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

NN := ideal<ZF | {5, 5, -w + 3}>;

primesArray := [
[2, 2, -w + 2],
[4, 2, -w^2 - w + 5],
[5, 5, -w + 3],
[7, 7, -w^2 - 2*w + 3],
[11, 11, -w^2 + 5],
[13, 13, w + 1],
[23, 23, w^2 - 3],
[25, 5, w^2 + 2*w - 1],
[27, 3, -3],
[29, 29, -3*w + 7],
[43, 43, -3*w^2 - 2*w + 17],
[49, 7, -2*w^2 + w + 11],
[67, 67, -2*w^2 - 4*w + 3],
[71, 71, -2*w^2 + w + 9],
[73, 73, w^2 + 2*w - 7],
[73, 73, -2*w^2 - 2*w + 11],
[73, 73, w - 5],
[89, 89, 2*w^2 + w - 9],
[89, 89, -w^2 - 2*w + 9],
[89, 89, -2*w - 1],
[101, 101, 4*w^2 + 4*w - 19],
[103, 103, 2*w + 3],
[107, 107, w + 5],
[109, 109, 3*w^2 + 2*w - 15],
[113, 113, 2*w - 7],
[121, 11, 2*w^2 + w - 7],
[127, 127, 2*w^2 - 9],
[127, 127, -4*w + 11],
[127, 127, -2*w^2 + 2*w + 9],
[131, 131, 3*w^2 + 4*w - 13],
[137, 137, -w^2 + 2*w - 3],
[149, 149, 2*w^2 - 6*w + 1],
[151, 151, 3*w - 1],
[157, 157, -3*w^2 - 6*w + 7],
[157, 157, -2*w^2 + 3*w + 3],
[157, 157, 2*w^2 + 2*w - 15],
[163, 163, 3*w^2 + 2*w - 19],
[163, 163, 4*w^2 + 5*w - 17],
[163, 163, -5*w^2 - 6*w + 19],
[167, 167, w^2 - 11],
[167, 167, -3*w - 7],
[167, 167, -w^2 - 2*w + 13],
[169, 13, w^2 - 2*w - 5],
[181, 181, -2*w - 7],
[181, 181, 2*w^2 - 7],
[181, 181, 2*w^2 - 2*w - 5],
[193, 193, 2*w^2 + 2*w - 3],
[197, 197, -4*w^2 - 7*w + 9],
[197, 197, 2*w^2 - 2*w - 19],
[197, 197, 2*w^2 + 3*w - 11],
[199, 199, -6*w^2 + 37],
[227, 227, 2*w^2 - 3*w - 7],
[233, 233, w^2 + 4*w - 15],
[251, 251, -4*w^2 + 23],
[257, 257, w^2 - 2*w - 7],
[269, 269, -4*w^2 - w + 23],
[271, 271, 2*w^2 - w - 19],
[277, 277, -3*w^2 - 2*w + 9],
[281, 281, w^2 - 2*w - 9],
[283, 283, 6*w^2 + 5*w - 31],
[293, 293, 2*w^2 - 2*w - 21],
[311, 311, -w^2 - 4*w - 1],
[317, 317, 3*w^2 + 2*w - 11],
[317, 317, 2*w^2 + 3*w - 13],
[317, 317, 2*w^2 - 2*w - 11],
[331, 331, 4*w^2 - w - 21],
[331, 331, 3*w + 5],
[331, 331, 4*w^2 + 2*w - 21],
[337, 337, -2*w^2 + 4*w + 5],
[349, 349, w^2 - 4*w - 1],
[353, 353, -w^2 - 4*w - 5],
[367, 367, 6*w^2 + 7*w - 23],
[379, 379, 2*w - 9],
[389, 389, 2*w^2 + 3*w - 15],
[389, 389, w^2 + 4*w - 13],
[389, 389, 2*w^2 + 4*w - 1],
[397, 397, 4*w - 1],
[409, 409, 2*w^2 + w - 19],
[409, 409, -w^2 + 6*w - 11],
[409, 409, 2*w^2 + 4*w - 21],
[419, 419, 4*w^2 + 4*w - 21],
[421, 421, -w^2 - 3],
[421, 421, -4*w^2 - 6*w + 17],
[421, 421, 4*w^2 + 3*w - 31],
[439, 439, 5*w^2 - 29],
[443, 443, w^2 - 13],
[449, 449, 4*w^2 - w - 27],
[457, 457, 2*w^2 - 19],
[467, 467, 4*w + 9],
[467, 467, 3*w^2 - 2*w - 17],
[467, 467, 2*w^2 - 2*w - 13],
[479, 479, -w^2 + 4*w + 19],
[487, 487, 4*w^2 + 5*w - 19],
[491, 491, 3*w - 11],
[503, 503, 4*w^2 + 3*w - 17],
[523, 523, -2*w^2 - 5*w - 1],
[529, 23, -4*w^2 - w + 21],
[547, 547, 5*w^2 + 4*w - 23],
[557, 557, 4*w^2 - 2*w - 33],
[563, 563, 2*w^2 + 2*w - 1],
[563, 563, 3*w^2 - 8*w - 1],
[563, 563, 3*w^2 + 4*w - 5],
[571, 571, 5*w^2 + 10*w - 11],
[577, 577, -4*w + 13],
[587, 587, 3*w^2 + 4*w - 17],
[593, 593, 3*w^2 - 11],
[593, 593, 4*w^2 + 2*w - 19],
[593, 593, w - 9],
[601, 601, -4*w^2 - 10*w + 3],
[607, 607, 3*w^2 + 4*w - 27],
[607, 607, -4*w^2 - 2*w + 33],
[607, 607, 2*w^2 - 7*w + 9],
[617, 617, -2*w^2 + 9*w - 5],
[619, 619, -4*w - 1],
[631, 631, -8*w^2 - 10*w + 29],
[641, 641, w^2 - 4*w - 3],
[653, 653, 3*w^2 - 2*w - 9],
[659, 659, 2*w^2 + 4*w - 19],
[661, 661, 6*w^2 + 6*w - 25],
[673, 673, 6*w^2 + 7*w - 27],
[683, 683, 5*w^2 + 2*w - 27],
[701, 701, 7*w - 13],
[709, 709, 6*w^2 + 8*w - 25],
[719, 719, -2*w^2 + 4*w + 7],
[719, 719, -7*w^2 - 10*w + 27],
[719, 719, 4*w^2 - w - 19],
[727, 727, 3*w^2 + 8*w + 1],
[733, 733, 3*w^2 - 2*w - 7],
[733, 733, 3*w^2 - 2*w - 29],
[739, 739, -w - 9],
[751, 751, -w^2 - 4*w + 19],
[757, 757, -4*w - 7],
[761, 761, -5*w^2 - 4*w + 29],
[787, 787, -2*w^2 - 6*w + 3],
[809, 809, -w^2 + 4*w - 9],
[809, 809, -4*w^2 + 3*w + 15],
[809, 809, 2*w^2 - 9*w + 13],
[821, 821, -4*w^2 - 9*w + 7],
[821, 821, 5*w - 1],
[821, 821, 8*w^2 + 11*w - 31],
[827, 827, 8*w^2 + 8*w - 35],
[829, 829, 2*w^2 - 5*w - 27],
[839, 839, -w^2 - 6*w - 7],
[841, 29, 3*w^2 + 4*w - 31],
[863, 863, 4*w^2 - 19],
[883, 883, -5*w^2 + 27],
[887, 887, 5*w^2 + 4*w - 21],
[907, 907, -6*w^2 + 2*w + 31],
[907, 907, 2*w^2 - 5*w - 29],
[907, 907, 2*w^2 + 5*w - 15],
[911, 911, w^2 - 15],
[919, 919, 8*w - 17],
[941, 941, -4*w^2 + 5*w + 15],
[947, 947, 4*w^2 + 2*w - 13],
[947, 947, 3*w^2 - 6*w - 7],
[947, 947, -5*w^2 - 4*w + 15]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^4 + x^3 - 8*x^2 - 4*x + 12;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, 1/2*e^3 + 1/2*e^2 - 2*e - 1, 1, -e - 1, -1/2*e^3 - 1/2*e^2 + 3*e + 3, e^2 - e - 4, -1/2*e^3 - 7/2*e^2 + 15, -3*e^2 - 2*e + 11, 1/2*e^3 + 3/2*e^2 - 4*e - 5, -3/2*e^3 - 3/2*e^2 + 9*e + 3, -e^3 + 7*e + 2, 1/2*e^3 + 1/2*e^2 - e + 5, 2*e^3 + e^2 - 11*e + 2, -e^3 + 7*e + 6, -e^3 - e^2 + 6*e + 8, -1/2*e^3 + 3/2*e^2 + 3*e - 1, 1/2*e^3 - 1/2*e^2 - 2*e + 5, 3*e^3 + 3*e^2 - 16*e - 6, -e^3 - 2*e^2 + 6*e + 9, e^3 + 2*e^2 - 7*e - 6, e^3 - e^2 - 10*e + 6, 2*e^3 - e^2 - 11*e + 8, e^3 + 2*e^2 - 5*e + 6, e^3 - 2*e^2 - 9*e + 8, e^3 + 3*e^2 + e - 15, -e^3 - e^2 + 2, -3/2*e^3 - 1/2*e^2 + 6*e - 7, 2*e^2 - 10, -2*e^3 - 6*e^2 + 8*e + 20, 4*e^2 + 3*e - 21, -2*e^3 - 2*e^2 + 11*e - 3, -2*e^3 - e^2 + 13*e - 6, -e^3 - 3*e^2 + e + 11, -2*e^3 - 2*e^2 + 6*e - 4, 2*e^3 - 8*e + 8, -e^3 - 4*e^2 + e + 26, -2*e^3 + 3*e^2 + 11*e - 22, 2*e^3 + e^2 - 6*e - 1, 3*e^3 + 2*e^2 - 15*e - 10, -e^3 - 4*e^2 + e + 12, -e^3 + 4*e^2 + 11*e - 18, -1/2*e^3 - 1/2*e^2 + 5*e + 9, 2*e^3 + 4*e^2 - 6*e - 22, -2*e^3 + e^2 + 17*e - 4, 2*e^3 - 3*e^2 - 9*e + 26, -2*e^3 + 3*e^2 + 12*e - 19, e^3 - 7*e - 4, 3*e^3 + 9*e^2 - 16*e - 24, e^3 - 5*e^2 - 7*e + 21, 4*e^2 + e - 21, 3*e^3 + 6*e^2 - 13*e - 16, 2*e^3 + 8*e^2 - 6*e - 24, 2*e^3 + e^2 - 5*e + 6, -3*e^3 - 7*e^2 + 16*e + 24, 3/2*e^3 - 1/2*e^2 - 7*e + 9, 1/2*e^3 + 15/2*e^2 - 4*e - 33, -e^2 - 5*e - 4, e^2 + 7*e - 4, -3/2*e^3 - 1/2*e^2 + 10*e - 9, -2*e^3 + 6*e - 16, 4*e^3 + 9*e^2 - 17*e - 24, 3*e^3 - 3*e^2 - 24*e + 6, -2*e^3 + e^2 + 17*e - 12, -e^3 - 6*e^2 + 5*e + 24, -5*e^3 - e^2 + 24*e - 6, -2*e^3 - e^2 + 16*e + 11, -5*e^2 - 3*e + 14, -2*e^3 - 7*e^2 + 11*e + 20, 4*e^3 + 9*e^2 - 17*e - 28, -3*e^3 + 2*e^2 + 19*e - 4, -3*e^3 - 3*e^2 + 18*e + 12, -3*e^3 - 2*e^2 + 20*e + 11, -5/2*e^3 - 7/2*e^2 + 4*e + 5, 4*e^3 - e^2 - 17*e + 18, e^3 - 4*e^2 - 15*e + 12, -e^3 - 4*e^2 + 5*e + 6, 1/2*e^3 - 11/2*e^2 - 17*e + 23, -3/2*e^3 - 11/2*e^2 + 3*e + 29, -3*e^3 + 6*e^2 + 19*e - 28, 2*e^3 + 3*e^2 - 11*e - 10, -1/2*e^3 - 1/2*e^2 + e - 15, 7/2*e^3 + 9/2*e^2 - 10*e - 25, 2*e^3 + e^2 - 13*e + 2, -5*e^3 - 7*e^2 + 18*e + 26, 7/2*e^3 + 5/2*e^2 - 22*e - 1, 2*e^3 + 2*e^2 - 7*e - 3, 3*e^3 + 3*e^2 - 16*e - 6, -3*e^3 + 17*e - 16, 2*e^3 + 5*e^2 - e - 24, 2*e^3 + 7*e^2 - 5*e - 30, -e^3 - 4*e^2 + 7*e + 6, -e^3 + 2*e^2 + e - 18, -3/2*e^3 - 17/2*e^2 + 35, 3*e^3 - 2*e^2 - 9*e + 24, -5*e^3 + 2*e^2 + 30*e - 9, -4*e^3 - 2*e^2 + 30*e + 8, -4*e^2 - 8*e + 14, -2*e^3 + 3*e^2 + 15*e - 10, -2*e^3 - 3*e^2 + 15*e - 18, -e^3 + 8*e^2 + 13*e - 24, 3*e^3 + 11*e^2 - 12*e - 48, 4*e - 24, -3*e^3 - 10*e^2 + 7*e + 26, -3*e^3 - e^2 + 18*e + 26, -4*e^3 - 7*e^2 + 17*e + 24, -5*e^2 + 3*e + 18, 9*e^2 + e - 30, -e^3 - 11*e^2 + 36, e^3 + 8*e^2 - 7*e - 28, e^3 - e^2 - 10*e + 32, 3*e^3 + 7*e^2 - 12*e - 40, -3*e^3 + 3*e^2 + 22*e + 8, 3/2*e^3 + 5/2*e^2 + 4*e - 15, -3*e^3 + 11*e^2 + 22*e - 52, -e^3 + 3*e^2 + 13*e - 31, 2*e^3 - 5*e^2 - 25*e + 24, -4*e^3 - e^2 + 13*e - 24, 7*e^3 - 48*e + 3, e^3 + e^2 - 16*e + 14, 2*e^3 - e^2 - 17*e - 4, -4*e^3 + 4*e^2 + 24*e - 36, -5*e^3 - 11*e^2 + 24*e + 36, -e^3 + 4*e^2 - e - 16, 5/2*e^3 + 15/2*e^2 - 18*e - 27, 6*e^3 + 3*e^2 - 25*e - 6, 2*e^3 - 7*e^2 - 15*e + 18, 8*e^2 + 2*e - 46, -7/2*e^3 - 15/2*e^2 + 13*e + 35, -5*e^3 - 10*e^2 + 33*e + 20, 3/2*e^3 - 7/2*e^2 - 22*e + 23, 3*e^3 - 5*e^2 - 16*e + 32, e^2 - 9*e + 20, -3/2*e^3 + 13/2*e^2 + 7*e - 51, -e^3 + 5*e^2 + 12*e - 22, e^3 + e^2 - 14*e - 12, -4*e^3 - 2*e^2 + 28*e - 12, -2*e^2 - 5*e + 15, 9/2*e^3 - 1/2*e^2 - 34*e - 3, -e^3 + 12*e^2 + 5*e - 66, 4*e^3 - 4*e^2 - 24*e + 30, -e^3 - 7*e^2 + 2*e + 12, -2*e^3 + 9*e^2 + 21*e - 16, 4*e^3 - 26*e + 18, -1/2*e^3 + 13/2*e^2 + 14*e - 19, 8*e^3 + 9*e^2 - 38*e - 39, 3*e^3 - 37*e + 2, -3*e^3 + 2*e^2 + 27*e + 6, -7*e^3 - 4*e^2 + 43*e + 20, -6*e^3 - 11*e^2 + 32*e + 17, 4*e^3 + 7*e^2 - 13*e - 28, 4*e^3 + 14*e^2 - 22*e - 48, -3*e^3 + 4*e^2 + e - 46, -3*e^3 - 4*e^2 + 17*e, 8*e^3 + 8*e^2 - 34*e - 30, -e^3 - 4*e^2 - 8*e - 9, -4*e^3 + 3*e^2 + 21*e - 6];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {5, 5, -w + 3}>] := -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;