/*
  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![-8, -14, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {8,4,-w - 2}>;

primesArray := [
[2, 2, -1/2*w^2 + 1/2*w + 8],
[2, 2, 1/2*w^2 - 3/2*w - 1],
[2, 2, w + 3],
[11, 11, -w^2 + 3*w + 7],
[11, 11, -2*w - 1],
[11, 11, -w^2 + w + 11],
[27, 3, 3],
[41, 41, 2*w + 5],
[41, 41, -w^2 + 3*w + 3],
[41, 41, -w^2 + w + 15],
[43, 43, -3*w^2 + 11*w + 11],
[47, 47, -w^2 - w + 5],
[47, 47, w^2 - 5*w - 3],
[47, 47, -2*w^2 + 4*w + 23],
[59, 59, -w^2 + w + 13],
[59, 59, w^2 - 3*w - 5],
[59, 59, -2*w - 3],
[97, 97, 7*w^2 - 11*w - 91],
[97, 97, -2*w^2 - 8*w - 5],
[97, 97, 5*w^2 - 19*w - 15],
[107, 107, w^2 + 3*w - 1],
[107, 107, -3*w^2 + 5*w + 37],
[107, 107, w^2 - 5*w - 23],
[113, 113, w^2 - 3*w - 1],
[113, 113, w^2 - w - 17],
[113, 113, -2*w - 7],
[125, 5, -5],
[127, 127, -w^2 - w - 1],
[127, 127, 2*w^2 - 4*w - 29],
[127, 127, -w^2 + 5*w - 3],
[131, 131, -w^2 + 5*w - 1],
[131, 131, 2*w^2 - 4*w - 27],
[131, 131, w^2 + w - 1],
[137, 137, 2*w^2 - 6*w - 5],
[137, 137, -4*w - 11],
[137, 137, 2*w^2 - 2*w - 31],
[151, 151, -2*w^2 + 8*w + 3],
[151, 151, 3*w^2 - 5*w - 41],
[151, 151, -w^2 - 3*w - 3],
[173, 173, 2*w^2 - 4*w - 25],
[173, 173, w^2 + w - 3],
[173, 173, -w^2 + 5*w + 1],
[193, 193, -6*w - 17],
[193, 193, 3*w^2 - 9*w - 7],
[193, 193, 3*w^2 - 3*w - 47],
[199, 199, w^2 + w - 7],
[199, 199, 2*w^2 - 4*w - 21],
[199, 199, -w^2 + 5*w + 5],
[211, 211, 2*w - 7],
[211, 211, w^2 - w - 3],
[211, 211, -w^2 + 3*w + 15],
[223, 223, w^2 - w - 7],
[223, 223, w^2 - 3*w - 11],
[223, 223, 2*w - 3],
[257, 257, -2*w^2 - 4*w - 1],
[257, 257, -3*w^2 + 13*w + 3],
[257, 257, 5*w^2 - 9*w - 67],
[269, 269, 2*w - 5],
[269, 269, w^2 - 3*w - 13],
[269, 269, w^2 - w - 5],
[293, 293, -13*w^2 + 51*w + 35],
[293, 293, -19*w^2 + 31*w + 247],
[293, 293, 6*w^2 + 20*w + 9],
[317, 317, 5*w^2 - 19*w - 17],
[317, 317, 7*w^2 - 11*w - 89],
[317, 317, 2*w^2 + 8*w + 3],
[343, 7, -7],
[379, 379, 3*w^2 - 11*w - 13],
[379, 379, 4*w^2 - 6*w - 49],
[379, 379, w^2 + 5*w + 1],
[383, 383, -4*w - 5],
[383, 383, -2*w^2 + 2*w + 25],
[383, 383, 2*w^2 - 6*w - 11],
[389, 389, 4*w^2 - 18*w - 13],
[389, 389, -7*w^2 + 13*w + 83],
[389, 389, 4*w^2 - 14*w - 11],
[409, 409, 2*w - 11],
[409, 409, w^2 - 3*w - 19],
[409, 409, -w^2 + w - 1],
[419, 419, 10*w^2 - 38*w - 31],
[419, 419, -14*w^2 + 22*w + 181],
[419, 419, 4*w^2 - 14*w - 17],
[431, 431, -4*w^2 + 16*w + 13],
[431, 431, 2*w^2 + 6*w - 1],
[431, 431, -6*w^2 + 10*w + 75],
[457, 457, -2*w^2 + 8*w + 9],
[457, 457, 3*w^2 - 5*w - 35],
[457, 457, w^2 + 3*w - 3],
[557, 557, 5*w^2 - 9*w - 63],
[557, 557, -3*w^2 + 13*w + 7],
[557, 557, 2*w^2 + 4*w - 3],
[563, 563, -9*w^2 + 35*w + 23],
[563, 563, 13*w^2 - 21*w - 171],
[563, 563, -4*w^2 - 14*w - 9],
[601, 601, 5*w^2 - 17*w - 13],
[601, 601, 6*w^2 - 8*w - 85],
[601, 601, -w^2 - 9*w - 17],
[613, 613, -2*w^2 + 15],
[613, 613, 3*w^2 - 7*w - 31],
[613, 613, 22*w^2 - 36*w - 285],
[641, 641, 3*w^2 - 5*w - 43],
[641, 641, -2*w^2 + 8*w + 1],
[641, 641, -w^2 - 3*w - 5],
[643, 643, 2*w^2 + 4*w - 1],
[643, 643, 5*w^2 - 9*w - 65],
[643, 643, -3*w^2 + 13*w + 5],
[647, 647, 9*w^2 - 35*w - 27],
[647, 647, 4*w^2 + 14*w + 5],
[647, 647, 13*w^2 - 21*w - 167],
[653, 653, -4*w^2 + 16*w + 7],
[653, 653, 6*w^2 - 10*w - 81],
[653, 653, -2*w^2 - 6*w - 5],
[661, 661, -w^2 - 7*w - 13],
[661, 661, -4*w^2 + 14*w + 9],
[661, 661, 5*w^2 - 7*w - 71],
[677, 677, -12*w^2 + 20*w + 155],
[677, 677, -4*w^2 - 12*w - 3],
[677, 677, -8*w^2 + 32*w + 21],
[709, 709, -2*w^2 + 10*w - 3],
[709, 709, 19*w^2 - 31*w - 249],
[709, 709, 4*w^2 - 8*w - 55],
[727, 727, -w^2 + 5*w + 21],
[727, 727, -23*w^2 + 37*w + 301],
[727, 727, -w^2 - w + 23],
[733, 733, 2*w^2 + 2*w - 9],
[733, 733, -2*w^2 + 10*w + 5],
[733, 733, 4*w^2 - 8*w - 47],
[739, 739, 2*w^2 - 6*w - 17],
[739, 739, 2*w^2 - 2*w - 19],
[739, 739, 4*w - 1],
[773, 773, 8*w^2 - 12*w - 103],
[773, 773, 6*w^2 - 22*w - 21],
[773, 773, -2*w^2 - 10*w - 7],
[809, 809, 4*w^2 + 16*w + 11],
[809, 809, w^2 - 9*w - 7],
[809, 809, 3*w^2 - w - 29],
[821, 821, 3*w^2 + w - 23],
[821, 821, 2*w^2 - 12*w - 9],
[821, 821, 5*w^2 - 11*w - 51],
[839, 839, 2*w^2 + 4*w - 7],
[839, 839, 5*w^2 - 9*w - 59],
[839, 839, 3*w^2 - 13*w - 11],
[859, 859, w^2 + w - 11],
[859, 859, 2*w^2 - 4*w - 17],
[859, 859, w^2 - 5*w - 9],
[881, 881, 4*w^2 - 14*w - 15],
[881, 881, 5*w^2 - 7*w - 65],
[881, 881, -w^2 - 7*w - 7],
[887, 887, 5*w^2 - 7*w - 67],
[887, 887, 4*w^2 - 14*w - 13],
[887, 887, -w^2 - 7*w - 9],
[907, 907, -3*w^2 - 11*w - 9],
[907, 907, 10*w^2 - 16*w - 133],
[907, 907, -7*w^2 + 27*w + 17],
[911, 911, w^2 - 7*w - 7],
[911, 911, 3*w^2 - 7*w - 27],
[911, 911, 2*w^2 - 19],
[919, 919, -3*w^2 - 5*w - 1],
[919, 919, -4*w^2 + 18*w + 1],
[919, 919, 7*w^2 - 13*w - 95],
[947, 947, 4*w^2 - 6*w - 57],
[947, 947, -3*w^2 + 11*w + 5],
[947, 947, -w^2 - 5*w - 9],
[967, 967, 8*w^2 - 30*w - 25],
[967, 967, -3*w^2 - 13*w - 9],
[967, 967, 11*w^2 - 17*w - 143],
[991, 991, 2*w^2 - 8*w - 11],
[991, 991, w^2 + 3*w - 5],
[991, 991, 3*w^2 - 5*w - 33],
[997, 997, 4*w^2 + 2*w - 27],
[997, 997, 7*w^2 - 15*w - 75],
[997, 997, -3*w^2 + 17*w + 11]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^3 - x^2 - 5*x + 3;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [-1, 0, e, e^2 - 3, e^2 - 3, e + 3, -2*e^2 + e + 7, 6, -2*e^2 + 6, 2*e^2 - e - 9, -e + 5, -e^2 - 5*e + 6, e^2 - 9, -2*e^2 - 2*e + 6, -e^2 - 2*e + 3, -e^2 - e, 2*e^2 - 2*e, e^2 + 6*e - 7, 6*e^2 - 4*e - 16, e^2 - 4*e - 1, 6, -3*e^2 + 7*e + 12, 2*e^2 - 2*e - 12, -3*e - 3, e^2 + 3*e - 6, e^2 + 4*e + 3, 2*e^2 - 4*e - 6, -2*e^2 + 20, -e^2 - e - 4, e^2 + 2*e - 1, -4*e^2 + 4*e + 18, -e^2 + 2*e + 9, 2*e^2 - 4*e - 6, -4*e^2 - 2*e + 24, -5*e^2 + 4*e + 9, -5*e^2 - 4*e + 21, -e^2 + 5*e - 4, 2*e^2 - 6*e - 10, -2*e^2 - 4*e + 14, 3*e^2 - 3*e - 6, 6*e^2 - 6*e - 18, -3*e + 9, -3*e^2 + 5*e + 14, -2*e^2 + 8*e + 8, 2*e^2 - e - 7, 2*e^2 - 8*e - 4, 6*e^2 + 6*e - 28, -5*e^2 - e + 8, -3*e^2 - 6*e + 11, 7*e - 1, -e^2 - 6*e + 5, -e^2 + 4*e + 11, 4*e^2 + e - 7, -11*e + 5, -4*e^2 - e + 3, -6*e^2 + 2*e + 30, 2*e^2 + 2*e - 12, -4*e^2 + 24, 8*e^2 - 30, 2*e + 6, 4*e - 6, 6*e^2 - 12*e - 24, 2*e^2 + 7*e - 9, 3*e^2 + 4*e - 33, e^2 + 10*e - 9, 2*e^2 + 6*e - 6, -e^2 - 6*e - 7, e^2 + 3*e + 8, -10*e^2 + 8*e + 26, -8*e^2 + 2*e + 32, -e^2 + 2*e + 15, 2*e^2 - 8*e - 6, -3*e^2 - 6*e + 3, -2*e^2 - 4*e + 12, -8*e^2 + 4*e + 18, -3*e^2 - 6*e + 21, -4*e^2 + 6*e + 8, -3*e^2 - 6*e + 5, -e^2 - 5*e - 10, -4*e^2 + 12*e + 18, -9*e^2 - 5*e + 36, -2*e^2 + 14*e + 6, 6*e^2 - 10*e - 24, 4*e^2 - 4*e - 36, -e^2 + 2*e - 3, -5*e^2 - 5*e + 26, -10*e^2 + 8*e + 32, -9*e^2 + 2*e + 35, -8*e^2 + 10*e + 12, -7*e^2 + 11*e + 42, 5*e^2 + 2*e - 15, 2*e^2 + 2*e + 12, 4*e^2 + 12, 6*e^2 - 6*e - 24, 10*e^2 - 10*e - 40, -12*e^2 - 8*e + 50, e^2 + 2*e - 1, 4*e^2 - 13*e - 13, 3*e^2 - 8*e + 5, 16*e + 2, 7*e^2 + 10*e - 57, 10*e^2 - 6*e - 30, 10*e - 12, -12*e^2 + 6*e + 44, 6*e^2 + 4*e - 34, -2*e^2 - 10*e + 32, -8*e^2 + 5*e + 27, -10*e^2 + 17*e + 33, 3*e^2 - 18*e - 3, -3*e^2 + 4*e + 9, -3*e^2 + 8*e - 15, -2*e^2 + 4*e + 30, -4*e^2 + 50, 4*e^2 + 6*e - 16, -2*e^2 - e + 29, 13*e^2 + 4*e - 45, -12*e^2 - 4*e + 42, 6*e + 12, 2*e^2 + 8*e + 14, -5*e^2 - 5*e + 26, -4*e^2 + 9*e + 23, 3*e^2 - e - 40, 10*e^2 - 6*e - 16, 4*e^2 - 4, 7*e^2 + 5*e - 34, -2*e^2 - 2*e - 28, -5*e + 35, 2*e^2 - 4, 3*e^2 - 14*e - 31, -8*e^2 + 20, -10*e^2 - 6*e + 30, 2*e^2 - 6*e + 18, 2*e^2 + 7*e + 9, 5*e^2 - 4*e + 3, 8*e^2 + 2*e - 12, 6*e^2 - 24, 7*e + 33, -e^2 + e + 30, 2*e^2 - 20*e, 5*e^2 - 15, 4*e^2 - 5*e - 9, -8*e^2 + e + 45, 6*e^2 - 10*e - 28, 10*e^2 - 2*e - 22, 8*e^2 - 4*e - 16, 17*e + 15, 3*e^2 - 7*e + 6, 6*e^2 + 2*e - 6, -2*e^2 - 10*e + 12, 11*e^2 - 22*e - 45, -2*e^2 + 20*e - 6, 8*e^2 + 4*e - 40, -3*e^2 - 9*e - 10, -16*e^2 + 2*e + 56, 9*e^2 - 2*e - 45, -8*e^2 - 11*e + 45, -6*e^2 - 19*e + 45, -17*e^2 + 4*e + 59, -5*e^2 - 10*e + 5, -3*e^2 - 8*e + 17, -10*e^2 - 4*e + 30, 8*e^2 + 16*e - 48, 2*e^2 - 8*e - 6, -2*e^2 - 10*e + 32, 10*e^2 - 4*e - 22, -10*e^2 - 18*e + 50, -18*e^2 + 2*e + 56, 14*e^2 - 18*e - 52, -6*e^2 + 10*e + 20, -7*e^2 + 14*e + 53, -2*e - 34, 2*e^2 + 4*e - 4];
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^2 - 3/2*w - 1}>] := -1;
ALEigenvalues[ideal<ZF | {2,2,-1/2*w^2 + 1/2*w + 8}>] := 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;