/*
  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![41, 13, -12, -2, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {19, 19, w + 1}>;

primesArray := [
[9, 3, -w^3 + 3*w^2 + 5*w - 15],
[9, 3, -w^3 + 8*w + 8],
[16, 2, 2],
[19, 19, w + 1],
[19, 19, -w^2 + 6],
[19, 19, -w^2 + 2*w + 5],
[19, 19, -w + 2],
[25, 5, 2*w^2 - 2*w - 13],
[29, 29, -w^2 + 9],
[29, 29, -w^2 + 2*w + 6],
[29, 29, w^2 - 7],
[29, 29, -w^2 + 2*w + 8],
[31, 31, -2*w^2 + w + 12],
[31, 31, 2*w^2 - 3*w - 11],
[41, 41, -w],
[41, 41, -w + 1],
[49, 7, w^3 + 2*w^2 - 10*w - 20],
[49, 7, w^3 - 5*w^2 - 3*w + 27],
[61, 61, 2*w^2 - 3*w - 14],
[61, 61, 2*w^2 - w - 15],
[71, 71, -w^3 + w^2 + 7*w - 5],
[71, 71, w^3 - 2*w^2 - 6*w + 2],
[79, 79, w^3 - 2*w^2 - 6*w + 4],
[79, 79, 3*w^2 - 2*w - 18],
[89, 89, 2*w^3 - 6*w^2 - 11*w + 33],
[89, 89, -4*w^2 + 3*w + 25],
[101, 101, w^3 + w^2 - 9*w - 12],
[101, 101, w^3 - 4*w^2 - 4*w + 19],
[109, 109, -w^3 + 3*w^2 + 6*w - 19],
[109, 109, -2*w^2 + w + 18],
[121, 11, -3*w^2 + 3*w + 19],
[121, 11, 3*w^2 - 3*w - 20],
[131, 131, 2*w^2 - 3*w - 15],
[131, 131, 2*w^2 - w - 16],
[139, 139, 3*w^2 - 4*w - 21],
[139, 139, -w^3 - w^2 + 7*w + 13],
[151, 151, 4*w^2 - 3*w - 27],
[151, 151, 2*w^3 - w^2 - 13*w - 8],
[151, 151, w^3 + 3*w^2 - 13*w - 27],
[151, 151, 4*w^2 - 5*w - 26],
[179, 179, w^3 - 3*w^2 - 5*w + 11],
[179, 179, 3*w^3 - 11*w^2 - 14*w + 61],
[179, 179, 2*w^3 - 2*w^2 - 12*w + 3],
[179, 179, w^3 - 8*w - 4],
[181, 181, -w - 4],
[181, 181, w^3 - 5*w^2 - 3*w + 25],
[181, 181, w^3 + 2*w^2 - 10*w - 18],
[181, 181, w - 5],
[229, 229, w^2 - 2*w - 11],
[229, 229, w^2 - 12],
[239, 239, 6*w^2 - 7*w - 42],
[239, 239, 2*w^3 - 2*w^2 - 14*w - 3],
[241, 241, w^3 - 4*w^2 - 4*w + 18],
[241, 241, 3*w^2 - 2*w - 16],
[269, 269, 3*w^2 - 4*w - 22],
[269, 269, -w^3 + 6*w + 8],
[281, 281, w^3 - w^2 - 9*w + 4],
[281, 281, 2*w^3 - 9*w^2 - 8*w + 49],
[311, 311, -2*w^3 - w^2 + 17*w + 19],
[311, 311, 2*w^3 - 7*w^2 - 9*w + 33],
[331, 331, w^3 + 2*w^2 - 9*w - 21],
[331, 331, w^3 - 5*w^2 - 2*w + 27],
[349, 349, 2*w^3 - w^2 - 15*w - 4],
[349, 349, -2*w^3 + 5*w^2 + 11*w - 18],
[359, 359, -w^3 + 6*w^2 - 31],
[359, 359, -w^3 - 3*w^2 + 9*w + 26],
[379, 379, 2*w^3 - 5*w^2 - 11*w + 20],
[379, 379, 2*w^3 - 14*w - 11],
[401, 401, w^3 - 3*w^2 - 4*w + 16],
[401, 401, -2*w^3 + 4*w^2 + 11*w - 16],
[401, 401, 2*w^3 - 2*w^2 - 13*w - 3],
[401, 401, w^3 - 7*w - 10],
[409, 409, w^3 - 4*w^2 - 4*w + 17],
[409, 409, 2*w^3 - 6*w^2 - 9*w + 22],
[409, 409, -2*w^3 + 15*w + 9],
[409, 409, -w^3 + 3*w^2 + 6*w - 12],
[419, 419, -w^3 + 5*w^2 + 3*w - 24],
[419, 419, w^3 + 2*w^2 - 10*w - 17],
[421, 421, w^3 + 3*w^2 - 12*w - 25],
[421, 421, -w^3 - w^2 + 9*w + 7],
[431, 431, -w^3 - 2*w^2 + 12*w + 20],
[431, 431, w^3 + 3*w^2 - 10*w - 28],
[431, 431, -w^3 + 6*w^2 + w - 34],
[431, 431, -w^3 + 5*w^2 + 5*w - 29],
[439, 439, w^3 - 7*w^2 - 2*w + 39],
[439, 439, w^3 + 4*w^2 - 13*w - 31],
[449, 449, -w^3 - 4*w^2 + 13*w + 34],
[449, 449, 6*w^2 - 5*w - 36],
[461, 461, -w^3 + 5*w^2 + w - 25],
[461, 461, 2*w^3 - 15*w - 12],
[461, 461, -2*w^3 + 6*w^2 + 9*w - 25],
[461, 461, w^3 + 2*w^2 - 8*w - 20],
[479, 479, -w^3 - 4*w^2 + 12*w + 31],
[479, 479, 2*w^3 - 15*w - 16],
[479, 479, -2*w^3 + 6*w^2 + 9*w - 29],
[479, 479, w^3 - 7*w^2 - w + 38],
[491, 491, w^2 - 3*w - 7],
[491, 491, -2*w^3 + 3*w^2 + 11*w - 2],
[491, 491, 4*w^2 - 2*w - 29],
[491, 491, w^2 + w - 9],
[499, 499, 5*w^2 - 6*w - 33],
[499, 499, 5*w^2 - 4*w - 34],
[509, 509, 2*w - 3],
[509, 509, -2*w - 1],
[541, 541, -w^3 - 4*w^2 + 12*w + 29],
[541, 541, -w^3 + 7*w^2 + w - 36],
[599, 599, -w^3 - w^2 + 8*w + 7],
[599, 599, w^3 - 4*w^2 - 3*w + 13],
[619, 619, w^2 - 3*w - 6],
[619, 619, w^2 + w - 8],
[641, 641, -2*w^3 + 16*w + 11],
[641, 641, -2*w^3 + 6*w^2 + 10*w - 25],
[659, 659, 2*w^3 + w^2 - 16*w - 16],
[659, 659, 2*w^3 - 7*w^2 - 8*w + 29],
[691, 691, -w^3 + 5*w^2 + 3*w - 23],
[691, 691, -w^3 + 3*w^2 + 6*w - 11],
[701, 701, w^3 - w^2 - 9*w + 3],
[701, 701, 2*w^3 - w^2 - 14*w - 5],
[709, 709, 2*w^3 - 3*w^2 - 11*w + 6],
[719, 719, w^2 - 2*w - 12],
[719, 719, -w^2 + 13],
[739, 739, -2*w^3 + 8*w^2 + 7*w - 36],
[739, 739, w^2 + 2*w - 13],
[751, 751, -2*w^3 + 5*w^2 + 10*w - 20],
[751, 751, 2*w^3 - w^2 - 14*w - 7],
[761, 761, w^3 + 4*w^2 - 12*w - 32],
[761, 761, -w^3 + 7*w^2 + w - 39],
[769, 769, -2*w^3 - 4*w^2 + 23*w + 45],
[769, 769, 2*w^3 - 5*w^2 - 9*w + 21],
[769, 769, 2*w^3 + 2*w^2 - 18*w - 31],
[769, 769, 2*w^3 - 10*w^2 - 9*w + 62],
[811, 811, -w^3 + 10*w + 5],
[811, 811, -5*w^2 + 8*w + 30],
[811, 811, w^3 - 8*w^2 + 2*w + 43],
[811, 811, 2*w^3 - 2*w^2 - 13*w + 2],
[829, 829, -2*w^3 - 2*w^2 + 18*w + 25],
[829, 829, -2*w^3 + 8*w^2 + 8*w - 39],
[859, 859, -w^3 - 4*w^2 + 12*w + 33],
[859, 859, -w^3 + 7*w^2 + w - 40],
[919, 919, w^3 - 4*w^2 - 3*w + 24],
[919, 919, -w^3 + 5*w^2 + w - 26],
[929, 929, 5*w^2 - 7*w - 28],
[929, 929, -w^3 + 2*w^2 + 6*w - 13],
[929, 929, w^3 - w^2 - 7*w - 6],
[929, 929, -2*w^3 + 2*w^2 + 14*w - 5],
[941, 941, -2*w^3 + 3*w^2 + 11*w - 1],
[941, 941, 2*w^3 - 3*w^2 - 11*w + 11],
[961, 31, 5*w^2 - 5*w - 33],
[971, 971, -w^3 - 3*w^2 + 11*w + 22],
[971, 971, 7*w^2 - 5*w - 45],
[971, 971, 7*w^2 - 9*w - 43],
[971, 971, w^3 - 6*w^2 - 2*w + 29],
[991, 991, w^3 - 3*w^2 - 3*w + 15],
[991, 991, -w^3 + 5*w^2 + 2*w - 29]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^3 - 4*x^2 - 12*x + 36;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, e, -1/2*e^2 + e + 5, -1, -1/2*e^2 + 9, -1/6*e^2 + 5/3*e - 1, e - 3, -1/3*e^2 + 1/3*e + 1, e - 2, 4, -1/3*e^2 - 2/3*e + 2, 1/3*e^2 - 1/3*e - 5, -1/2*e^2 + 7, 2*e - 2, 1/6*e^2 + 4/3*e - 3, -1/3*e^2 - 2/3*e + 6, -1/3*e^2 + 4/3*e + 2, 2/3*e^2 + 1/3*e - 14, -1/3*e^2 + 7/3*e + 2, -2/3*e^2 + 5/3*e, 1/3*e^2 + 5/3*e - 4, -e^2 + 2*e + 10, -1/6*e^2 - 4/3*e + 11, 0, -1/3*e^2 - 5/3*e + 8, 2/3*e^2 + 7/3*e - 10, 4/3*e^2 - 10/3*e - 14, -5/3*e^2 + 2/3*e + 22, -1/2*e^2 + 17, 1/2*e^2 - 11, 4/3*e^2 - 4/3*e - 18, 2/3*e^2 - 5/3*e - 16, -1/3*e^2 + 1/3*e - 4, e^2 - e - 6, 1/3*e^2 + 2/3*e + 10, -5/3*e^2 + 8/3*e + 13, -1/3*e^2 + 7/3*e + 12, -2/3*e^2 - 7/3*e + 19, 4/3*e^2 - 7/3*e - 8, -1/3*e^2 - 5/3*e + 2, 1/3*e^2 - 13/3*e - 5, 12, 1/2*e^2 - e - 19, 2/3*e^2 + 7/3*e - 17, -e^2 + 6, 1/3*e^2 + 2/3*e - 2, -e^2 - e + 7, -4/3*e^2 + 4/3*e + 9, 4*e, -e^2 - 3*e + 25, -2/3*e^2 + 20/3*e + 4, 2*e^2 - 4*e - 20, 2/3*e^2 - 5/3*e - 28, -1/3*e^2 - 5/3*e, 2*e^2 + e - 40, 3/2*e^2 - 2*e - 19, -7/6*e^2 + 11/3*e - 5, 4/3*e^2 - 13/3*e - 8, -5/2*e^2 + 4*e + 23, -4*e + 20, -5/3*e^2 - 10/3*e + 22, 11/3*e^2 - 11/3*e - 46, -1/3*e^2 + 16/3*e + 3, 3*e^2 - 6*e - 22, -1/3*e^2 + 22/3*e - 9, -4*e - 8, 2*e^2 + e - 34, -1/2*e^2 + 5*e + 17, e^2 - 37, 2*e + 2, -1/3*e^2 + 4/3*e, e^2 + 6*e - 22, -1/2*e^2 - 2*e + 15, 5/6*e^2 + 11/3*e - 19, -3/2*e^2 + 5*e + 11, 11/3*e^2 - 2/3*e - 50, -4/3*e^2 + 7/3*e + 29, 1/3*e^2 - 16/3*e - 3, 8/3*e^2 - 2/3*e - 38, e^2 - 3*e - 2, -1/3*e^2 - 5/3*e - 7, -1/3*e^2 - 17/3*e + 14, 1/3*e^2 - 7/3*e - 6, 8/3*e^2 - 17/3*e - 14, 2/3*e^2 + 4/3*e + 6, -11/6*e^2 + 25/3*e + 21, -19/6*e^2 + 5/3*e + 41, 2/3*e^2 + 7/3*e - 26, -1/3*e^2 + 16/3*e - 3, -5/3*e^2 + 26/3*e + 10, 1/3*e^2 - 1/3*e + 4, 4*e - 6, 7/3*e^2 - 31/3*e - 18, 4/3*e^2 + 5/3*e - 24, -1/3*e^2 - 2/3*e + 12, 3*e^2 - 7*e - 38, -5/3*e^2 - 7/3*e + 50, -8/3*e^2 + 8/3*e + 43, -2*e^2 + 6*e + 23, 17/6*e^2 - 7/3*e - 47, 7/3*e^2 - 7/3*e - 24, -5*e + 25, 8/3*e^2 - 8/3*e - 34, -4/3*e^2 + 22/3*e + 16, e^2 - 4*e - 14, -2*e^2 + 10*e + 18, 4/3*e^2 + 8/3*e - 24, 1/6*e^2 + 7/3*e + 3, 10/3*e^2 - 16/3*e - 32, -5/3*e^2 + 5/3*e - 5, -5/3*e^2 + 29/3*e + 8, -3*e^2 + 9*e + 34, -e^2 - e, e^2 - 3*e + 26, 11/6*e^2 + 14/3*e - 53, 2/3*e^2 - 23/3*e + 2, 10/3*e^2 - 10/3*e - 50, 11/6*e^2 - 19/3*e - 1, 2*e^2 - 9*e - 23, 10/3*e^2 - 4/3*e - 40, e^2 + e - 22, -7/3*e^2 - 5/3*e + 60, 1/3*e^2 + 17/3*e - 6, -1/6*e^2 - 19/3*e + 29, -25/6*e^2 + 5/3*e + 45, 2/3*e^2 - 5/3*e - 27, -1/3*e^2 - 17/3*e + 20, 8/3*e^2 + 4/3*e - 65, 2/3*e^2 - 2/3*e - 55, 8/3*e^2 - 17/3*e - 22, 4/3*e^2 - 16/3*e + 4, 2*e^2 - 2*e - 18, 3*e + 8, 2/3*e^2 + 7/3*e - 22, -13/3*e^2 - 2/3*e + 66, -1/3*e^2 - 23/3*e + 14, -2*e^2 - e + 58, 2/3*e^2 - 29/3*e - 4, 3*e^2 - 5*e - 48, 3*e^2 - 12*e - 36, -16/3*e^2 + 7/3*e + 64, -14/3*e^2 + 2/3*e + 70, -5*e^2 + 8*e + 46, -5/3*e^2 + 14/3*e + 12, 4/3*e^2 - 19/3*e - 14, 11/3*e^2 + 16/3*e - 68, -1/3*e^2 + 4/3*e + 34, 13/3*e^2 + 8/3*e - 59, 2/3*e^2 - 17/3*e - 14, -10/3*e^2 + 7/3*e + 34, 8/3*e^2 - 11/3*e - 4, -17/6*e^2 + 7/3*e + 17, 8/3*e^2 - 20/3*e - 16, 11/3*e^2 - 35/3*e - 13];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {19, 19, 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;