/*
  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![-3, -6, 0, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {23, 23, -w^2 + 2}>;

primesArray := [
[2, 2, w + 1],
[3, 3, w],
[4, 2, -w^2 + w + 5],
[7, 7, -w + 2],
[19, 19, -2*w^2 + w + 10],
[23, 23, -w^2 + 8],
[23, 23, -w^2 + 2],
[29, 29, -w^2 + 2*w + 4],
[37, 37, w^2 - 2*w - 8],
[41, 41, 4*w^2 - 3*w - 20],
[43, 43, -w - 4],
[47, 47, 2*w - 1],
[49, 7, -2*w^2 + 3*w + 4],
[59, 59, 2*w^2 - 13],
[61, 61, -3*w^2 + 4*w + 10],
[67, 67, 2*w^2 - 2*w - 7],
[71, 71, -3*w - 4],
[79, 79, 2*w^2 - 3*w - 8],
[83, 83, 2*w^2 - w - 14],
[83, 83, -4*w^2 + 3*w + 22],
[83, 83, 2*w^2 - w - 8],
[97, 97, 4*w^2 - 2*w - 25],
[101, 101, 3*w^2 - 2*w - 14],
[103, 103, 2*w^2 - 7],
[109, 109, 2*w^2 + 2*w - 1],
[113, 113, w^2 - 4*w - 4],
[113, 113, 4*w^2 - 4*w - 17],
[113, 113, 2*w^2 - 2*w - 5],
[125, 5, -5],
[131, 131, -5*w^2 + 2*w + 26],
[137, 137, 3*w^2 - 2*w - 20],
[137, 137, -w^2 + 2*w - 2],
[137, 137, -w^2 - 2],
[149, 149, w^2 + 2*w - 4],
[149, 149, 2*w^2 + w - 2],
[149, 149, 4*w^2 - 4*w - 19],
[151, 151, w^2 - 10],
[151, 151, -6*w^2 + 4*w + 31],
[151, 151, 2*w^2 - 3*w - 14],
[157, 157, 2*w^2 + w - 4],
[163, 163, 4*w + 5],
[163, 163, 2*w^2 - w - 4],
[163, 163, 3*w^2 - 20],
[167, 167, 2*w^2 - 3*w - 10],
[173, 173, 2*w^2 - 5],
[179, 179, -3*w^2 + 16],
[181, 181, 3*w - 2],
[197, 197, 3*w^2 - 4*w - 8],
[199, 199, -2*w - 7],
[229, 229, -w^2 - 2*w - 4],
[233, 233, 3*w - 4],
[239, 239, 5*w^2 - 2*w - 32],
[241, 241, 2*w^2 + 1],
[257, 257, w^2 + 2*w - 10],
[269, 269, -5*w^2 + 4*w + 28],
[283, 283, 2*w^2 + w - 14],
[307, 307, -6*w - 5],
[307, 307, -3*w^2 + 4*w + 14],
[307, 307, 2*w^2 - w - 16],
[311, 311, -w^2 - 2*w + 14],
[313, 313, -2*w^2 + 4*w - 1],
[317, 317, 3*w^2 + 4*w - 2],
[337, 337, w^2 - 4*w - 8],
[347, 347, -8*w^2 + 3*w + 46],
[353, 353, 2*w^2 - 17],
[361, 19, -3*w^2 + 2*w + 4],
[367, 367, 2*w^2 + 5*w - 2],
[373, 373, -2*w^2 + 2*w - 1],
[379, 379, 3*w^2 - 4*w - 20],
[397, 397, -w^2 + 2*w - 4],
[397, 397, 3*w^2 - 10],
[397, 397, 3*w^2 - 2*w - 10],
[409, 409, -w^2 - 4],
[409, 409, -4*w^2 + 4*w + 23],
[409, 409, 2*w^2 - 5*w - 8],
[421, 421, 2*w^2 - 4*w - 11],
[433, 433, 8*w^2 - 4*w - 49],
[443, 443, -3*w^2 + 4*w + 16],
[449, 449, 6*w^2 - 3*w - 38],
[457, 457, -6*w^2 + 5*w + 32],
[461, 461, w - 8],
[467, 467, -w - 8],
[467, 467, 2*w^2 - 4*w - 13],
[467, 467, -4*w - 11],
[491, 491, 2*w^2 + 6*w - 1],
[509, 509, 8*w^2 - 6*w - 43],
[521, 521, 4*w^2 - 2*w - 17],
[521, 521, 6*w + 7],
[521, 521, 4*w - 7],
[523, 523, 5*w^2 - 4*w - 22],
[541, 541, 6*w^2 - w - 32],
[541, 541, 4*w^2 - 8*w - 11],
[541, 541, 3*w^2 - 4],
[547, 547, 2*w^2 + 2*w - 17],
[547, 547, 4*w - 5],
[547, 547, -3*w^2 + 8*w + 8],
[557, 557, 3*w^2 - 8],
[557, 557, 6*w^2 - 4*w - 29],
[557, 557, -2*w^2 + w - 2],
[571, 571, -2*w^2 - 4*w + 5],
[587, 587, 3*w^2 + 2*w + 2],
[593, 593, -6*w^2 + 5*w + 34],
[599, 599, -w^2 - 8*w - 10],
[601, 601, -4*w^2 - 6*w + 1],
[601, 601, 4*w^2 - 3*w - 28],
[601, 601, 2*w^2 + 2*w - 11],
[613, 613, 6*w^2 - w - 38],
[617, 617, 2*w^2 - 5*w - 10],
[617, 617, -10*w^2 + 6*w + 53],
[617, 617, 7*w^2 - 4*w - 44],
[619, 619, -3*w^2 - 4*w + 8],
[631, 631, 6*w^2 - 6*w - 29],
[641, 641, w^2 + 4*w - 4],
[641, 641, -6*w^2 + 2*w + 29],
[641, 641, -w^2 - 4*w - 8],
[643, 643, 4*w^2 - w - 16],
[647, 647, 3*w^2 + 6*w - 4],
[653, 653, 7*w^2 - 6*w - 32],
[661, 661, 4*w^2 - 4*w - 13],
[673, 673, 2*w^2 + 8*w + 1],
[673, 673, -8*w^2 + 6*w + 47],
[673, 673, 2*w^2 + 3*w - 8],
[683, 683, 8*w^2 - 7*w - 40],
[709, 709, 5*w^2 - 26],
[719, 719, 2*w^2 + 2*w - 13],
[727, 727, -w^2 + 6*w - 2],
[733, 733, w^2 - 6*w - 8],
[751, 751, 4*w^2 - 7*w - 14],
[757, 757, 4*w^2 - 5*w - 26],
[761, 761, 2*w^2 - 5*w - 16],
[769, 769, -2*w^2 + 8*w + 1],
[787, 787, 5*w^2 - 2*w - 34],
[797, 797, 2*w^2 + w - 20],
[797, 797, 4*w^2 - 3*w - 14],
[797, 797, -8*w^2 + 2*w + 53],
[809, 809, -5*w^2 - 2*w + 40],
[811, 811, -7*w - 8],
[811, 811, 3*w^2 + 2*w - 14],
[811, 811, -6*w^2 + 8*w + 19],
[821, 821, -6*w^2 + w + 34],
[823, 823, 4*w^2 + 2*w - 17],
[823, 823, 4*w^2 - 4*w - 7],
[823, 823, 4*w^2 - 6*w - 17],
[841, 29, 2*w^2 - 5*w - 14],
[857, 857, -8*w^2 + 4*w + 41],
[859, 859, -10*w^2 + 7*w + 56],
[859, 859, 2*w^2 - 19],
[859, 859, 5*w^2 - 2*w - 22],
[863, 863, 6*w - 1],
[887, 887, w^2 - 14],
[907, 907, 10*w^2 - 9*w - 46],
[911, 911, -7*w - 10],
[911, 911, 6*w^2 - 5*w - 26],
[911, 911, 5*w - 4],
[919, 919, -2*w^2 - 11*w - 8],
[929, 929, w^2 - 6*w - 10],
[937, 937, w - 10],
[947, 947, 5*w^2 - 4*w - 20],
[967, 967, -7*w^2 + 6*w + 38],
[967, 967, 2*w^2 - 2*w - 19],
[967, 967, 8*w^2 - 5*w - 40],
[977, 977, 2*w^2 + 3*w - 10],
[977, 977, 2*w^2 + 4*w - 7],
[977, 977, 2*w^2 - 6*w - 19],
[997, 997, -12*w^2 + 7*w + 64],
[997, 997, 4*w^2 - 13],
[997, 997, 5*w^2 - 32]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^4 + x^3 - 5*x^2 - 4*x + 3;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, e^2 - 3, -1, -e - 1, e^3 - e^2 - 4*e + 2, -2*e^3 + 7*e - 3, -1, -e^2 + 3, -e^3 - e^2 + 6*e + 2, 2*e^3 + 2*e^2 - 6*e - 6, -2*e^3 + 2*e^2 + 8*e - 4, 2*e^3 - 10*e, -2*e^2 - 4*e + 8, -2*e^3 + e^2 + 6*e + 3, 4*e^3 - 2*e^2 - 16*e + 2, 2*e^3 - 3*e^2 - 8*e + 5, -4*e^2 + 4*e + 12, -3*e^3 - 3*e^2 + 14*e + 8, 2*e^3 - 10*e - 6, 6*e^3 - 2*e^2 - 22*e + 6, e^3 - e^2 - 6*e, 2*e^3 - 4*e + 2, -4*e^3 - e^2 + 20*e + 3, 2*e^3 - 8*e + 2, -2*e^3 + 2*e^2 + 12*e - 4, 4*e^3 + 2*e^2 - 13*e - 3, 2*e^3 - 2*e^2 - 5*e + 3, 2*e^3 - 8*e + 6, -2*e^3 + 4*e, e^3 - 3*e^2 - 8*e + 6, -4*e^2 + 18, 2*e^3 - 4*e^2 - 8*e + 6, -6*e^2 - 2*e + 12, -4*e^3 - 4*e^2 + 8*e + 12, -4*e^3 + 6*e^2 + 18*e - 18, -2*e^3 + 6*e^2 + 8*e - 12, -2*e^3 + 2*e^2 + 8*e - 16, 4*e^3 + 2*e^2 - 12*e - 10, 2*e^3 + e^2 - 4*e - 7, 2*e^3 + 2*e^2 - 7*e - 7, -4*e^3 + 4*e^2 + 12*e - 16, -8*e^3 - 4*e^2 + 28*e + 8, -3*e^3 + e^2 + 16*e - 10, -4*e^3 + 14*e + 6, -8*e^3 + 2*e^2 + 32*e, 6*e^3 - 21*e - 9, -2*e^3 - 2*e^2 + 5*e + 5, -4*e^3 + 4*e^2 + 18*e, 4*e^3 + 4*e^2 - 12*e - 16, -6*e^3 + 2*e^2 + 20*e - 10, -4*e^3 - 2*e^2 + 23*e + 9, -2*e^3 + 4*e^2 + 2*e - 12, e^2 - 4*e - 19, -2*e^3 + 2*e^2 + 7*e - 9, -4*e^3 + 6*e^2 + 9*e - 33, -2*e^3 + 14*e - 16, 4*e^3 + 3*e^2 - 14*e + 5, 6*e^3 - 2*e^2 - 16*e + 20, 6*e^3 + 6*e^2 - 22*e - 10, -7*e^3 + 5*e^2 + 22*e - 24, -2*e^3 + 2*e^2 + 10*e - 4, 3*e^3 + 3*e^2 - 16*e - 12, -2*e^3 - 4*e^2 + 14*e + 8, -4*e^3 + 3*e^2 + 10*e - 27, 4*e^3 + 6*e^2 - 12*e - 12, -6*e^3 - 4*e^2 + 28*e + 8, -8*e^2 - 10*e + 26, 8*e^3 + e^2 - 46*e - 13, 4*e^2 + 7*e - 19, 8*e^3 - 5*e^2 - 36*e + 11, 12*e^3 + 2*e^2 - 48*e - 10, -12*e^3 + 4*e^2 + 40*e - 16, -7*e^2 + 2*e - 1, -8*e^3 + 4*e^2 + 26*e - 10, -4*e^3 - 8*e^2 + 22*e + 14, 12*e^3 + 5*e^2 - 48*e - 19, -2*e^3 + 8*e^2 + 16*e - 16, 7*e^3 - 5*e^2 - 22*e + 12, 12*e^3 - 2*e^2 - 46*e + 6, -11*e^2 + 6*e + 35, 4*e^3 - 20*e, 6*e^3 - 32*e + 6, -6*e^3 + 8*e^2 + 29*e - 15, -e^3 + e^2 - 4*e - 6, -4*e^3 + 2*e^2 + 16*e - 6, -6*e^3 + 2*e^2 + 32*e - 6, -7*e^3 + 5*e^2 + 16*e - 24, -4*e^3 + 10*e^2 + 30*e - 30, 5*e^3 + e^2 - 38*e - 6, -2*e^3 + 12*e + 2, 10*e^3 - 12*e^2 - 40*e + 38, 4*e^3 + 6*e^2 - 7*e - 25, -2*e^3 + 4*e + 8, 4*e^2 - 14*e - 22, 6*e^3 - 4*e^2 - 10*e + 20, -12*e^3 - 4*e^2 + 55*e + 17, -9*e^3 + 7*e^2 + 26*e - 30, -3*e^3 + 9*e^2 + 12*e - 24, -2*e^3 + 14*e^2 + 10*e - 36, 6*e^3 - 3*e^2 - 18*e + 23, 6*e^3 - e^2 - 30*e + 9, -8*e^3 - 8*e^2 + 24*e + 18, 4*e^3 - e + 15, -4*e^3 + 18*e - 22, 2*e^3 + 4*e^2 - 2*e - 4, -2*e^3 + 2*e^2 - e - 25, 7*e^3 + 3*e^2 - 28*e - 28, 2*e^3 + 6*e^2 - 3*e - 15, 10*e^3 + 10*e^2 - 42*e - 6, 4*e^3 + 7*e^2 - 10*e - 27, -6*e^3 + 7*e^2 + 32*e - 25, -6*e^3 - 8*e^2 + 30*e + 20, 11*e^3 + 3*e^2 - 54*e - 18, 5*e^3 + 5*e^2 - 18*e - 6, -6*e^3 - 12*e^2 + 22*e + 30, -2*e^3 + 8*e^2 - 4*e - 22, 4*e^2 + 10*e - 6, 2*e^3 - 4*e^2 - 12*e - 12, 4*e^3 - 16*e + 2, -5*e^3 + 15*e^2 + 28*e - 52, -4*e^3 - 6*e^2 + 6*e + 14, 2*e^3 - 6*e + 8, 12*e^3 + 8*e^2 - 39*e - 27, -11*e^3 + 5*e^2 + 48*e + 8, -e^3 - 7*e^2 + 6*e - 12, -12*e^3 + 7*e^2 + 50*e - 19, -2*e^3 + 2*e^2 - 11*e - 7, -8*e^3 - 18*e^2 + 26*e + 44, -4*e^3 + 2*e^2 + 21*e - 13, 4*e^3 - 3*e^2 - 6*e + 3, 4*e^3 + 6*e^2 + e - 25, -16*e^3 + 2*e^2 + 60*e + 14, 8*e^3 - 14*e^2 - 34*e + 30, 4*e^3 - 2*e^2 - 10*e - 12, -8*e^3 + 12*e^2 + 30*e - 54, 12*e^3 - 6*e^2 - 48*e + 18, 10*e^3 - 64*e - 10, -6*e^3 + 4*e^2 + 22*e - 10, 8*e^3 + 6*e^2 - 20*e - 22, 8*e^3 - 24*e + 18, -8*e^3 + 16*e^2 + 28*e - 52, 6*e^3 + 8*e^2 - 8*e - 34, -11*e^3 - 9*e^2 + 50*e + 8, -3*e^3 + 13*e^2 + 2*e - 58, 6*e^3 + 6*e^2 - 32*e - 6, 2*e^3 - 24*e^2 - 10*e + 68, -8*e^2 - e - 1, 4*e^3 - 4*e^2 - 28*e + 8, 8*e^2 - 4*e, 4*e^3 + 8*e^2 - 26*e + 6, 2*e^3 + 8*e^2 - 22*e - 28, -6*e^3 - 4*e^2 + 11*e + 9, -2*e^3 - 12*e^2 + 14*e + 36, -e^3 + e^2 + 20*e + 18, 8*e^3 - 4*e^2 - 47*e - 1, -e^3 - 25*e^2 + 4*e + 60, -16*e^3 - 10*e^2 + 56*e + 14, -8*e^3 - 10*e^2 + 32*e + 30, -8*e^3 + 2*e^2 + 20*e - 10, 7*e^3 + 3*e^2 - 22*e - 4, 6*e^3 + 2*e^2 - 30*e + 14, -12*e^3 - 10*e^2 + 44*e + 18, 8*e^3 - 5*e^2 - 38*e + 21, -4*e - 30, 16*e^3 + 8*e^2 - 52*e - 4, -14*e^3 + 8*e^2 + 54*e - 10, 14*e^3 + 6*e^2 - 35*e - 19];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {23, 23, -w^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;