/*
  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![1, 0, -4, 0, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {49,7,-2*w^3 + 6*w - 1}>;

primesArray := [
[2, 2, w^3 - 4*w + 1],
[9, 3, -w^2 + 2],
[23, 23, w^3 - w^2 - 4*w + 1],
[23, 23, w^2 - w - 3],
[23, 23, -w^2 - w + 3],
[23, 23, -w^3 - w^2 + 4*w + 1],
[25, 5, w^3 - 5*w + 1],
[25, 5, w^3 - 5*w - 1],
[47, 47, -3*w^3 + 2*w^2 + 12*w - 8],
[47, 47, -2*w^3 - w^2 + 6*w],
[47, 47, 2*w^3 - w^2 - 6*w],
[47, 47, w^2 + w - 5],
[49, 7, 2*w^3 - 6*w - 1],
[49, 7, -2*w^3 + 6*w - 1],
[71, 71, 2*w^3 - w^2 - 7*w + 1],
[71, 71, 3*w^3 - w^2 - 11*w + 2],
[71, 71, 4*w^3 - 2*w^2 - 14*w + 5],
[71, 71, -3*w^3 + 2*w^2 + 10*w - 4],
[73, 73, -w^3 - 2*w^2 + 3*w + 5],
[73, 73, -w^3 + 2*w^2 + 3*w - 3],
[73, 73, w^3 + 2*w^2 - 3*w - 3],
[73, 73, w^3 - 2*w^2 - 3*w + 5],
[97, 97, 2*w^2 + w - 4],
[97, 97, w^3 + 2*w^2 - 4*w - 4],
[97, 97, -w^3 + 2*w^2 + 4*w - 4],
[97, 97, 2*w^2 - w - 4],
[121, 11, 2*w^3 - 5*w],
[121, 11, 3*w^3 - 10*w],
[167, 167, -2*w^3 + 3*w^2 + 5*w - 7],
[167, 167, -2*w^3 - w^2 + 10*w - 2],
[167, 167, 3*w^3 - 11*w + 3],
[167, 167, 3*w^3 - 2*w^2 - 10*w + 8],
[169, 13, 2*w^3 - 9*w],
[169, 13, w^3 - 6*w],
[191, 191, w^2 - 2*w - 4],
[191, 191, 2*w^3 + w^2 - 8*w],
[191, 191, -2*w^3 + w^2 + 8*w],
[191, 191, w^2 + 2*w - 4],
[193, 193, -w - 4],
[193, 193, -w^3 + 4*w - 4],
[193, 193, w^3 - 4*w - 4],
[193, 193, w - 4],
[239, 239, 4*w^3 - 3*w^2 - 16*w + 12],
[239, 239, -5*w^3 + w^2 + 19*w - 6],
[239, 239, 4*w^3 - 16*w + 3],
[239, 239, w^3 - 2*w^2 - 5*w + 9],
[241, 241, 2*w^3 + w^2 - 10*w - 4],
[241, 241, 2*w^3 - w^2 - 10*w],
[241, 241, -2*w^3 - w^2 + 10*w],
[241, 241, -2*w^3 + w^2 + 10*w - 4],
[263, 263, -2*w^3 + 2*w^2 + 6*w - 7],
[263, 263, -4*w^3 + 3*w^2 + 13*w - 9],
[263, 263, -w^3 + 2*w^2 + 2*w - 6],
[263, 263, -3*w^3 + 3*w^2 + 9*w - 8],
[289, 17, 3*w^3 - 9*w + 1],
[289, 17, -3*w^3 + 9*w + 1],
[311, 311, 2*w^3 + w^2 - 8*w + 2],
[311, 311, -w^2 - 2*w + 6],
[311, 311, 3*w^3 - 2*w^2 - 13*w + 9],
[311, 311, -2*w^3 + w^2 + 8*w + 2],
[313, 313, -w^3 - w^2 + 4*w - 3],
[313, 313, w^2 - w - 7],
[313, 313, w^2 + w - 7],
[313, 313, w^3 - w^2 - 4*w - 3],
[337, 337, 3*w^3 - 8*w - 2],
[337, 337, -4*w^3 + 13*w - 2],
[337, 337, 4*w^3 - 13*w - 2],
[337, 337, 3*w^3 - 8*w + 2],
[359, 359, w^3 + 2*w^2 - 6*w - 6],
[359, 359, -2*w^3 + 2*w^2 + 9*w - 2],
[359, 359, 2*w^3 + 2*w^2 - 9*w - 2],
[359, 359, -w^3 + 2*w^2 + 6*w - 6],
[361, 19, -w^3 + 5*w - 5],
[361, 19, w^3 - 5*w - 5],
[383, 383, -w^3 + w^2 - 3],
[383, 383, 4*w^3 + w^2 - 15*w - 1],
[383, 383, -4*w^3 + w^2 + 15*w - 1],
[383, 383, w^3 + w^2 - 3],
[409, 409, 3*w^3 + w^2 - 8*w - 1],
[409, 409, 4*w^3 - w^2 - 13*w + 3],
[409, 409, 4*w^3 + w^2 - 13*w - 3],
[409, 409, -3*w^3 + w^2 + 8*w - 1],
[431, 431, -3*w^3 - 2*w^2 + 11*w + 3],
[431, 431, -w^3 + 2*w^2 + w - 5],
[431, 431, w^3 + 2*w^2 - w - 5],
[431, 431, 3*w^3 - 2*w^2 - 11*w + 3],
[433, 433, 3*w^3 - 13*w + 1],
[433, 433, w^3 + w^2 - 4*w - 7],
[433, 433, -w^3 + w^2 + 4*w - 7],
[433, 433, 3*w^3 - 13*w - 1],
[457, 457, 2*w^3 - 9*w - 6],
[457, 457, w^3 - 6*w - 6],
[457, 457, -w^3 + 6*w - 6],
[457, 457, -2*w^3 + 9*w - 6],
[479, 479, -2*w^3 + w^2 + 4*w - 4],
[479, 479, 4*w^3 + w^2 - 14*w],
[479, 479, -4*w^3 + w^2 + 14*w],
[479, 479, 2*w^3 + w^2 - 4*w - 4],
[503, 503, -3*w^3 - 2*w^2 + 10*w + 2],
[503, 503, 2*w^3 + 2*w^2 - 5*w - 6],
[503, 503, -2*w^3 + 2*w^2 + 5*w - 6],
[503, 503, 3*w^3 - 2*w^2 - 10*w + 2],
[577, 577, 3*w^3 + w^2 - 11*w - 8],
[577, 577, -w^3 + w^2 + 6*w - 9],
[577, 577, -4*w^3 + 4*w^2 + 15*w - 12],
[577, 577, -4*w^3 + w^2 + 12*w],
[599, 599, 2*w^3 + 2*w^2 - 8*w - 1],
[599, 599, 2*w^2 - 2*w - 7],
[599, 599, 2*w^2 + 2*w - 7],
[599, 599, -2*w^3 + 2*w^2 + 8*w - 1],
[601, 601, -w^3 + 3*w^2 + 4*w - 7],
[601, 601, 3*w^2 - w - 5],
[601, 601, 3*w^2 + w - 5],
[601, 601, w^3 + 3*w^2 - 4*w - 7],
[647, 647, -6*w^3 + 3*w^2 + 21*w - 7],
[647, 647, -5*w^3 + 3*w^2 + 17*w - 6],
[647, 647, 4*w^3 - 2*w^2 - 14*w + 3],
[647, 647, 5*w^3 - 2*w^2 - 18*w + 4],
[673, 673, -5*w^3 + 2*w^2 + 16*w - 6],
[673, 673, 3*w^3 - w^2 - 14*w + 5],
[673, 673, -3*w^3 - w^2 + 14*w + 5],
[673, 673, 5*w^3 + 2*w^2 - 16*w - 6],
[719, 719, 3*w^3 - w^2 - 13*w - 2],
[719, 719, 6*w^3 - w^2 - 23*w + 7],
[719, 719, -5*w^3 + 20*w - 4],
[719, 719, -3*w^3 - w^2 + 13*w - 2],
[743, 743, -3*w^3 + w^2 + 11*w + 2],
[743, 743, w^3 + w^2 - w - 6],
[743, 743, -w^3 + w^2 + w - 6],
[743, 743, 3*w^3 + w^2 - 11*w + 2],
[769, 769, -4*w^3 + 17*w - 2],
[769, 769, -2*w^3 + w^2 + 8*w - 8],
[769, 769, 2*w^3 + w^2 - 8*w - 8],
[769, 769, 4*w^3 - 17*w - 2],
[839, 839, w^3 + 3*w^2 - 6*w - 9],
[839, 839, -4*w^3 - 2*w^2 + 16*w + 3],
[839, 839, 4*w^3 - 2*w^2 - 16*w + 3],
[839, 839, -w^3 + 3*w^2 + 6*w - 9],
[841, 29, -4*w^3 + w^2 + 12*w - 2],
[841, 29, 4*w^3 + w^2 - 12*w - 2],
[863, 863, 2*w^2 + 3*w - 6],
[863, 863, 3*w^3 - 2*w^2 - 12*w + 2],
[863, 863, 3*w^3 + 2*w^2 - 12*w - 2],
[863, 863, 2*w^2 - 3*w - 6],
[887, 887, w^3 + 2*w^2 - 6*w - 8],
[887, 887, 2*w^2 - 2*w - 9],
[887, 887, 2*w^2 + 2*w - 9],
[887, 887, -w^3 + 2*w^2 + 6*w - 8],
[911, 911, -3*w^3 - 3*w^2 + 11*w + 4],
[911, 911, w^3 + 3*w^2 - w - 8],
[911, 911, -w^3 + 3*w^2 + w - 8],
[911, 911, 3*w^3 - 3*w^2 - 11*w + 4],
[937, 937, -w^3 - w^2 + 8*w + 1],
[937, 937, 4*w^3 - w^2 - 17*w + 3],
[937, 937, -4*w^3 - w^2 + 17*w + 3],
[937, 937, w^3 - w^2 - 8*w + 1],
[961, 31, -4*w^3 + 12*w - 1],
[961, 31, 4*w^3 - 12*w - 1],
[983, 983, w^2 + 3*w - 7],
[983, 983, 3*w^3 - w^2 - 12*w - 3],
[983, 983, 3*w^3 + w^2 - 12*w + 3],
[983, 983, w^2 - 3*w - 7]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^6 - 14*x^4 + 53*x^2 - 32;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, -e^4 + 7*e^2 - 2, -2*e, 1/2*e^5 - 9/2*e^3 + 8*e, -2*e, 1/2*e^5 - 9/2*e^3 + 8*e, 1/2*e^4 - 9/2*e^2 + 6, 1/2*e^4 - 9/2*e^2 + 6, 1/2*e^5 - 9/2*e^3 + 6*e, -e^5 + 9*e^3 - 14*e, 1/2*e^5 - 9/2*e^3 + 6*e, -e^5 + 9*e^3 - 14*e, -e^4 + 11*e^2 - 18, -1, 2*e^3 - 14*e, -2*e^3 + 12*e, 2*e^3 - 14*e, -2*e^3 + 12*e, -2*e^2 + 2, -2*e^2 + 2, -e^4 + 7*e^2 + 2, -e^4 + 7*e^2 + 2, -1/2*e^4 + 5/2*e^2 + 6, 3*e^4 - 23*e^2 + 22, -1/2*e^4 + 5/2*e^2 + 6, 3*e^4 - 23*e^2 + 22, 2*e^4 - 16*e^2 + 14, 2*e^4 - 16*e^2 + 14, -e^5 + 5*e^3 + 12*e, -3/2*e^5 + 27/2*e^3 - 24*e, -e^5 + 5*e^3 + 12*e, -3/2*e^5 + 27/2*e^3 - 24*e, 3/2*e^4 - 27/2*e^2 + 10, 3/2*e^4 - 27/2*e^2 + 10, -e^5 + 9*e^3 - 16*e, -e^5 + 9*e^3 - 16*e, -e^5 + 7*e^3 + 2*e, -e^5 + 7*e^3 + 2*e, -7/2*e^4 + 59/2*e^2 - 38, -7/2*e^4 + 59/2*e^2 - 38, 5/2*e^4 - 41/2*e^2 + 26, 5/2*e^4 - 41/2*e^2 + 26, -1/2*e^5 + 1/2*e^3 + 26*e, -1/2*e^5 + 1/2*e^3 + 26*e, e^5 - 5*e^3 - 16*e, e^5 - 5*e^3 - 16*e, 4*e^4 - 32*e^2 + 26, 1/2*e^4 - 13/2*e^2 + 10, 4*e^4 - 32*e^2 + 26, 1/2*e^4 - 13/2*e^2 + 10, -e^5 + 15*e^3 - 46*e, -e^5 + 15*e^3 - 46*e, -e^5 + 9*e^3 - 12*e, -e^5 + 9*e^3 - 12*e, -4*e^4 + 28*e^2 + 2, e^4 + e^2 - 30, 4*e^3 - 22*e, e^5 - 7*e^3 + 6*e, 4*e^3 - 22*e, e^5 - 7*e^3 + 6*e, -5/2*e^4 + 45/2*e^2 - 26, -5/2*e^4 + 45/2*e^2 - 26, -2*e^4 + 14*e^2 - 10, -2*e^4 + 14*e^2 - 10, -e^4 + 13*e^2 - 18, -4*e^2 + 14, -e^4 + 13*e^2 - 18, -4*e^2 + 14, -e^5 + 5*e^3 + 6*e, 3/2*e^5 - 27/2*e^3 + 20*e, -e^5 + 5*e^3 + 6*e, 3/2*e^5 - 27/2*e^3 + 20*e, 2*e^4 - 12*e^2 - 2, 2*e^4 - 12*e^2 - 2, 3/2*e^5 - 27/2*e^3 + 18*e, 3/2*e^5 - 27/2*e^3 + 18*e, e^5 - 11*e^3 + 26*e, e^5 - 11*e^3 + 26*e, 1/2*e^4 - 1/2*e^2 - 6, 1/2*e^4 - 1/2*e^2 - 6, e^4 - 9*e^2 + 10, e^4 - 9*e^2 + 10, -1/2*e^5 + 1/2*e^3 + 30*e, -1/2*e^5 + 1/2*e^3 + 30*e, -4*e^3 + 20*e, -4*e^3 + 20*e, -2*e^4 + 18*e^2 - 26, -2*e^4 + 18*e^2 - 26, 3/2*e^4 - 15/2*e^2 - 10, 3/2*e^4 - 15/2*e^2 - 10, -3/2*e^4 + 27/2*e^2 - 22, -3/2*e^4 + 27/2*e^2 - 22, -3/2*e^4 + 27/2*e^2 - 22, -3/2*e^4 + 27/2*e^2 - 22, 1/2*e^5 + 7/2*e^3 - 42*e, 1/2*e^5 + 7/2*e^3 - 42*e, -e^5 + 15*e^3 - 54*e, -e^5 + 15*e^3 - 54*e, -4*e^3 + 34*e, e^5 - 15*e^3 + 42*e, -4*e^3 + 34*e, e^5 - 15*e^3 + 42*e, 3*e^4 - 33*e^2 + 62, 8*e^2 - 34, 3*e^4 - 33*e^2 + 62, 8*e^2 - 34, 3*e^5 - 27*e^3 + 40*e, 3*e^5 - 27*e^3 + 40*e, -2*e^3 - 2*e, -2*e^3 - 2*e, -2*e^4 + 14*e^2 + 18, 3*e^4 - 21*e^2 + 18, -2*e^4 + 14*e^2 + 18, 3*e^4 - 21*e^2 + 18, -3/2*e^5 + 35/2*e^3 - 60*e, 2*e^5 - 16*e^3 + 8*e, -3/2*e^5 + 35/2*e^3 - 60*e, 2*e^5 - 16*e^3 + 8*e, 5*e^4 - 37*e^2 + 22, 5*e^4 - 37*e^2 + 22, -3*e^4 + 31*e^2 - 42, -3*e^4 + 31*e^2 - 42, 3*e^5 - 31*e^3 + 66*e, 3*e^5 - 31*e^3 + 66*e, -5/2*e^5 + 37/2*e^3 - 14*e, -5/2*e^5 + 37/2*e^3 - 14*e, -2*e^5 + 18*e^3 - 22*e, -2*e^5 + 18*e^3 - 22*e, 1/2*e^5 + 7/2*e^3 - 44*e, 1/2*e^5 + 7/2*e^3 - 44*e, -5*e^4 + 41*e^2 - 54, -7/2*e^4 + 67/2*e^2 - 38, -5*e^4 + 41*e^2 - 54, -7/2*e^4 + 67/2*e^2 - 38, -e^5 + 5*e^3 + 16*e, -1/2*e^5 + 17/2*e^3 - 36*e, -e^5 + 5*e^3 + 16*e, -1/2*e^5 + 17/2*e^3 - 36*e, e^4 - 13*e^2 + 38, e^4 - 13*e^2 + 38, -3*e^5 + 27*e^3 - 42*e, -3*e^5 + 27*e^3 - 42*e, 2*e^5 - 22*e^3 + 64*e, 2*e^5 - 22*e^3 + 64*e, 5/2*e^5 - 53/2*e^3 + 52*e, -2*e^3 + 24*e, 5/2*e^5 - 53/2*e^3 + 52*e, -2*e^3 + 24*e, 2*e^5 - 22*e^3 + 60*e, 5/2*e^5 - 53/2*e^3 + 50*e, 2*e^5 - 22*e^3 + 60*e, 5/2*e^5 - 53/2*e^3 + 50*e, -6*e^2 + 62, -2*e^4 + 10*e^2 + 30, -6*e^2 + 62, -2*e^4 + 10*e^2 + 30, -4*e^4 + 32*e^2 - 2, -4*e^4 + 32*e^2 - 2, e^5 - 13*e^3 + 44*e, e^5 - 13*e^3 + 44*e, 1/2*e^5 - 9/2*e^3 + 28*e, 1/2*e^5 - 9/2*e^3 + 28*e];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {49,7,-2*w^3 + 6*w - 1}>] := 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;