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

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

primesArray := [
[2, 2, w + 2],
[3, 3, w + 1],
[4, 2, -w^2 + 3*w + 3],
[9, 3, -w^2 + 2*w + 7],
[13, 13, w + 3],
[13, 13, -w + 3],
[13, 13, -w + 1],
[17, 17, -w^2 - 2*w + 1],
[23, 23, w^2 - 2*w - 5],
[31, 31, 2*w + 3],
[31, 31, -2*w^2 + 3*w + 15],
[31, 31, 3*w + 7],
[37, 37, 3*w^2 - 4*w - 27],
[41, 41, -2*w^2 + 7*w + 1],
[59, 59, w^2 - 3],
[61, 61, 4*w^2 - 12*w - 11],
[71, 71, w^2 - 2*w - 11],
[97, 97, 3*w + 5],
[101, 101, 2*w^2 - 6*w - 7],
[103, 103, 2*w^2 - 3*w - 19],
[107, 107, -3*w^2 + 10*w + 3],
[109, 109, -w - 5],
[109, 109, 5*w^2 - 8*w - 41],
[109, 109, 2*w^2 - 2*w - 21],
[113, 113, -4*w - 3],
[125, 5, -5],
[127, 127, 2*w^2 - 2*w - 17],
[131, 131, 2*w - 3],
[137, 137, 2*w^2 - 5*w - 5],
[137, 137, w^2 - 4*w - 13],
[137, 137, 2*w - 5],
[139, 139, 2*w^2 - 3*w - 13],
[151, 151, 2*w^2 - 9],
[163, 163, 2*w^2 - 5*w - 11],
[173, 173, -2*w^2 + 8*w - 1],
[179, 179, w^2 - 15],
[181, 181, -w^2 + 11],
[191, 191, 3*w - 1],
[193, 193, -6*w^2 + 16*w + 23],
[197, 197, 2*w^2 - 8*w + 3],
[199, 199, 4*w^2 - 5*w - 35],
[211, 211, 2*w - 9],
[223, 223, 3*w^2 - 8*w - 13],
[227, 227, 3*w^2 + 2*w - 11],
[227, 227, w^2 - 4*w - 15],
[227, 227, w - 7],
[229, 229, -4*w^2 + 15*w - 1],
[239, 239, -4*w - 5],
[257, 257, 2*w^2 + w - 9],
[263, 263, -w^2 - 6*w - 7],
[269, 269, w^2 - 4*w - 7],
[277, 277, 4*w^2 - 7*w - 33],
[281, 281, 6*w^2 - 17*w - 19],
[281, 281, 3*w^2 - 8*w - 9],
[281, 281, -5*w - 1],
[289, 17, -2*w^2 + 9*w - 5],
[293, 293, -3*w^2 + 8*w + 7],
[293, 293, 2*w^2 + 5*w - 1],
[293, 293, 6*w^2 - 10*w - 47],
[307, 307, 2*w^2 - 2*w - 3],
[313, 313, 2*w^2 - 4*w - 7],
[317, 317, 3*w^2 - 6*w - 17],
[317, 317, 2*w^2 - 7*w - 7],
[317, 317, 2*w^2 + 2*w - 7],
[331, 331, 2*w^2 + 7*w + 7],
[337, 337, w^2 - 4*w - 9],
[337, 337, 3*w^2 - 4*w - 29],
[337, 337, w^2 - 6*w - 5],
[343, 7, -7],
[347, 347, 2*w^2 + 4*w - 3],
[349, 349, 2*w^2 - w - 25],
[359, 359, -2*w^2 - w + 5],
[379, 379, 4*w^2 - 9*w - 23],
[379, 379, -5*w^2 + 18*w + 3],
[379, 379, 2*w^2 - 3*w - 21],
[383, 383, -w^2 + 6*w - 7],
[397, 397, 2*w^2 - 7],
[401, 401, 9*w^2 - 28*w - 21],
[419, 419, 3*w^2 - 6*w - 25],
[419, 419, 4*w^2 - 7*w - 27],
[419, 419, 4*w^2 - 6*w - 29],
[433, 433, 8*w^2 - 13*w - 61],
[433, 433, -6*w^2 + 23*w - 3],
[433, 433, -w^2 - 3],
[443, 443, -2*w^2 + 5*w + 15],
[449, 449, -7*w^2 + 12*w + 51],
[457, 457, 2*w^2 - 2*w - 9],
[457, 457, -w^2 - 6*w - 1],
[457, 457, 6*w^2 - 11*w - 43],
[461, 461, -5*w^2 + 6*w + 45],
[463, 463, 2*w^2 - 5*w - 17],
[463, 463, 3*w - 5],
[463, 463, 2*w^2 - 3*w - 7],
[467, 467, -3*w^2 + 10*w + 9],
[479, 479, 3*w - 7],
[487, 487, 7*w^2 - 20*w - 23],
[487, 487, -4*w - 11],
[487, 487, 5*w + 7],
[509, 509, 2*w^2 - 2*w - 5],
[523, 523, -8*w^2 + 28*w + 7],
[529, 23, 2*w^2 - w - 7],
[541, 541, 3*w^2 + 2*w - 9],
[547, 547, -6*w - 1],
[557, 557, 2*w^2 - 13],
[563, 563, w - 9],
[569, 569, -8*w^2 + 27*w + 9],
[571, 571, 5*w^2 - 14*w - 15],
[577, 577, w^2 + 4*w + 7],
[587, 587, 4*w^2 - 4*w - 27],
[593, 593, 2*w^2 + 5*w + 5],
[599, 599, -2*w - 9],
[599, 599, -w^2 + 6*w - 1],
[599, 599, 2*w^2 - w - 19],
[607, 607, 4*w - 13],
[617, 617, -5*w - 13],
[617, 617, -w^2 - 2*w - 5],
[617, 617, 4*w^2 - 16*w + 5],
[619, 619, 3*w^2 - 8*w - 15],
[643, 643, -6*w - 11],
[647, 647, -w^2 + 17],
[647, 647, 2*w^2 - 9*w - 7],
[647, 647, 2*w^2 - 6*w - 21],
[659, 659, 4*w^2 - 8*w - 23],
[661, 661, 2*w - 11],
[673, 673, 3*w^2 - 13],
[677, 677, 5*w^2 - 16*w - 13],
[677, 677, -3*w^2 - 4*w + 3],
[677, 677, 5*w^2 - 6*w - 43],
[691, 691, 6*w^2 - 12*w - 37],
[701, 701, -3*w^2 + 12*w - 1],
[709, 709, -6*w^2 + 21*w + 7],
[719, 719, -w^2 + 6*w - 3],
[733, 733, -w - 9],
[733, 733, 2*w^2 + w - 11],
[733, 733, -2*w^2 + 9*w - 3],
[739, 739, 6*w^2 - 17*w - 21],
[743, 743, -2*w^2 - 6*w + 1],
[751, 751, 4*w^2 - 8*w - 29],
[751, 751, -w^2 - 8*w - 11],
[751, 751, 4*w^2 - 21],
[757, 757, 2*w^2 - 9*w - 1],
[761, 761, -4*w^2 + 10*w + 15],
[809, 809, w^2 + 2*w - 19],
[811, 811, -7*w - 17],
[823, 823, 3*w^2 - 17],
[827, 827, -9*w - 19],
[857, 857, 4*w^2 + 14*w + 9],
[857, 857, 4*w^2 - 16*w + 3],
[857, 857, 4*w^2 - 6*w - 27],
[859, 859, -2*w^2 + 5*w - 1],
[863, 863, 12*w^2 - 37*w - 29],
[877, 877, 3*w^2 - 6*w - 7],
[877, 877, 3*w^2 - 6*w - 31],
[877, 877, 3*w^2 - 6*w - 13],
[907, 907, 8*w^2 - 14*w - 57],
[907, 907, 4*w^2 - 7*w - 25],
[907, 907, 8*w^2 - 24*w - 19],
[911, 911, -3*w - 11],
[919, 919, -16*w^2 + 53*w + 23],
[929, 929, 2*w^2 - 6*w - 19],
[937, 937, w^2 - 6*w - 15],
[937, 937, 3*w^2 - 4*w - 17],
[937, 937, -3*w^2 + 12*w + 1],
[941, 941, -2*w^2 + 6*w + 15],
[947, 947, -2*w^2 - 3*w + 7],
[953, 953, 2*w^2 - 25],
[971, 971, 3*w^2 - 10*w - 11],
[971, 971, 3*w^2 - 12*w + 5],
[971, 971, -4*w^2 + 10*w + 9],
[977, 977, 2*w^2 - 8*w - 9],
[983, 983, -5*w^2 + 8*w + 35],
[991, 991, 4*w^2 + 5*w - 7]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^3 + x^2 - 5*x - 1;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, 1/2*e^2 - 5/2, -1, 1/2*e^2 + 2*e - 3/2, 1/2*e^2 + e + 3/2, -e - 2, -1/2*e^2 - 2*e - 1/2, -3, -1/2*e^2 - e - 1/2, 2*e - 2, -e^2 + 7, -1/2*e^2 + 13/2, 2*e^2 + e - 9, -2*e^2 - 3*e + 2, 1/2*e^2 - 2*e - 5/2, 3/2*e^2 - e - 15/2, e^2 + 1, 1/2*e^2 - e - 17/2, -1/2*e^2 - 6*e - 9/2, 5/2*e^2 - 17/2, -3/2*e^2 - 4*e + 39/2, -1/2*e^2 + 5*e - 3/2, 2*e^2 + e - 17, -1/2*e^2 + 2*e - 5/2, -5*e^2 - 8*e + 14, -2*e^2 - e - 3, 3/2*e^2 + e - 5/2, -5*e^2 - 6*e + 13, -11/2*e^2 - 5*e + 31/2, e^2 - 8, -1/2*e^2 + 2*e + 15/2, -2*e + 18, 2*e^2 + 2*e - 8, -3/2*e^2 - 8*e + 11/2, e^2 - 2*e - 2, 1/2*e^2 - e - 3/2, -e^2 - 4*e - 8, -13/2*e^2 - 9*e + 31/2, -e^2 + 7*e + 8, -5*e^2 - 4*e + 22, -4*e^2 - 6*e + 4, 5/2*e^2 + e + 9/2, -9/2*e^2 + 41/2, 3*e^2 - 2*e - 13, -13/2*e^2 - 5*e + 31/2, -1/2*e^2 - 3*e - 9/2, -3*e^2 - 2*e + 10, -11/2*e^2 + e + 29/2, -15/2*e^2 - 12*e + 49/2, -17/2*e^2 - 7*e + 27/2, -4*e^2 - 4*e + 2, 5/2*e^2 + 7*e - 17/2, 1/2*e^2 + e - 9/2, 3/2*e^2 + e - 3/2, -2*e^2 + 7*e + 21, -6*e^2 - 9*e + 16, -1/2*e^2 + 15/2, 4*e^2 + 7*e - 8, 4*e^2 + 9*e - 17, 6*e^2 + 12*e - 28, e^2 - 9*e - 18, -5/2*e^2 - e - 15/2, -2*e^2 - 7*e - 6, -4*e^2 - 6*e + 24, 2*e^2 + 10*e - 2, 3*e^2 + 6*e - 23, 6*e^2 + 13*e - 22, 1/2*e^2 + 9*e - 9/2, 1/2*e^2 - e - 3/2, 3/2*e^2 + 12*e - 11/2, -5*e - 19, 4*e^2 + 10*e - 12, 9*e^2 + 6*e - 21, 3/2*e^2 - 2*e + 5/2, -4*e^2 - 2*e + 12, -3/2*e^2 + 9*e + 9/2, -3/2*e^2 + 8*e + 25/2, -4*e^2 + 8*e + 21, -e^2 - 10*e + 9, 17/2*e^2 + 8*e - 13/2, 8*e^2 - 2*e - 40, -7*e^2 - 3*e + 36, -15/2*e^2 - 8*e + 41/2, -3/2*e^2 + 12*e + 33/2, 4*e^2 + 18*e - 18, -3*e^2 - 6*e - 5, 3/2*e^2 + 7*e + 17/2, 10*e^2 + 7*e - 29, 8*e^2 - 4*e - 33, e^2 - 9*e - 25, -9*e^2 + 27, -1/2*e^2 + 8*e - 19/2, 5*e^2 + 4*e - 5, 3/2*e^2 + 9*e - 37/2, 15/2*e^2 + e - 41/2, -7/2*e^2 - 16*e + 31/2, 9/2*e^2 + 12*e - 17/2, -19/2*e^2 - 5*e + 81/2, 9*e^2 + 2*e - 29, 3*e^2 - 6*e + 3, 7*e^2 + 6*e - 2, -4*e^2 - 10*e + 16, -15/2*e^2 - 8*e + 47/2, 9/2*e^2 + 11*e - 65/2, -9/2*e^2 + 3*e + 15/2, -e^2 + 8*e - 17, 5/2*e^2 + 8*e + 27/2, 3*e^2 - 3*e - 1, -9/2*e^2 + 3*e + 27/2, 9/2*e^2 + 15*e - 41/2, -6*e^2 - 2*e + 26, 8*e^2 + 12*e - 22, 11/2*e^2 + 11*e - 45/2, -9/2*e^2 - 7*e + 19/2, -4*e^2 - 3*e - 18, 2*e^2 + 3*e + 11, 23/2*e^2 + 5*e - 95/2, 8*e^2 + 22*e - 28, 21/2*e^2 + 9*e - 19/2, 2*e^2 + 14*e - 8, 11/2*e^2 - 6*e - 39/2, 5/2*e^2 - 2*e - 21/2, 2*e^2 - 6*e - 10, 5/2*e^2 + 3*e + 47/2, 2*e^2 + e, 4*e^2 - 4*e - 42, 2*e^2 + 4*e - 3, -13/2*e^2 - 9*e + 21/2, 3/2*e^2 + 17*e + 19/2, 4*e^2 + 18*e - 9, 6*e^2 + 10*e - 25, -19/2*e^2 - 8*e + 39/2, 8*e^2 - 5*e - 46, 8*e^2 + 19*e - 35, 3/2*e^2 + 3*e - 23/2, -9/2*e^2 + 6*e + 77/2, -3/2*e^2 + 13*e + 57/2, 9/2*e^2 - 7*e - 7/2, 5/2*e^2 - e + 41/2, 3*e^2 + 14*e - 31, -6*e^2 - 11*e - 9, -19/2*e^2 - 27*e + 67/2, 10*e^2 - 11*e - 52, -10*e^2 - 16*e + 44, 35/2*e^2 + 15*e - 97/2, -5/2*e^2 + 7*e + 39/2, 2*e^2 + 3*e - 19, 8*e^2 + 2*e - 23, e^2 - 3*e - 31, -1/2*e^2 - 4*e - 91/2, 1/2*e^2 - 4*e + 19/2, -5*e^2 + 11*e + 15, 3*e^2 - 5*e - 27, 2*e^2 - 13*e - 11, 13/2*e^2 + 13*e + 9/2, -5*e^2 + 2*e + 31, 10*e^2 + 10*e - 44, -7/2*e^2 - 3*e - 15/2, 3/2*e^2 - 12*e - 7/2, -12*e^2 - 4*e + 38, 17/2*e^2 + 21*e - 53/2, 6*e^2 + 7*e - 29, -4*e^2 - 20*e + 13, -19/2*e^2 + 3*e + 59/2, 9*e^2 + 16*e - 13, -5*e^2 - 6*e + 1, -17/2*e^2 - 17*e + 7/2, -1/2*e^2 + 15*e + 27/2, -7/2*e^2 + e + 57/2, 27/2*e^2 + 9*e - 79/2, 15/2*e^2 + 5*e - 13/2, -3*e^2 + 18*e + 15];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {4, 2, -w^2 + 3*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;