/*
  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![4, 8, -7, -2, 1];
F<w> := NumberField(g);
ZF := Integers(F);

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

primesArray := [
[2, 2, 1/2*w^2 + 1/2*w - 1],
[2, 2, -1/2*w^2 + 3/2*w],
[9, 3, -1/2*w^2 + 1/2*w + 2],
[13, 13, 1/2*w^3 - w^2 - 5/2*w + 2],
[13, 13, -1/2*w^3 + 1/2*w^2 + 3*w - 1],
[23, 23, 1/2*w^3 - w^2 - 5/2*w],
[23, 23, -1/2*w^3 + 1/2*w^2 + 3*w - 3],
[37, 37, 1/2*w^3 - 1/2*w^2 - 3*w - 3],
[37, 37, 1/2*w^3 - w^2 - 5/2*w + 6],
[59, 59, -1/2*w^3 + 7/2*w],
[59, 59, -1/2*w^3 + 3/2*w^2 + 2*w - 3],
[61, 61, -w^3 + 2*w^2 + 7*w - 7],
[61, 61, w^3 - w^2 - 6*w + 3],
[61, 61, w^3 - 2*w^2 - 5*w + 3],
[61, 61, w^3 - w^2 - 8*w + 1],
[71, 71, -1/2*w^3 + 1/2*w^2 + 5*w - 5],
[71, 71, -w^3 + 3/2*w^2 + 15/2*w - 6],
[83, 83, -1/2*w^3 + 3/2*w^2 + 2*w - 7],
[83, 83, 1/2*w^3 - 7/2*w - 4],
[97, 97, 2*w - 1],
[107, 107, 1/2*w^3 + 1/2*w^2 - 4*w - 1],
[107, 107, 3/2*w^3 - 7/2*w^2 - 9*w + 13],
[109, 109, -3/2*w^2 + 7/2*w + 4],
[109, 109, 1/2*w^3 + 1/2*w^2 - 4*w - 5],
[109, 109, -1/2*w^3 + 2*w^2 + 3/2*w - 8],
[109, 109, 3/2*w^2 + 1/2*w - 6],
[121, 11, w^2 - w - 5],
[121, 11, w^2 - w - 3],
[131, 131, -w^3 + 11*w - 1],
[131, 131, -w^3 + 3*w^2 + 8*w - 9],
[157, 157, -w^3 + w^2 + 6*w + 1],
[157, 157, 1/2*w^3 - 2*w^2 - 11/2*w + 4],
[167, 167, w^2 - 3*w - 3],
[167, 167, 1/2*w^3 - 3/2*w^2 - 2*w + 1],
[167, 167, 3/2*w^3 - 5/2*w^2 - 10*w + 9],
[167, 167, w^2 + w - 5],
[169, 13, w^2 - w - 9],
[179, 179, w^3 - 7*w - 5],
[179, 179, 1/2*w^3 + 1/2*w^2 - 2*w - 3],
[181, 181, -1/2*w^3 + 1/2*w^2 + 3*w - 5],
[181, 181, 1/2*w^3 - w^2 - 5/2*w - 2],
[191, 191, -1/2*w^3 + 3/2*w + 2],
[191, 191, 2*w - 5],
[191, 191, -2*w - 3],
[191, 191, -1/2*w^3 + 2*w^2 + 3/2*w - 10],
[193, 193, -1/2*w^3 + w^2 + 9/2*w - 4],
[193, 193, 1/2*w^3 - 5/2*w^2 + w + 5],
[193, 193, -1/2*w^3 - w^2 + 5/2*w + 4],
[193, 193, 1/2*w^3 - 1/2*w^2 - 5*w + 1],
[227, 227, -1/2*w^3 + 3/2*w^2 + 4*w - 9],
[227, 227, w^3 - 2*w^2 - 7*w + 5],
[227, 227, w^3 - w^2 - 8*w + 3],
[227, 227, 1/2*w^3 - 11/2*w - 4],
[239, 239, 1/2*w^3 - 3/2*w^2 - 4*w + 3],
[239, 239, 1/2*w^3 - 11/2*w + 2],
[251, 251, 1/2*w^3 - w^2 - 1/2*w - 2],
[251, 251, -1/2*w^3 + 1/2*w^2 + w - 3],
[263, 263, -1/2*w^3 - w^2 + 17/2*w],
[263, 263, 1/2*w^3 - 5/2*w^2 - 5*w + 7],
[277, 277, w^3 - w^2 - 6*w + 1],
[277, 277, -w^3 + 2*w^2 + 5*w - 5],
[311, 311, -w^3 + 5/2*w^2 + 13/2*w - 8],
[311, 311, -w^3 + 1/2*w^2 + 17/2*w],
[337, 337, 2*w^3 - 3*w^2 - 13*w + 9],
[337, 337, 5/2*w^2 + 3/2*w - 8],
[347, 347, 2*w^3 - 7/2*w^2 - 25/2*w + 14],
[347, 347, 2*w^3 - 5/2*w^2 - 27/2*w],
[349, 349, w^3 - 3/2*w^2 - 11/2*w + 4],
[349, 349, -w^3 + 3/2*w^2 + 11/2*w - 2],
[359, 359, -1/2*w^3 + w^2 + 1/2*w + 4],
[359, 359, 1/2*w^3 - 1/2*w^2 - w + 5],
[373, 373, 1/2*w^3 - 3/2*w + 4],
[373, 373, 1/2*w^3 - 3/2*w^2 - 3],
[383, 383, w^3 - 3/2*w^2 - 15/2*w + 8],
[383, 383, -2*w^2 + 1],
[409, 409, 1/2*w^3 + w^2 - 13/2*w - 4],
[409, 409, -1/2*w^3 + 5/2*w^2 + 3*w - 9],
[431, 431, 3/2*w^3 - 5/2*w^2 - 10*w + 5],
[431, 431, 1/2*w^3 - 5/2*w^2 - 3*w + 13],
[431, 431, 1/2*w^3 + w^2 - 13/2*w - 8],
[431, 431, 3/2*w^3 - 2*w^2 - 21/2*w + 6],
[433, 433, 2*w^3 - 4*w^2 - 14*w + 13],
[433, 433, w^3 - 1/2*w^2 - 9/2*w + 6],
[443, 443, 2*w^3 - 11/2*w^2 - 21/2*w + 24],
[443, 443, w^3 - 5*w - 3],
[457, 457, -2*w^3 + 4*w^2 + 12*w - 13],
[457, 457, -w^3 + 1/2*w^2 + 9/2*w],
[467, 467, -3/2*w^3 + 7/2*w^2 + 7*w - 7],
[467, 467, 3/2*w^2 + 1/2*w - 8],
[467, 467, w^3 - 7/2*w^2 - 15/2*w + 12],
[467, 467, 3/2*w^3 - w^2 - 19/2*w + 2],
[503, 503, -2*w^2 + 11],
[503, 503, 2*w^3 - 2*w^2 - 12*w + 7],
[503, 503, 2*w^3 - 4*w^2 - 10*w + 5],
[503, 503, -2*w^2 + 4*w + 9],
[529, 23, 3/2*w^2 - 3/2*w - 8],
[541, 541, -1/2*w^3 + w^2 + 1/2*w + 6],
[541, 541, 1/2*w^3 - 1/2*w^2 - w + 7],
[563, 563, 1/2*w^2 + 3/2*w - 6],
[563, 563, 1/2*w^2 - 5/2*w - 4],
[577, 577, -1/2*w^3 + 5/2*w^2 + w - 11],
[577, 577, -1/2*w^3 - w^2 + 9/2*w + 8],
[587, 587, -w^3 + 1/2*w^2 + 13/2*w - 2],
[587, 587, -w^3 + 5/2*w^2 + 9/2*w - 4],
[599, 599, 1/2*w^3 + w^2 - 9/2*w - 12],
[599, 599, -1/2*w^3 + 5/2*w^2 + w - 15],
[601, 601, 1/2*w^3 - 1/2*w^2 - 3*w - 5],
[601, 601, -1/2*w^3 + w^2 + 5/2*w - 8],
[625, 5, -5],
[659, 659, 3/2*w^3 - 1/2*w^2 - 8*w + 3],
[659, 659, 3/2*w^3 - 4*w^2 - 9/2*w + 4],
[673, 673, 2*w^3 - 2*w^2 - 16*w - 1],
[673, 673, w^3 - 3/2*w^2 - 7/2*w - 8],
[673, 673, w^3 - 3/2*w^2 - 7/2*w + 12],
[673, 673, 2*w^3 - 4*w^2 - 14*w + 17],
[683, 683, -1/2*w^3 - w^2 + 9/2*w - 2],
[683, 683, -w^3 + 5/2*w^2 + 9/2*w - 12],
[683, 683, -w^3 + 1/2*w^2 + 13/2*w + 6],
[683, 683, 3/2*w^3 - w^2 - 11/2*w + 10],
[709, 709, -1/2*w^3 + w^2 + 9/2*w - 10],
[709, 709, 1/2*w^3 - 1/2*w^2 - 5*w - 5],
[719, 719, -3/2*w^3 + 3/2*w^2 + 11*w - 3],
[719, 719, -3/2*w^3 + 3*w^2 + 19/2*w - 8],
[733, 733, w^3 - 3*w^2 - 8*w + 13],
[733, 733, -w^3 + 7/2*w^2 + 7/2*w - 12],
[733, 733, w^3 + 1/2*w^2 - 15/2*w - 6],
[733, 733, -3/2*w^3 + 9/2*w^2 + 8*w - 19],
[757, 757, w^3 - w^2 - 4*w + 1],
[757, 757, -2*w^3 + 7/2*w^2 + 25/2*w - 10],
[769, 769, -2*w^2 + 4*w + 7],
[769, 769, -2*w^2 + 9],
[827, 827, -1/2*w^3 - 1/2*w^2 + 6*w - 1],
[827, 827, -1/2*w^3 + 2*w^2 + 7/2*w - 4],
[829, 829, 1/2*w^3 + 2*w^2 - 7/2*w - 12],
[829, 829, 7/2*w^3 - 8*w^2 - 41/2*w + 32],
[829, 829, 3/2*w^3 - w^2 - 15/2*w - 2],
[829, 829, -w^3 + w^2 + 8*w + 5],
[839, 839, 3/2*w^3 - w^2 - 23/2*w],
[839, 839, -3/2*w^3 + 7/2*w^2 + 9*w - 11],
[853, 853, 1/2*w^3 - 3/2*w^2 - 2*w + 11],
[853, 853, -1/2*w^3 + 7/2*w + 8],
[863, 863, 2*w^3 - 11/2*w^2 - 25/2*w + 26],
[863, 863, 3/2*w^3 - 5/2*w^2 - 10*w + 13],
[887, 887, 2*w^3 - 7/2*w^2 - 29/2*w + 8],
[887, 887, 2*w^3 - 5/2*w^2 - 31/2*w + 8],
[911, 911, 2*w^3 - w^2 - 15*w - 3],
[911, 911, 2*w^3 - 5*w^2 - 11*w + 17],
[947, 947, -w^3 + 3*w^2 + 6*w - 9],
[947, 947, w^3 - 9*w - 1],
[961, 31, -3/2*w^3 + 3*w^2 + 23/2*w - 8],
[961, 31, 3/2*w^3 - 3/2*w^2 - 13*w + 5],
[983, 983, -1/2*w^3 + 2*w^2 - 1/2*w - 6],
[983, 983, 1/2*w^3 + 1/2*w^2 - 2*w - 5],
[997, 997, w^3 - 13*w + 7],
[997, 997, 1/2*w^3 - 5/2*w^2 - w + 9],
[997, 997, 1/2*w^3 + w^2 - 9/2*w - 6],
[997, 997, -w^3 + 3*w^2 + 10*w - 5]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^4 + x^3 - 29*x^2 - 51*x - 18;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [0, 0, e, 1/3*e^3 + 1/3*e^2 - 29/3*e - 12, 1/3*e^3 + 1/3*e^2 - 29/3*e - 12, 0, 0, e^3 - 30*e - 26, e^3 - 30*e - 26, 0, 0, -2/3*e^3 + 1/3*e^2 + 52/3*e + 10, e^3 - 27*e - 20, e^3 - 27*e - 20, -2/3*e^3 + 1/3*e^2 + 52/3*e + 10, 0, 0, 0, 0, -2*e^3 + 54*e + 46, 0, 0, -4/3*e^3 + 2/3*e^2 + 113/3*e + 24, -e^2 + 2*e + 22, -e^2 + 2*e + 22, -4/3*e^3 + 2/3*e^2 + 113/3*e + 24, -1/3*e^3 + 2/3*e^2 + 32/3*e - 10, e^3 - 27*e - 32, 0, 0, -2/3*e^3 - 2/3*e^2 + 67/3*e + 36, -2/3*e^3 - 2/3*e^2 + 67/3*e + 36, 0, 0, 0, 0, 26, 0, 0, -1/3*e^3 - 4/3*e^2 + 26/3*e + 30, -1/3*e^3 - 4/3*e^2 + 26/3*e + 30, 0, 0, 0, 0, 2/3*e^3 - 4/3*e^2 - 55/3*e - 4, 2/3*e^3 - 1/3*e^2 - 70/3*e - 6, 2/3*e^3 - 1/3*e^2 - 70/3*e - 6, 2/3*e^3 - 4/3*e^2 - 55/3*e - 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/3*e^3 + 4/3*e^2 - 44/3*e - 38, 1/3*e^3 + 4/3*e^2 - 44/3*e - 38, 0, 0, 5/3*e^3 + 2/3*e^2 - 166/3*e - 54, 5/3*e^3 + 2/3*e^2 - 166/3*e - 54, 0, 0, e^3 - 27*e - 20, e^3 - 27*e - 20, 0, 0, 8/3*e^3 + 2/3*e^2 - 247/3*e - 64, 8/3*e^3 + 2/3*e^2 - 247/3*e - 64, 0, 0, 5/3*e^3 - 1/3*e^2 - 151/3*e - 52, 5/3*e^3 - 1/3*e^2 - 151/3*e - 52, 0, 0, 0, 0, 1/3*e^3 - 5/3*e^2 - 35/3*e + 28, 1/3*e^3 - 5/3*e^2 - 35/3*e + 28, 0, 0, -3*e^3 + 81*e + 56, -3*e^3 + 81*e + 56, 0, 0, 0, 0, 0, 0, 0, 0, e^3 - 27*e - 56, -3*e^3 + 81*e + 68, -3*e^3 + 81*e + 68, 0, 0, -5/3*e^3 + 7/3*e^2 + 139/3*e + 16, -5/3*e^3 + 7/3*e^2 + 139/3*e + 16, 0, 0, 0, 0, e^3 - 27*e + 16, e^3 - 27*e + 16, 50, 0, 0, 4*e^3 + e^2 - 116*e - 106, -8/3*e^3 - 5/3*e^2 + 244/3*e + 70, -8/3*e^3 - 5/3*e^2 + 244/3*e + 70, 4*e^3 + e^2 - 116*e - 106, 0, 0, 0, 0, -3*e^3 + 81*e + 92, -3*e^3 + 81*e + 92, 0, 0, 10/3*e^3 + 1/3*e^2 - 308/3*e - 102, 2*e^3 + 2*e^2 - 55*e - 76, 2*e^3 + 2*e^2 - 55*e - 76, 10/3*e^3 + 1/3*e^2 - 308/3*e - 102, -14/3*e^3 - 2/3*e^2 + 427/3*e + 124, -14/3*e^3 - 2/3*e^2 + 427/3*e + 124, -11/3*e^3 - 2/3*e^2 + 346/3*e + 90, -11/3*e^3 - 2/3*e^2 + 346/3*e + 90, 0, 0, 4*e^3 - 108*e - 74, -2*e^3 + 54*e + 58, -2*e^3 + 54*e + 58, 4*e^3 - 108*e - 74, 0, 0, -7/3*e^3 - 4/3*e^2 + 188/3*e + 74, -7/3*e^3 - 4/3*e^2 + 188/3*e + 74, 0, 0, 0, 0, 0, 0, 0, 0, e^3 - 27*e - 56, e^3 - 27*e - 56, 0, 0, -e^3 - e^2 + 41*e + 44, 6*e^3 - 162*e - 142, 6*e^3 - 162*e - 142, -e^3 - e^2 + 41*e + 44];
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 + 1/2*w - 1}>] := -1;
ALEigenvalues[ideal<ZF | {2, 2, -1/2*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;