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

NN := ideal<ZF | {9, 3, -w^2 + 2*w + 4}>;

primesArray := [
[3, 3, w + 1],
[7, 7, w - 1],
[8, 2, 2],
[9, 3, -w^2 + 2*w + 4],
[11, 11, -w^2 + 2*w + 2],
[13, 13, -w^2 + w + 4],
[19, 19, w + 3],
[19, 19, -w^2 + 2*w + 5],
[19, 19, -w^2 + 3*w + 2],
[23, 23, w^2 - w - 3],
[23, 23, -w^2 + 2],
[23, 23, -w + 4],
[31, 31, w^2 - 5],
[43, 43, w^2 - 3*w - 3],
[49, 7, w^2 - 6],
[53, 53, 2*w - 5],
[61, 61, 2*w^2 - 2*w - 9],
[71, 71, 2*w - 3],
[73, 73, 2*w^2 - 5*w - 5],
[83, 83, w^2 - w - 9],
[97, 97, 2*w^2 - 3*w - 7],
[103, 103, w^2 - 3*w - 6],
[109, 109, w^2 - 4*w - 3],
[121, 11, 2*w^2 - w - 7],
[125, 5, -5],
[127, 127, w^2 + w - 5],
[131, 131, w^2 - 11],
[137, 137, 3*w^2 - 7*w - 8],
[137, 137, 2*w^2 - 5*w - 6],
[137, 137, -w^2 + w - 2],
[139, 139, 3*w^2 - 2*w - 16],
[139, 139, 3*w - 2],
[139, 139, -w^2 + 3*w - 3],
[151, 151, w^2 + w - 11],
[157, 157, 2*w^2 - 3*w - 3],
[163, 163, w^2 - w - 10],
[169, 13, 2*w^2 - 3*w - 4],
[173, 173, -3*w^2 + 6*w + 8],
[191, 191, -4*w^2 + 7*w + 15],
[193, 193, 2*w^2 - 3],
[197, 197, w^2 - 3*w - 11],
[199, 199, 3*w^2 - 5*w - 16],
[211, 211, w^2 - 4*w - 9],
[223, 223, 2*w^2 - w - 4],
[227, 227, 3*w - 4],
[229, 229, w^2 + w - 8],
[241, 241, w^2 - 4*w - 6],
[251, 251, w - 7],
[257, 257, w^2 - 4*w - 7],
[263, 263, 3*w^2 - 5*w - 10],
[263, 263, 2*w^2 - 5*w - 15],
[263, 263, -w^2 + 6*w - 1],
[269, 269, -w^2 - w + 14],
[277, 277, 2*w^2 - 5*w - 8],
[277, 277, w^2 + 3*w - 3],
[277, 277, 3*w^2 - 4*w - 12],
[281, 281, -2*w - 7],
[293, 293, 3*w^2 - 2*w - 12],
[307, 307, 2*w^2 + w - 8],
[313, 313, w^2 + 4*w - 2],
[317, 317, 3*w^2 - 6*w - 5],
[331, 331, 4*w^2 - 4*w - 19],
[337, 337, -5*w^2 + 8*w + 21],
[347, 347, 3*w^2 - 6*w - 13],
[347, 347, 3*w^2 - 5*w - 22],
[347, 347, 3*w^2 - 5*w - 9],
[349, 349, w^2 - 5*w - 11],
[353, 353, -w^2 + 6*w - 7],
[359, 359, -4*w^2 + 8*w + 11],
[367, 367, 3*w^2 - 4*w - 11],
[373, 373, w^2 + 2*w - 6],
[379, 379, 5*w^2 - 9*w - 18],
[409, 409, -w^2 - 4],
[439, 439, 3*w^2 - 6*w - 14],
[443, 443, -w^2 - 2*w + 13],
[443, 443, 2*w^2 - 15],
[443, 443, 3*w^2 - 13],
[449, 449, w^2 + 2*w - 7],
[449, 449, 2*w^2 - 6*w - 7],
[449, 449, 3*w^2 - 3*w - 11],
[461, 461, -w^2 - 2*w - 5],
[467, 467, 3*w^2 - 5*w - 7],
[479, 479, -w^2 + 6*w - 4],
[491, 491, 3*w^2 - 5*w - 6],
[499, 499, w^2 - 5*w - 8],
[509, 509, 3*w^2 - 6*w - 19],
[521, 521, 3*w^2 - 2*w - 10],
[523, 523, 4*w - 9],
[541, 541, 3*w^2 - 3*w - 10],
[547, 547, 2*w^2 + w - 20],
[547, 547, w^2 - 6*w - 5],
[547, 547, 6*w^2 - 11*w - 24],
[557, 557, -5*w^2 + 11*w + 11],
[563, 563, 3*w^2 - 6*w - 16],
[569, 569, 4*w^2 - 5*w - 17],
[587, 587, 5*w^2 - 8*w - 20],
[593, 593, 5*w - 3],
[593, 593, 3*w^2 - w - 17],
[593, 593, w - 9],
[613, 613, 2*w^2 - w - 19],
[613, 613, 3*w^2 - 4*w - 5],
[613, 613, 3*w^2 - w - 8],
[617, 617, 3*w^2 - 2*w - 9],
[617, 617, -w^2 + w - 5],
[617, 617, 4*w^2 - 6*w - 15],
[619, 619, -w^2 + 3*w - 6],
[631, 631, 4*w^2 - 7*w - 21],
[641, 641, -w^2 - 3*w + 20],
[643, 643, 3*w^2 - 14],
[643, 643, w^2 - w - 13],
[643, 643, 2*w^2 - 9*w + 2],
[647, 647, 2*w^2 - 6*w - 9],
[653, 653, 3*w^2 - 5],
[659, 659, 6*w^2 - 11*w - 21],
[661, 661, w^2 - 6*w - 6],
[661, 661, w^2 + 3*w - 6],
[661, 661, 5*w^2 - 7*w - 22],
[673, 673, w^2 - 6*w - 12],
[677, 677, 4*w^2 - 7*w - 22],
[677, 677, 3*w^2 - 7*w - 12],
[677, 677, -2*w^2 - 3],
[683, 683, -2*w - 9],
[683, 683, 4*w^2 - 7*w - 12],
[683, 683, 3*w^2 - 3*w - 8],
[691, 691, 2*w^2 + w - 11],
[701, 701, 3*w^2 - 3*w - 5],
[709, 709, -w^2 - 2*w - 6],
[727, 727, 3*w^2 - w - 6],
[727, 727, 3*w^2 - w - 27],
[727, 727, 3*w^2 - w - 5],
[733, 733, -2*w^2 + 9*w - 5],
[739, 739, 3*w^2 - 4*w - 23],
[743, 743, 2*w^2 - 7*w - 7],
[743, 743, 3*w^2 - w - 20],
[743, 743, 3*w^2 - 2*w - 7],
[757, 757, -w - 9],
[757, 757, 4*w^2 - 2*w - 29],
[757, 757, 2*w^2 - 3*w - 18],
[761, 761, 3*w^2 - 2*w - 6],
[761, 761, 3*w^2 - 2*w - 25],
[773, 773, -5*w - 12],
[787, 787, -3*w - 10],
[809, 809, 4*w^2 - w - 19],
[821, 821, 5*w^2 - 10*w - 19],
[823, 823, -4*w - 11],
[827, 827, -7*w - 5],
[827, 827, 2*w^2 - 6*w - 13],
[827, 827, -w^2 - 2*w + 18],
[839, 839, 6*w^2 - 9*w - 26],
[839, 839, 5*w^2 - 12*w - 13],
[839, 839, w - 10],
[857, 857, -w^2 + 4*w - 8],
[859, 859, 3*w^2 - 8*w - 10],
[859, 859, 4*w^2 - 9*w - 14],
[859, 859, 3*w^2 - 26],
[863, 863, 2*w^2 + 3*w - 7],
[877, 877, -5*w^2 + 11*w + 9],
[877, 877, -5*w^2 + 10*w + 13],
[877, 877, -w^2 + w - 6],
[881, 881, w^2 - 3*w - 14],
[883, 883, -w^2 + 3*w - 7],
[887, 887, -3*w^2 + 6*w - 2],
[907, 907, w^2 + 3*w - 8],
[911, 911, 4*w^2 - 6*w - 29],
[937, 937, -6*w^2 + 11*w + 25],
[953, 953, 6*w^2 - 6*w - 29],
[961, 31, -w^2 + 7*w - 5],
[967, 967, 2*w^2 + w - 14],
[971, 971, 5*w^2 - 3*w - 27],
[971, 971, 4*w - 15],
[971, 971, w^2 - 2*w - 14],
[977, 977, 3*w^2 - 5*w - 24],
[983, 983, w^2 - 4*w - 15],
[991, 991, 6*w^2 - 10*w - 23],
[991, 991, 5*w^2 - 6*w - 22],
[991, 991, 3*w^2 + w - 13]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^3 - 2*x^2 - 4*x + 4;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, 1/2*e^2, -e^2 + e + 1, 1, e^2 - e, 1/2*e^2 - e, -1/2*e^2 + 2*e, -e^2 + 6, -2*e^2 + e + 8, -1/2*e^2 + 3*e, 1/2*e^2 - 3*e - 2, -e - 2, -2*e, -5/2*e^2 + e + 10, -1/2*e^2 - 4*e + 4, 1/2*e^2 - 4, e^2 + 6, -2*e^2 + 4, e^2 - e + 2, -e^2 + 2*e + 2, -e^2 + 4*e + 8, -1/2*e^2 - e + 6, e^2 - e - 2, -e^2 + e - 6, 3*e^2 - 5*e - 12, -e^2 + 4*e + 8, -3/2*e^2 + 6*e + 12, 1/2*e^2 + e - 4, -e^2 - 5*e + 14, 3*e^2 - 10*e - 10, 3*e^2 - 18, 3*e^2 - 6*e - 4, -4*e^2 - e + 18, -5/2*e^2 + 6*e + 18, 4*e^2 - 8*e - 14, -3*e^2 - 2*e + 20, -5/2*e^2 + 11*e + 6, -e^2 + e + 6, 5*e^2 - 8*e - 16, 2*e^2 + 5*e - 18, -11/2*e^2 + 10*e + 6, 3*e^2 - 2*e - 10, -5*e^2 + e + 28, 2*e^2 - 3*e - 16, e^2 - 8*e - 6, e^2 - 8*e + 6, 3*e^2 - 9*e - 10, -e^2 + 9*e + 2, 5*e - 2, -2*e^2 - 3*e + 8, e^2 - 7*e - 4, -5*e^2 + 14*e + 8, 7*e^2 - 10*e - 22, e^2 + 3*e + 10, 7/2*e^2 - 10*e - 18, e^2 + 12, 5*e^2 - 8*e - 26, 6*e - 20, 3*e^2 - 10*e, -3/2*e^2 - 2*e + 26, 7*e^2 - 9*e - 18, e^2 + 6*e + 2, 7/2*e^2 - 7*e - 10, -e^2 + 8*e + 12, 5*e^2 + 6*e - 36, 13/2*e^2 - 9*e - 20, 1/2*e^2 + 9*e - 4, -7*e^2 + 5*e + 16, 2*e^2 + e - 16, -7*e^2 + 13*e + 28, 4*e^2 - 4*e - 14, -4*e^2 + 4*e + 8, 6*e^2 - 16*e - 12, 9/2*e^2 - e - 8, 6*e^2 - 16*e - 8, -9/2*e^2 + 4*e, 3/2*e^2 + 2*e - 8, -10*e^2 + 11*e + 16, -5*e^2 + 6*e + 26, -4*e^2 - 2*e + 22, -7*e^2 + 8*e + 12, 3*e^2 - 2*e + 8, 9*e^2 - 13*e - 22, -17/2*e^2 + 9*e + 12, 21/2*e^2 - 9*e - 42, -4*e^2 + 4*e + 8, -e^2 + 3*e + 26, -7*e^2 + 21*e + 22, 5*e^2 - 14*e - 2, -e^2 - 12*e + 12, 2*e^2 + 5*e - 4, 3*e^2 - 13*e - 8, 9/2*e^2 + 9*e - 40, 10*e^2 - 18*e - 24, 15*e^2 - 19*e - 34, 15*e^2 - 16*e - 38, 6*e^2 - 4*e - 26, -7/2*e^2 + 8*e + 8, -11/2*e^2 + 4*e + 26, 18*e - 18, -2*e^2 + 3*e - 6, -5*e^2 + 13*e + 22, 11*e^2 - 12*e - 30, -11/2*e^2 + 8*e + 34, -8*e^2 + 3*e + 34, 7*e^2 - 12, 8*e^2 - 13*e - 24, -19/2*e^2 - 4*e + 50, 19/2*e^2 - 4*e - 32, -2*e^2 + 6*e + 22, -3*e^2 - 6*e + 16, -4*e^2 + 4*e + 26, -9*e^2 + 5*e + 38, -3/2*e^2 - e - 12, 1/2*e^2 - 11*e + 8, -3*e^2 + 16*e + 16, 2*e^2 - 7*e + 18, -4*e - 20, -9*e^2 + 12*e + 14, -2*e^2 - 7*e + 18, e^2 - 10*e - 18, 9/2*e^2 - 2*e - 10, 3*e^2 - 14, 5*e^2 - 13*e - 4, 5*e^2 - 10*e + 12, 4*e^2 - 17*e - 14, 5/2*e^2 + 8*e - 12, 1/2*e^2 + 9*e - 6, 7*e^2 - 2*e - 6, -2*e^2 - 5*e, 10*e + 14, -7*e^2 + 22*e + 4, -7*e^2 - 2*e + 54, 7*e^2 - 10*e - 8, -9*e^2 + 12*e + 40, 15*e^2 - 24*e - 36, 5/2*e^2 - 14*e + 14, -2*e^2 - 10, 7/2*e^2 - 7*e - 6, 8*e^2 - 8*e - 14, -4*e^2 + 10*e + 18, -9*e^2 + 15*e + 2, 8*e - 22, -6*e - 4, 4*e^2 + 4*e - 16, 9/2*e^2 - 9*e - 8, 8*e^2 + 9*e - 56, 15/2*e^2 + 8*e - 54, 19*e - 16, 5*e, 4*e^2 + e + 2, 4*e^2 + 16*e - 30, -9/2*e^2 + 17*e + 4, -7*e^2 + 12*e + 4, 9/2*e^2 - 4*e - 26, -7*e^2 + 8*e + 4, -10*e^2 + 16*e + 14, e^2 - 2*e - 10, -6*e^2 - 4*e + 20, 19/2*e^2 - 24*e - 34, 8*e^2 - 6*e - 40, -3*e^2 + 8*e - 16, -9*e^2 + 26*e + 28, -7/2*e^2 + 4*e - 8, -5/2*e^2 + 9*e + 22, -e^2 + 21*e - 14, -2*e^2 - 10*e - 6, 2*e + 40, -7/2*e^2 - 6*e - 8, -18*e^2 + 18*e + 48, -8*e + 4, -10*e^2 + 8*e + 56, -1/2*e^2 - 4*e + 18, 4*e^2 + 10*e - 32, -17/2*e^2 - 7*e + 56, -19/2*e^2 + 6*e + 20];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {9, 3, -w^2 + 2*w + 4}>] := -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;