 5/2*w - 2],
[81, 3, -3],
[83, 83, 2*w^3 - 2*w^2 - 12*w + 5],
[83, 83, -1/2*w^3 - w^2 + 1/2*w + 2],
[89, 89, -3/2*w^3 + 2*w^2 + 17/2*w - 2],
[89, 89, 3/2*w^3 - 2*w^2 - 21/2*w + 2],
[97, 97, 1/2*w^3 - w^2 - 1/2*w - 2],
[97, 97, -w^3 + w^2 + 8*w + 1],
[103, 103, 2*w^3 - 2*w^2 - 14*w + 1],
[103, 103, w^2 - w - 3],
[107, 107, -3/2*w^3 + 2*w^2 + 21/2*w - 4],
[109, 109, 5/2*w^3 - 3*w^2 - 33/2*w + 4],
[109, 109, -1/2*w^3 + w^2 + 9/2*w - 4],
[113, 113, -w^3 + w^2 + 8*w - 3],
[113, 113, 1/2*w^3 - w^2 - 1/2*w + 2],
[121, 11, 5/2*w^3 - 2*w^2 - 31/2*w + 2],
[127, 127, -3/2*w^3 + w^2 + 23/2*w],
[127, 127, -1/2*w^3 + 11/2*w],
[131, 131, w^2 - 3*w - 3],
[137, 137, 1/2*w^3 - w^2 - 5/2*w + 6],
[139, 139, -3/2*w^3 + 2*w^2 + 13/2*w + 2],
[151, 151, -1/2*w^3 + 2*w^2 + 3/2*w - 10],
[157, 157, 7/2*w^3 - 5*w^2 - 47/2*w + 10],
[157, 157, 7/2*w^3 - 4*w^2 - 45/2*w + 4],
[167, 167, 1/2*w^3 + w^2 - 5/2*w - 4],
[167, 167, 2*w + 5],
[173, 173, -w^3 + w^2 + 4*w - 7],
[181, 181, w^3 - 2*w^2 - 9*w + 1],
[193, 193, -1/2*w^3 + 2*w^2 + 3/2*w - 8],
[197, 197, -1/2*w^3 - w^2 + 13/2*w + 2],
[197, 197, -w^3 + w^2 + 4*w - 1],
[197, 197, w^3 - 2*w^2 - 7*w + 7],
[197, 197, -2*w^3 + 3*w^2 + 13*w - 11],
[199, 199, -2*w^3 + 3*w^2 + 11*w - 5],
[239, 239, -w^3 + 2*w^2 + 5*w - 1],
[239, 239, 7/2*w^3 - 4*w^2 - 45/2*w + 6],
[241, 241, 1/2*w^3 - 2*w^2 - 7/2*w + 8],
[257, 257, 3/2*w^3 - 2*w^2 - 17/2*w - 4],
[257, 257, 1/2*w^3 - 7/2*w - 6],
[257, 257, 7/2*w^3 - 5*w^2 - 43/2*w + 10],
[257, 257, 3*w^3 - 4*w^2 - 19*w + 7],
[269, 269, 1/2*w^3 + w^2 + 3/2*w + 2],
[269, 269, -2*w^3 + 3*w^2 + 11*w - 11],
[277, 277, 5/2*w^3 - 4*w^2 - 35/2*w + 8],
[281, 281, -3/2*w^3 + w^2 + 23/2*w - 2],
[283, 283, -w^3 - w^2 + 8*w + 11],
[283, 283, -3*w^2 + 3*w + 19],
[311, 311, 3/2*w^3 - 2*w^2 - 17/2*w - 2],
[331, 331, -4*w^3 + 7*w^2 + 25*w - 21],
[337, 337, -5/2*w^3 + 2*w^2 + 35/2*w + 4],
[337, 337, -2*w^3 + 3*w^2 + 11*w - 9],
[347, 347, 2*w^3 - 4*w^2 - 12*w + 13],
[347, 347, 5/2*w^3 - 5*w^2 - 29/2*w + 20],
[347, 347, -1/2*w^3 - 1/2*w - 6],
[347, 347, -1/2*w^3 + 3*w^2 + 1/2*w - 16],
[349, 349, 1/2*w^3 + w^2 - 9/2*w - 4],
[353, 353, -1/2*w^3 + 2*w^2 + 7/2*w],
[359, 359, 3/2*w^3 - w^2 - 19/2*w + 2],
[373, 373, -2*w^3 + 2*w^2 + 12*w + 1],
[373, 373, -1/2*w^3 + w^2 + 9/2*w - 6],
[379, 379, w^2 - w - 9],
[379, 379, -2*w^3 + 3*w^2 + 13*w - 5],
[383, 383, w^2 + w - 5],
[389, 389, 9/2*w^3 - 6*w^2 - 59/2*w + 14],
[397, 397, 2*w^2 - 2*w - 15],
[401, 401, -1/2*w^3 + 2*w^2 + 3/2*w - 4],
[419, 419, -3/2*w^3 + w^2 + 15/2*w - 2],
[419, 419, w^3 + w^2 - 8*w - 15],
[431, 431, -5/2*w^3 + 4*w^2 + 27/2*w - 16],
[431, 431, 2*w^3 - 4*w^2 - 10*w + 11],
[443, 443, 1/2*w^3 - 3*w^2 - 5/2*w + 14],
[449, 449, 1/2*w^3 + w^2 - 9/2*w - 6],
[457, 457, -w^3 + 2*w^2 + 7*w - 9],
[461, 461, 4*w^3 - 5*w^2 - 27*w + 7],
[487, 487, 2*w^2 - 9],
[499, 499, 3/2*w^3 - w^2 - 15/2*w - 2],
[499, 499, 3/2*w^3 - 4*w^2 - 17/2*w + 16],
[503, 503, -5/2*w^3 + 3*w^2 + 37/2*w],
[509, 509, -3*w^3 + 3*w^2 + 18*w + 7],
[521, 521, -1/2*w^3 - 2*w^2 + 11/2*w + 18],
[521, 521, -5/2*w^3 + 4*w^2 + 31/2*w - 8],
[523, 523, -1/2*w^3 + 2*w^2 + 11/2*w],
[523, 523, -1/2*w^3 - w^2 + 9/2*w],
[547, 547, -7/2*w^3 + 6*w^2 + 45/2*w - 18],
[547, 547, -w^3 - w^2 + 12*w + 5],
[547, 547, 3/2*w^3 - w^2 - 19/2*w + 4],
[547, 547, w^3 - w^2 - 6*w - 5],
[557, 557, 1/2*w^3 + w^2 - 9/2*w - 12],
[557, 557, 3/2*w^3 - 2*w^2 - 21/2*w],
[557, 557, 2*w^3 - 2*w^2 - 12*w + 1],
[557, 557, w^3 - 3*w^2 - 6*w + 11],
[569, 569, w^3 - 2*w^2 - 7*w + 1],
[577, 577, w^3 + w^2 - 6*w - 5],
[599, 599, -3/2*w^3 + 3*w^2 + 15/2*w - 10],
[607, 607, -5/2*w^3 + 3*w^2 + 25/2*w + 4],
[613, 613, w^3 - 9*w + 1],
[619, 619, 3*w^2 - w - 17],
[619, 619, w^3 - 3*w^2 - 6*w + 13],
[625, 5, -5],
[631, 631, 11/2*w^3 - 7*w^2 - 71/2*w + 16],
[631, 631, -3/2*w^3 + 2*w^2 + 13/2*w],
[643, 643, w^2 - 3*w - 5],
[653, 653, 2*w^2 - 2*w - 11],
[659, 659, 5*w^3 - 3*w^2 - 36*w - 9],
[659, 659, -w^3 + 7*w - 1],
[673, 673, 5/2*w^3 - 2*w^2 - 31/2*w - 4],
[677, 677, 13/2*w^3 - 6*w^2 - 87/2*w + 2],
[691, 691, -3/2*w^3 + 2*w^2 + 13/2*w - 2],
[701, 701, -w^3 + 3*w^2 + 4*w - 13],
[701, 701, w^3 - 2*w^2 - 9*w + 3],
[709, 709, -w^2 - 5*w - 1],
[739, 739, 3/2*w^3 - w^2 - 15/2*w],
[743, 743, -2*w^3 + w^2 + 15*w + 1],
[751, 751, -2*w^3 + 4*w^2 + 12*w - 11],
[757, 757, -6*w^3 + 9*w^2 + 39*w - 23],
[757, 757, -w^3 - w^2 + 8*w + 3],
[757, 757, 13/2*w^3 - 8*w^2 - 83/2*w + 16],
[757, 757, 3*w^3 - 4*w^2 - 17*w + 3],
[761, 761, 7/2*w^3 - 5*w^2 - 47/2*w + 12],
[769, 769, -3*w^2 + 7*w + 3],
[773, 773, -3/2*w^3 - w^2 + 19/2*w + 4],
[787, 787, -4*w^3 + 5*w^2 + 23*w - 17],
[787, 787, 1/2*w^3 - w^2 - 9/2*w - 4],
[809, 809, -2*w^3 + 2*w^2 + 16*w + 3],
[809, 809, -4*w^3 + 5*w^2 + 25*w - 9],
[811, 811, 3*w^3 - 3*w^2 - 20*w + 5],
[811, 811, w^3 - 3*w^2 - 6*w + 17],
[823, 823, 5/2*w^3 - w^2 - 33/2*w - 10],
[829, 829, 5/2*w^3 - 3*w^2 - 33/2*w + 2],
[829, 829, -3*w^3 + 2*w^2 + 23*w - 1],
[839, 839, -11/2*w^3 + 7*w^2 + 67/2*w - 18],
[853, 853, -w^3 + 2*w^2 + 3*w - 5],
[853, 853, -5*w^3 + 7*w^2 + 32*w - 15],
[857, 857, 3/2*w^3 - 5*w^2 - 15/2*w + 26],
[859, 859, -w^3 + 3*w^2 + 4*w - 17],
[863, 863, 4*w^3 - 4*w^2 - 26*w + 3],
[877, 877, w^3 - 5*w - 5],
[887, 887, 3/2*w^3 - w^2 - 11/2*w + 8],
[907, 907, -9/2*w^3 + 7*w^2 + 53/2*w - 24],
[919, 919, 7/2*w^3 - 6*w^2 - 41/2*w + 22],
[929, 929, -2*w^3 + 3*w^2 + 11*w - 1],
[937, 937, 2*w^3 - 4*w^2 - 12*w + 7],
[947, 947, 9/2*w^3 - 5*w^2 - 61/2*w + 10],
[953, 953, -1/2*w^3 + 2*w^2 + 3/2*w + 2],
[961, 31, -1/2*w^3 + 2*w^2 - 1/2*w - 6],
[977, 977, 5/2*w^3 - 2*w^2 - 31/2*w],
[983, 983, -5/2*w^3 + 6*w^2 + 27/2*w - 24],
[983, 983, -3/2*w^3 + 2*w^2 + 21/2*w - 10],
[983, 983, -w^3 + 4*w^2 + 7*w - 15],
[983, 983, -1/2*w^3 + 3*w^2 - 3/2*w - 4],
[997, 997, 1/2*w^3 - w^2 - 5/2*w - 4],
[997, 997, 1/2*w^3 + w^2 - 17/2*w - 2]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^2 + x - 1;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, 0, -1, -2*e, -2*e - 3, -1, -5, 4*e + 4, 2*e - 4, 2*e + 5, 6*e + 7, 2*e + 4, -2*e - 6, 10*e, 4*e + 1, 4*e + 11, -2*e - 7, -4*e - 8, -8*e, -14*e - 6, 6*e - 2, 8*e + 4, -4*e + 6, 2*e + 19, 10*e + 8, -4*e - 6, -4*e - 8, -10, -4*e - 4, -4*e + 3, -6*e + 6, -10*e - 7, 18*e + 9, 8*e + 5, -12*e - 13, -4, 4*e - 10, 12*e - 4, -12, -6*e + 16, -10*e - 5, -14*e - 8, -11, -12*e - 2, -8*e - 5, -12*e + 3, 8*e + 20, -4*e + 24, -14*e - 22, 14*e + 2, 2*e + 8, -2*e - 13, -13, 10*e - 6, 4*e - 7, -14*e + 4, -8*e - 14, 22*e + 17, 6*e + 14, 24*e + 15, -26*e - 13, 26*e + 20, 2*e + 2, -4*e - 14, -30, -16*e - 14, -2*e - 28, 24*e + 14, -2*e - 15, 24*e + 16, 4*e + 11, -10*e + 14, 8*e - 4, 4*e - 9, -4*e - 12, 12*e + 2, -14, -4*e + 16, 14*e + 22, 14*e, 20*e + 2, -14*e - 2, -12*e - 4, 22*e + 2, -10*e - 6, -6*e - 19, -12*e + 1, -16*e - 23, -16*e - 20, -26*e - 22, -6*e + 10, 20*e + 10, -6*e - 6, -8*e + 21, 4*e + 13, -22*e - 2, 2*e - 27, 14*e + 19, 20*e + 2, 18*e + 2, -22*e - 3, 18*e + 14, -6*e + 20, -30*e - 13, 16*e + 19, -18*e - 9, 14*e + 18, 4*e + 28, -16*e + 6, 16*e + 12, 4*e - 12, 14*e - 5, 4*e + 12, 10*e - 7, 30*e + 10, 22*e + 2, -16*e - 35, 12*e - 12, -36*e - 14, 4*e + 2, -10*e - 27, 2, -40*e - 22, -2*e - 34, 20*e - 8, 38*e + 13, -6*e - 6, -4*e - 16, 2*e - 17, -28*e - 26, -18*e + 10, 2*e - 3, 20*e - 14, -4*e + 9, -4*e - 24, 18*e + 44, -12*e - 38, -8*e - 6, 6*e - 3, -11, 14*e - 20, 28*e + 15, -36*e - 17, 26*e + 3, 8*e - 18, 28*e + 8, -28*e - 37, 4*e + 12, 14*e + 42, -26*e - 20, -12*e - 40, -16*e + 9, -6*e + 9, -4*e + 10, 6*e + 25, 2*e + 10, -6*e - 2, -6*e - 34, -18*e - 44, 14*e + 17, 6*e + 23, -10*e - 13, 32*e + 8, -16*e - 38, -4*e - 40];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {2, 2, -1/2*w^3 + w^2 + 3/2*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;