/*
  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![9, -1, -7, 0, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {10, 10, w^3 - w^2 - 4*w + 4}>;

primesArray := [
[2, 2, -w^2 + w + 4],
[3, 3, -w^2 + w + 3],
[5, 5, w^2 - 4],
[8, 2, w^3 - w^2 - 4*w + 5],
[17, 17, -w^2 + w + 5],
[27, 3, -w^3 + 4*w - 2],
[29, 29, -w^2 + w + 1],
[29, 29, -w^3 + 4*w + 2],
[37, 37, -2*w^3 + 3*w^2 + 9*w - 13],
[41, 41, w^3 - w^2 - 5*w + 2],
[43, 43, -w^3 + w^2 + 3*w - 4],
[61, 61, w^2 - 2*w - 2],
[61, 61, 2*w^2 - w - 8],
[67, 67, -4*w^3 + 5*w^2 + 18*w - 22],
[73, 73, -w^2 - 2*w + 2],
[79, 79, w^2 - 2*w - 4],
[89, 89, w^2 + w - 5],
[89, 89, 2*w^3 - 2*w^2 - 9*w + 8],
[89, 89, w^3 - 5*w - 5],
[89, 89, -w^3 + 2*w^2 + 4*w - 4],
[97, 97, w^3 + w^2 - 5*w - 4],
[101, 101, -2*w^3 + 2*w^2 + 8*w - 11],
[103, 103, w^3 - 5*w + 1],
[109, 109, 2*w - 1],
[109, 109, -w^3 + 2*w^2 + 3*w - 7],
[113, 113, 2*w^2 - 7],
[125, 5, -w^3 + 3*w^2 + 3*w - 8],
[127, 127, 2*w^3 - 4*w^2 - 11*w + 16],
[131, 131, 3*w^2 + 2*w - 8],
[131, 131, -2*w^3 + 4*w^2 + 8*w - 13],
[137, 137, w^3 - 3*w^2 - 7*w + 10],
[149, 149, -w^3 + 4*w - 4],
[149, 149, w^2 - 2*w - 8],
[157, 157, -w - 4],
[163, 163, -w^3 + 2*w^2 + 6*w - 10],
[167, 167, -w^3 + 2*w^2 + 5*w - 11],
[167, 167, -3*w^2 - 2*w + 10],
[167, 167, -w^2 - w + 7],
[167, 167, 2*w^2 - 2*w - 5],
[173, 173, -3*w^3 + 2*w^2 + 13*w - 11],
[179, 179, w^3 - 3*w^2 - 5*w + 10],
[191, 191, -2*w^3 - 3*w^2 + 10*w + 14],
[191, 191, 3*w^2 + w - 7],
[193, 193, w^2 - w + 1],
[193, 193, -w^2 - w - 1],
[227, 227, 4*w^3 - 4*w^2 - 17*w + 20],
[227, 227, -3*w^3 + 4*w^2 + 13*w - 17],
[229, 229, -w^3 + 6*w - 2],
[233, 233, -3*w^3 + 3*w^2 + 13*w - 16],
[241, 241, -w^2 + 2*w - 2],
[263, 263, 2*w^3 - 5*w^2 - 12*w + 16],
[263, 263, -3*w^3 + 4*w^2 + 13*w - 19],
[269, 269, w^3 + 2*w^2 - 5*w - 7],
[269, 269, -w^3 - 2*w^2 + w + 5],
[281, 281, 2*w^3 - 2*w^2 - 8*w + 1],
[281, 281, w^2 - 8],
[289, 17, 2*w^3 - 10*w - 7],
[307, 307, w^3 + 2*w^2 - 6*w - 8],
[307, 307, -2*w^3 + 2*w^2 + 8*w - 7],
[313, 313, 5*w^3 - 6*w^2 - 24*w + 28],
[313, 313, 2*w^3 - 2*w^2 - 10*w + 5],
[337, 337, w^3 - 2*w - 2],
[347, 347, -w^3 + 4*w^2 + w - 13],
[349, 349, -2*w^3 + w^2 + 9*w - 7],
[353, 353, 2*w^3 + w^2 - 7*w + 1],
[359, 359, -3*w^2 - 3*w + 5],
[359, 359, 3*w^3 - 5*w^2 - 13*w + 16],
[367, 367, 3*w^3 - 6*w^2 - 15*w + 23],
[367, 367, -w^3 + 2*w^2 + 6*w - 4],
[367, 367, -2*w^3 + 2*w^2 + 8*w - 5],
[367, 367, 2*w^3 - 3*w^2 - 12*w + 16],
[373, 373, -2*w^3 + 4*w^2 + 12*w - 17],
[373, 373, 3*w^2 + 4*w - 8],
[383, 383, -w^3 + 2*w^2 + 3*w - 13],
[383, 383, 2*w^2 - 2*w - 11],
[389, 389, 5*w^3 - 6*w^2 - 24*w + 26],
[389, 389, -w^3 + 3*w - 5],
[401, 401, w^3 + 2*w^2 - 4*w - 10],
[401, 401, 2*w^3 - w^2 - 7*w + 5],
[409, 409, -2*w^2 + 3*w + 10],
[419, 419, w^2 - 3*w - 1],
[419, 419, -w^3 + 5*w^2 + 9*w - 14],
[421, 421, 2*w^3 - 8*w - 7],
[431, 431, -3*w^2 + 16],
[431, 431, -3*w^3 + 11*w - 7],
[433, 433, 4*w^3 - 7*w^2 - 18*w + 26],
[439, 439, 3*w - 2],
[439, 439, -w^3 + w^2 + 5*w - 8],
[443, 443, 2*w^3 - w^2 - 6*w + 4],
[443, 443, w^2 + 3*w - 1],
[449, 449, -2*w^3 + 5*w^2 + 7*w - 17],
[457, 457, 4*w^3 - 6*w^2 - 19*w + 22],
[479, 479, 3*w^2 - 3*w - 13],
[479, 479, -4*w^3 + 8*w^2 + 22*w - 31],
[479, 479, 3*w^3 - 7*w^2 - 17*w + 26],
[479, 479, -6*w^3 + 7*w^2 + 28*w - 32],
[487, 487, -2*w^3 + 4*w^2 + 7*w - 8],
[491, 491, 3*w^2 - w - 11],
[491, 491, 4*w^3 - 5*w^2 - 18*w + 20],
[521, 521, -3*w^3 + 11*w - 5],
[521, 521, 3*w^3 - 3*w^2 - 15*w + 14],
[523, 523, -2*w^3 + 5*w^2 + 8*w - 14],
[529, 23, -2*w^3 + 3*w^2 + 8*w - 14],
[529, 23, 3*w^3 - 2*w^2 - 14*w + 4],
[541, 541, -w^3 + 2*w^2 + 7*w - 11],
[541, 541, -w^3 - w^2 + 5*w - 2],
[547, 547, -3*w - 2],
[547, 547, w^3 - 2*w^2 - 7*w + 7],
[547, 547, w^3 - 4*w^2 - 8*w + 10],
[547, 547, 2*w^3 - 3*w^2 - 7*w + 11],
[563, 563, 2*w^3 - 4*w^2 - 12*w + 13],
[563, 563, -w^3 + 4*w^2 + w - 7],
[571, 571, -w^3 + 2*w^2 + 2*w - 8],
[571, 571, 3*w^2 - 10],
[577, 577, 2*w^3 + w^2 - 7*w - 5],
[599, 599, 2*w^3 - 7*w + 2],
[599, 599, -4*w^3 + 7*w^2 + 21*w - 29],
[617, 617, -2*w^3 + w^2 + 11*w + 1],
[631, 631, 3*w^2 + w - 13],
[631, 631, 3*w^2 - 4*w - 10],
[647, 647, 3*w^3 - 4*w^2 - 13*w + 11],
[677, 677, -2*w^3 + 4*w^2 + 8*w - 17],
[691, 691, -w^3 + 3*w^2 + 3*w - 14],
[733, 733, -w^3 + 4*w^2 + 3*w - 11],
[733, 733, 3*w^3 - w^2 - 15*w - 2],
[733, 733, -w^3 + w^2 + 7*w - 2],
[733, 733, 3*w^3 - 4*w^2 - 16*w + 16],
[739, 739, -3*w^3 + w^2 + 11*w - 8],
[743, 743, -3*w^3 + 2*w^2 + 12*w - 14],
[751, 751, -2*w^3 + w^2 + 8*w - 2],
[761, 761, -2*w^3 + 2*w^2 + 7*w - 10],
[761, 761, 2*w^3 - 4*w^2 - 3*w + 8],
[769, 769, -w^3 + w^2 + 7*w - 4],
[769, 769, 2*w^3 + 2*w^2 - 8*w - 11],
[787, 787, -2*w^3 + 3*w^2 + 6*w - 10],
[787, 787, -2*w^2 - 3*w + 8],
[797, 797, -2*w^3 + w^2 + 7*w - 1],
[797, 797, -2*w^3 - 2*w^2 + 11*w + 10],
[809, 809, -2*w^3 + 5*w^2 + 7*w - 19],
[811, 811, 2*w^3 - 2*w^2 - 7*w - 2],
[811, 811, 3*w^3 - 2*w^2 - 11*w - 1],
[827, 827, 3*w^3 - 2*w^2 - 14*w + 10],
[829, 829, 2*w^3 - 2*w^2 - 9*w + 2],
[829, 829, 4*w^2 + 2*w - 11],
[841, 29, 2*w^3 + w^2 - 6*w - 2],
[863, 863, 2*w^3 - 8*w - 1],
[883, 883, 3*w^3 - 4*w^2 - 15*w + 13],
[887, 887, 3*w^3 - 2*w^2 - 12*w + 10],
[907, 907, 2*w^3 - 3*w^2 - 9*w + 7],
[911, 911, 2*w^3 + w^2 - 11*w - 5],
[919, 919, -2*w^3 + 5*w - 8],
[929, 929, -w^3 + 4*w^2 + 3*w - 13],
[937, 937, w^3 - 5*w - 7],
[937, 937, -2*w^3 + 3*w^2 + 10*w - 8],
[971, 971, -2*w^3 + 4*w^2 + 11*w - 20],
[971, 971, -4*w^3 + 4*w^2 + 19*w - 20],
[977, 977, -4*w^3 + 6*w^2 + 19*w - 28],
[983, 983, -4*w^2 - 4*w + 7],
[991, 991, -5*w^3 + 8*w^2 + 26*w - 32],
[991, 991, -6*w^3 + 6*w^2 + 26*w - 31],
[997, 997, -3*w^3 + 8*w^2 + 18*w - 26]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^3 - 3*x^2 + 1;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [-1, e, -1, 2*e^2 - 4*e, 3*e^2 - 7*e - 1, -e^2 + 5*e - 1, -e^2 + 4, 2*e^2 - 3*e - 8, -5*e^2 + 10*e + 4, 5*e^2 - 7*e - 9, -2*e^2 + 10*e - 2, 4*e^2 - 12*e - 6, -2*e^2 + 2*e + 2, -6*e^2 + 21*e - 2, 8*e^2 - 22*e + 2, 3*e + 2, 6*e^2 - 22*e - 4, 2*e^2 - 6*e - 8, 2*e^2 - 14*e + 8, -2*e^2 + 6*e + 2, -e^2 + e + 15, 2*e^2 - 3*e + 4, -e^2 - 6*e + 6, -13*e^2 + 32*e + 4, 8*e^2 - 20*e, -8*e^2 + 22*e + 6, -6*e + 6, -2*e^2 + 4*e - 8, -4*e^2 + 8*e - 6, -6*e^2 + 16*e - 8, 2*e^2 - 9*e - 8, -8*e^2 + 22*e, 8*e^2 - 22*e + 6, 5*e^2 - 14*e - 8, -9*e^2 + 16*e + 14, 8*e^2 - 30*e + 4, -10*e^2 + 36*e - 2, -7*e^2 + 15*e + 1, 2*e^2 - 8*e + 14, 17*e^2 - 37*e - 9, 10*e^2 - 28*e - 8, -5*e^2 + 16*e, 12*e^2 - 30*e + 6, -12*e^2 + 30*e + 8, -7*e^2 + 28*e + 4, -7*e^2 + 15*e + 19, 2*e^2 - 14*e + 20, 2*e - 18, 12*e^2 - 36*e, -5*e^2 + 10*e + 16, 4*e^2 - 16*e + 4, -6*e^2 + 14*e + 2, 12*e^2 - 42*e, -9*e^2 + 23*e - 7, -12*e^2 + 42*e + 6, 9*e^2 - 16*e + 2, 14*e^2 - 42*e + 2, -5*e^2 + 8*e + 18, -6*e^2 + 18*e - 8, -8*e^2 + 20*e + 6, -6*e + 4, 4*e^2 - 2*e - 4, -9*e^2 + 8*e + 26, -10, -10*e^2 + 36*e - 2, 4*e^2 - 11*e, -11*e^2 + 14*e + 28, -12*e^2 + 33*e + 10, -8*e^2 + 20*e + 6, -6*e^2 - 2*e + 36, 14*e^2 - 32*e - 18, e^2 + 9*e - 17, -8*e^2 + 16*e - 16, -8*e^2 + 13*e + 18, -11*e^2 + 27*e + 23, -3*e^2 + 7*e + 13, -12*e^2 + 44*e - 10, 15*e^2 - 35*e - 11, -2*e^2 - 17*e + 30, 2*e^2 - 17*e + 18, -2*e^2 + 2*e - 2, 5*e^2 - 29*e + 17, -9*e^2 + 15*e + 25, -8*e^2 + 16*e + 12, -10*e^2 + 44*e - 12, 6*e^2 - 19*e + 18, 14*e^2 - 29*e + 4, -5*e^2 + 23*e + 11, 9*e^2 - 30*e + 12, -3*e + 24, 2*e^2 + 6*e - 8, -6*e^2 + 6*e + 14, -10*e^2 + 14*e + 6, 8*e^2 - 20*e - 28, -5*e^2 + 14*e + 16, 28*e^2 - 68*e - 6, -e^2 + 8*e + 4, -6*e^2 + 6*e + 18, 4*e^2 - 20*e - 6, -11*e^2 + 16*e + 6, 14*e^2 - 36*e - 20, -3*e^2 + 3*e - 5, 13*e^2 - 34*e - 6, 14*e^2 - 40*e + 4, -12*e^2 + 36*e - 10, -2*e^2 - 10*e + 26, -11*e^2 + 24*e + 24, 5*e^2 - 38*e + 24, 8*e^2 - 38*e + 16, 14*e^2 - 28*e - 28, 4*e^2 - 20*e + 6, 12*e - 12, 12*e^2 - 9*e - 34, 4*e^2 - 32, -16*e^2 + 44*e + 4, -6*e^2 - 8*e + 34, 10*e^2 - 43*e + 16, 2*e^2 - 19*e + 12, 18*e^2 - 42*e - 20, -10*e^2 + 20*e + 2, 20*e^2 - 69*e - 2, -11*e^2 + 36*e + 8, 11*e^2 - 34*e + 8, 12*e^2 - 24*e - 8, -20*e^2 + 41*e + 24, 4*e^2 - 22*e + 30, 2*e^2 + 13*e - 6, -2*e^2 + 3*e + 16, 11*e^2 - 33*e + 13, 2*e^2 + 12*e - 6, 20*e^2 - 60*e + 4, -6*e^2 + 4*e + 52, -26*e^2 + 71*e - 4, -2*e^2 + 22, -16*e^2 + 46*e - 24, -20*e^2 + 43*e - 4, -4*e^2 + 16*e + 2, 8*e^2 - 14*e + 8, -28*e^2 + 68*e - 6, -15*e^2 + 26*e, 5*e^2 + e - 17, 25*e^2 - 63*e + 11, 14*e^2 - 32*e + 24, -7*e - 12, -14*e^2 + 20*e + 38, 8*e^2 + 4*e - 22, -5*e^2 + 20*e - 36, 5*e^2 - 9*e + 19, -11*e^2 + 23*e - 9, -16*e^2 + 56*e - 12, 8*e^2 - 34*e + 14, 18*e^2 - 52*e + 8, -8*e^2 + 58, -6*e^2 + 8*e + 12, 30*e^2 - 74*e - 2, -14*e^2 + 48*e + 14, -10*e^2 + 30*e + 4, -7*e^2 + 28*e - 4, 8*e^2 - 20*e + 4, -15*e^2 + 54*e - 22, -16*e^2 + 40*e - 18];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {2, 2, -w^2 + w + 4}>] := 1;
ALEigenvalues[ideal<ZF | {5, 5, w^2 - 4}>] := 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;