/*
  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![11, 9, -9, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {16, 2, 2}>;

primesArray := [
[5, 5, -2*w^3 - 2*w^2 + 13*w + 8],
[11, 11, 2*w^3 + 3*w^2 - 12*w - 11],
[11, 11, -w^3 - 2*w^2 + 6*w + 9],
[11, 11, w^3 + w^2 - 7*w - 4],
[11, 11, w - 1],
[16, 2, 2],
[19, 19, -w^2 + 4],
[19, 19, 2*w^3 + 2*w^2 - 13*w - 6],
[19, 19, 3*w^3 + 4*w^2 - 18*w - 16],
[19, 19, -w^3 - w^2 + 5*w + 3],
[49, 7, 2*w^3 + 3*w^2 - 13*w - 15],
[59, 59, -3*w^3 - 4*w^2 + 19*w + 14],
[59, 59, 3*w^3 + 4*w^2 - 18*w - 17],
[59, 59, 2*w^3 + 3*w^2 - 11*w - 13],
[59, 59, -w^3 - 2*w^2 + 7*w + 9],
[71, 71, 3*w^3 + 3*w^2 - 17*w - 10],
[71, 71, -2*w^3 - w^2 + 14*w + 2],
[71, 71, 5*w^3 + 7*w^2 - 30*w - 25],
[71, 71, -3*w^3 - 4*w^2 + 16*w + 16],
[81, 3, -3],
[89, 89, -2*w^3 - 2*w^2 + 13*w + 4],
[89, 89, 3*w^3 + 4*w^2 - 17*w - 16],
[89, 89, 3*w^3 + 4*w^2 - 18*w - 18],
[89, 89, -4*w^3 - 5*w^2 + 25*w + 18],
[139, 139, -w^3 + 6*w + 1],
[139, 139, -3*w^3 - 3*w^2 + 19*w + 8],
[139, 139, 4*w^3 + 5*w^2 - 24*w - 21],
[139, 139, -2*w^3 - 2*w^2 + 11*w + 5],
[151, 151, -6*w^3 - 7*w^2 + 37*w + 27],
[151, 151, 3*w^3 + 5*w^2 - 19*w - 17],
[151, 151, -6*w^3 - 8*w^2 + 37*w + 31],
[151, 151, 4*w^3 + 6*w^2 - 25*w - 26],
[191, 191, 5*w^3 + 7*w^2 - 31*w - 26],
[191, 191, -2*w^3 - 3*w^2 + 10*w + 13],
[191, 191, -5*w^3 - 7*w^2 + 31*w + 29],
[191, 191, 2*w^3 + 4*w^2 - 13*w - 17],
[199, 199, 2*w^3 + 4*w^2 - 11*w - 20],
[199, 199, -2*w^3 - 2*w^2 + 11*w + 3],
[199, 199, 3*w^3 + 5*w^2 - 17*w - 17],
[199, 199, 3*w^3 + 3*w^2 - 19*w - 6],
[211, 211, 3*w^3 + 5*w^2 - 18*w - 19],
[211, 211, -3*w^3 - 5*w^2 + 18*w + 21],
[211, 211, 2*w^3 + 4*w^2 - 12*w - 17],
[211, 211, -4*w^3 - 6*w^2 + 24*w + 23],
[229, 229, 2*w^3 + 4*w^2 - 13*w - 19],
[229, 229, -2*w^3 - 3*w^2 + 10*w + 15],
[229, 229, -5*w^3 - 7*w^2 + 31*w + 24],
[229, 229, 2*w^3 + 2*w^2 - 14*w - 3],
[269, 269, 2*w^3 + w^2 - 11*w - 1],
[269, 269, -6*w^3 - 7*w^2 + 37*w + 24],
[269, 269, 5*w^3 + 6*w^2 - 29*w - 24],
[269, 269, 3*w^3 + 2*w^2 - 19*w - 7],
[281, 281, -4*w^3 - 6*w^2 + 25*w + 25],
[281, 281, -5*w^3 - 7*w^2 + 31*w + 27],
[281, 281, w^2 + 2*w - 7],
[281, 281, 3*w^3 + 5*w^2 - 19*w - 18],
[331, 331, -2*w^3 - 3*w^2 + 12*w + 7],
[331, 331, w^3 + w^2 - 7*w - 8],
[331, 331, w^3 + 2*w^2 - 6*w - 13],
[331, 331, w - 5],
[349, 349, 2*w^3 + 5*w^2 - 16*w - 12],
[349, 349, 2*w^3 + 4*w^2 - 15*w - 10],
[349, 349, 8*w^3 + 9*w^2 - 50*w - 32],
[349, 349, 2*w^3 + 4*w^2 - 15*w - 9],
[401, 401, 4*w^3 + 3*w^2 - 21*w - 12],
[401, 401, 3*w^3 + 2*w^2 - 16*w - 9],
[401, 401, -3*w^3 - 6*w^2 + 20*w + 19],
[401, 401, -4*w^3 - 3*w^2 + 27*w + 9],
[409, 409, -5*w^3 - 6*w^2 + 31*w + 20],
[409, 409, 3*w^3 + 4*w^2 - 20*w - 12],
[409, 409, -3*w^3 - 5*w^2 + 17*w + 23],
[409, 409, 4*w^3 + 5*w^2 - 23*w - 21],
[419, 419, -4*w^3 - 4*w^2 + 21*w + 18],
[419, 419, -4*w^3 - 4*w^2 + 20*w + 19],
[419, 419, -3*w^3 - 4*w^2 + 16*w + 18],
[419, 419, -2*w^3 - w^2 + 8*w + 10],
[421, 421, -6*w^3 - 7*w^2 + 36*w + 26],
[421, 421, 5*w^3 + 5*w^2 - 31*w - 17],
[421, 421, 3*w^3 + 2*w^2 - 18*w - 6],
[421, 421, 4*w^3 + 4*w^2 - 23*w - 14],
[431, 431, 6*w^3 + 7*w^2 - 36*w - 25],
[431, 431, -3*w^3 - 2*w^2 + 18*w + 5],
[431, 431, 4*w^3 + 4*w^2 - 23*w - 15],
[431, 431, -5*w^3 - 5*w^2 + 31*w + 18],
[439, 439, w^3 + w^2 - 4*w - 1],
[439, 439, w^3 + 3*w^2 - 6*w - 10],
[439, 439, 3*w^3 + 3*w^2 - 20*w - 7],
[439, 439, 5*w^3 + 7*w^2 - 30*w - 30],
[479, 479, -4*w^3 - 5*w^2 + 25*w + 15],
[479, 479, -5*w^3 - 6*w^2 + 32*w + 21],
[479, 479, 5*w^3 + 7*w^2 - 29*w - 28],
[479, 479, 2*w^2 + w - 9],
[491, 491, 7*w^3 + 8*w^2 - 43*w - 31],
[491, 491, 6*w^3 + 6*w^2 - 37*w - 24],
[491, 491, 4*w^3 + 3*w^2 - 24*w - 6],
[491, 491, 5*w^3 + 5*w^2 - 29*w - 21],
[509, 509, -4*w^3 - 5*w^2 + 25*w + 14],
[509, 509, -3*w^3 - 4*w^2 + 17*w + 20],
[509, 509, -6*w^3 - 7*w^2 + 38*w + 25],
[509, 509, -6*w^3 - 8*w^2 + 35*w + 31],
[541, 541, 9*w^3 + 12*w^2 - 55*w - 46],
[541, 541, w^3 + 4*w^2 - 7*w - 17],
[541, 541, -4*w^3 - 5*w^2 + 21*w + 19],
[541, 541, -4*w^3 - 3*w^2 + 27*w + 8],
[571, 571, -3*w^3 - 6*w^2 + 18*w + 23],
[571, 571, 6*w^3 + 9*w^2 - 36*w - 37],
[571, 571, 7*w^3 + 10*w^2 - 42*w - 36],
[571, 571, 2*w^3 + 5*w^2 - 12*w - 24],
[619, 619, -9*w^3 - 11*w^2 + 55*w + 39],
[619, 619, -5*w^3 - 4*w^2 + 32*w + 14],
[619, 619, 7*w^3 + 9*w^2 - 42*w - 36],
[619, 619, 4*w^3 + 6*w^2 - 26*w - 27],
[631, 631, 2*w^3 + 5*w^2 - 12*w - 21],
[631, 631, -7*w^3 - 10*w^2 + 42*w + 39],
[631, 631, 4*w^3 + 4*w^2 - 27*w - 13],
[631, 631, w^3 + w^2 - 3*w - 4],
[641, 641, -9*w^3 - 10*w^2 + 56*w + 36],
[641, 641, -5*w^3 - 9*w^2 + 30*w + 32],
[641, 641, -3*w^3 - 6*w^2 + 19*w + 21],
[641, 641, 4*w^3 + 3*w^2 - 22*w - 10],
[701, 701, -2*w^3 - 5*w^2 + 14*w + 20],
[701, 701, -6*w^3 - 7*w^2 + 36*w + 34],
[701, 701, 4*w^3 + 6*w^2 - 23*w - 17],
[701, 701, -9*w^3 - 12*w^2 + 56*w + 46],
[719, 719, 3*w^3 + 5*w^2 - 17*w - 25],
[719, 719, -3*w^3 - 2*w^2 + 17*w + 3],
[719, 719, 7*w^3 + 8*w^2 - 43*w - 26],
[719, 719, -6*w^3 - 7*w^2 + 35*w + 29],
[751, 751, 2*w^3 + 2*w^2 - 15*w - 4],
[751, 751, 5*w^3 + 8*w^2 - 30*w - 34],
[751, 751, w^3 + w^2 - 9*w - 5],
[751, 751, -4*w^3 - 7*w^2 + 24*w + 26],
[769, 769, 6*w^3 + 8*w^2 - 37*w - 27],
[769, 769, -w^3 - 3*w^2 + 7*w + 16],
[769, 769, 2*w^3 + w^2 - 14*w - 5],
[769, 769, -3*w^3 - 4*w^2 + 16*w + 19],
[821, 821, 6*w^3 + 6*w^2 - 37*w - 23],
[821, 821, 5*w^3 + 5*w^2 - 29*w - 20],
[821, 821, -7*w^3 - 8*w^2 + 42*w + 27],
[821, 821, 4*w^3 + 3*w^2 - 24*w - 7],
[829, 829, 7*w^3 + 8*w^2 - 44*w - 28],
[829, 829, 7*w^3 + 9*w^2 - 41*w - 35],
[829, 829, -2*w^3 - w^2 + 10*w + 2],
[829, 829, 2*w^3 - 13*w + 2],
[839, 839, 4*w^3 + 8*w^2 - 25*w - 26],
[839, 839, -3*w^3 - 3*w^2 + 15*w + 16],
[839, 839, -w^3 - 4*w^2 + 8*w + 10],
[839, 839, -3*w^3 - 7*w^2 + 18*w + 26],
[841, 29, -5*w^3 - 5*w^2 + 30*w + 16],
[841, 29, -5*w^3 - 5*w^2 + 30*w + 19],
[859, 859, 5*w^3 + 6*w^2 - 32*w - 20],
[859, 859, -4*w^3 - 5*w^2 + 23*w + 23],
[859, 859, -5*w^3 - 6*w^2 + 31*w + 18],
[859, 859, -2*w^3 - w^2 + 13*w + 6],
[911, 911, 9*w^3 + 12*w^2 - 56*w - 48],
[911, 911, 6*w^3 + 5*w^2 - 38*w - 14],
[911, 911, 7*w^3 + 8*w^2 - 40*w - 28],
[911, 911, 10*w^3 + 12*w^2 - 61*w - 46],
[929, 929, -7*w^3 - 10*w^2 + 39*w + 39],
[929, 929, -10*w^3 - 13*w^2 + 60*w + 45],
[929, 929, -6*w^3 - 7*w^2 + 32*w + 29],
[929, 929, -3*w^3 - w^2 + 12*w + 12],
[961, 31, -5*w^3 - 5*w^2 + 30*w + 17],
[961, 31, -5*w^3 - 5*w^2 + 30*w + 18],
[991, 991, 10*w^3 + 12*w^2 - 61*w - 45],
[991, 991, -6*w^3 - 9*w^2 + 38*w + 38],
[991, 991, 6*w^3 + 8*w^2 - 37*w - 38],
[991, 991, -8*w^3 - 11*w^2 + 49*w + 43]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^2 - x - 23;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [-4, e, e, -e + 1, -e + 1, -1, -e + 3, e + 2, -e + 3, e + 2, 14, -e - 7, e - 8, e - 8, -e - 7, 6, 6, 6, 6, 2, -e, e - 1, e - 1, -e, 0, 0, 0, 0, 2*e + 8, -2*e + 10, 2*e + 8, -2*e + 10, -18, -18, -18, -18, 2*e + 2, -2*e + 4, 2*e + 2, -2*e + 4, -3*e + 2, -3*e + 2, 3*e - 1, 3*e - 1, -4*e + 4, 4*e, -4*e + 4, 4*e, -2*e + 10, -2*e + 10, 2*e + 8, 2*e + 8, -e - 16, e - 17, e - 17, -e - 16, e - 2, -e - 1, e - 2, -e - 1, 2, 2, 2, 2, -3*e + 17, -3*e + 17, 3*e + 14, 3*e + 14, -3*e + 25, -3*e + 25, 3*e + 22, 3*e + 22, -4*e, 4*e - 4, 4*e - 4, -4*e, -4*e - 12, 4*e - 16, -4*e - 12, 4*e - 16, -2*e + 2, -2*e + 2, 2*e, 2*e, 2*e + 16, -2*e + 18, 2*e + 16, -2*e + 18, 22, 22, 22, 22, -3*e - 2, 3*e - 5, -3*e - 2, 3*e - 5, -2*e - 14, 2*e - 16, -2*e - 14, 2*e - 16, 2*e - 4, 2*e - 4, -2*e - 2, -2*e - 2, -e + 27, -e + 27, e + 26, e + 26, 3*e + 1, -3*e + 4, 3*e + 1, -3*e + 4, -4*e + 22, -4*e + 22, 4*e + 18, 4*e + 18, -7*e + 7, 7*e, 7*e, -7*e + 7, -34, -34, -34, -34, 2*e + 12, -2*e + 14, -2*e + 14, 2*e + 12, -2*e - 10, 2*e - 12, -2*e - 10, 2*e - 12, e - 33, e - 33, -e - 32, -e - 32, -2*e + 20, -2*e + 20, 2*e + 18, 2*e + 18, -4*e - 16, 4*e - 20, -4*e - 16, 4*e - 20, -4*e + 36, -4*e + 36, 4*e + 32, 4*e + 32, -3*e + 34, 3*e + 31, 5*e - 24, -5*e - 19, 5*e - 24, -5*e - 19, -4*e + 32, 4*e + 28, 4*e + 28, -4*e + 32, 4*e + 2, -4*e + 6, -4*e + 6, 4*e + 2, -11*e + 14, 11*e + 3, 4*e - 28, -4*e - 24, -4*e - 24, 4*e - 28];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {16, 2, 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;