/*
  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![19, 7, -10, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {16,4,w - 1}>;

primesArray := [
[4, 2, -1/4*w^3 + 1/2*w + 3/4],
[4, 2, 1/2*w^3 - 4*w - 1/2],
[9, 3, 1/2*w^3 + w^2 - 3*w - 7/2],
[11, 11, -1/4*w^3 + 1/2*w + 7/4],
[11, 11, 1/2*w^3 - 4*w - 3/2],
[19, 19, w],
[19, 19, 1/4*w^3 - 5/2*w + 1/4],
[25, 5, 1/2*w^3 - 3*w - 1/2],
[29, 29, 3/4*w^3 + w^2 - 9/2*w - 25/4],
[29, 29, 1/2*w^3 - w^2 - 3*w + 9/2],
[41, 41, -w^3 + 7*w + 3],
[41, 41, -3/4*w^3 + 11/2*w - 3/4],
[41, 41, 1/4*w^3 + w^2 - 5/2*w - 11/4],
[71, 71, 1/2*w^3 - 4*w + 5/2],
[71, 71, 3/4*w^3 + w^2 - 11/2*w - 21/4],
[79, 79, 1/4*w^3 - 2*w^2 + 1/2*w + 25/4],
[79, 79, -3/4*w^3 + 13/2*w + 9/4],
[89, 89, -7/4*w^3 - w^2 + 27/2*w + 41/4],
[89, 89, -7/4*w^3 - w^2 + 27/2*w + 33/4],
[101, 101, 1/4*w^3 + w^2 - 1/2*w - 27/4],
[101, 101, -1/4*w^3 + w^2 + 5/2*w - 17/4],
[109, 109, 1/2*w^3 + w^2 - 3*w - 3/2],
[109, 109, -w^3 + 7*w + 1],
[121, 11, 3/4*w^3 - 9/2*w - 5/4],
[139, 139, 3/4*w^3 + 2*w^2 - 9/2*w - 45/4],
[139, 139, 5/4*w^3 + w^2 - 17/2*w - 27/4],
[139, 139, w^3 - w^2 - 6*w + 3],
[139, 139, -1/4*w^3 + 2*w^2 + 3/2*w - 65/4],
[149, 149, 3/4*w^3 + 2*w^2 - 13/2*w - 61/4],
[149, 149, -1/4*w^3 - 2*w^2 + 7/2*w + 23/4],
[151, 151, 5/4*w^3 + w^2 - 17/2*w - 31/4],
[151, 151, w^3 + 2*w^2 - 7*w - 13],
[179, 179, 3/4*w^3 + w^2 - 11/2*w - 17/4],
[179, 179, -1/4*w^3 + w^2 + 1/2*w - 25/4],
[181, 181, 1/4*w^3 + 2*w^2 - 3/2*w - 35/4],
[181, 181, 1/4*w^3 + 2*w^2 - 3/2*w - 51/4],
[191, 191, -1/4*w^3 - 2*w^2 + 3/2*w + 27/4],
[191, 191, -5/4*w^3 + 19/2*w + 11/4],
[199, 199, 2*w^2 - 13],
[199, 199, 1/2*w^3 + 2*w^2 - 3*w - 17/2],
[211, 211, 3/4*w^3 + 2*w^2 - 13/2*w - 41/4],
[211, 211, -1/4*w^3 + 2*w^2 + 3/2*w - 49/4],
[229, 229, -1/4*w^3 + 2*w^2 + 3/2*w - 45/4],
[229, 229, -3/4*w^3 - 2*w^2 + 9/2*w + 41/4],
[239, 239, 3/4*w^3 + w^2 - 9/2*w - 33/4],
[239, 239, 1/4*w^3 + w^2 - 1/2*w - 31/4],
[241, 241, -1/4*w^3 + 2*w^2 + 1/2*w - 41/4],
[241, 241, -2*w^2 + w + 12],
[241, 241, -w^3 - 2*w^2 + 7*w + 11],
[241, 241, -3/4*w^3 - 2*w^2 + 11/2*w + 37/4],
[251, 251, -2*w^2 + 3*w + 4],
[251, 251, -1/4*w^3 - 1/2*w - 9/4],
[251, 251, -3/2*w^3 - w^2 + 11*w + 13/2],
[251, 251, -3/4*w^3 + 13/2*w - 11/4],
[271, 271, -3/4*w^3 + w^2 + 13/2*w - 23/4],
[271, 271, w^2 - 10],
[271, 271, 1/2*w^3 + w^2 - w - 11/2],
[271, 271, 1/4*w^3 + w^2 - 3/2*w - 3/4],
[281, 281, -5/4*w^3 - 2*w^2 + 11/2*w + 39/4],
[281, 281, 3/2*w^3 - 11*w - 7/2],
[311, 311, w^2 - 2*w - 6],
[311, 311, -1/4*w^3 + w^2 + 1/2*w - 29/4],
[331, 331, 2*w^2 - 11],
[331, 331, 1/2*w^3 + 2*w^2 - 3*w - 21/2],
[361, 19, w^3 - 6*w - 2],
[389, 389, 1/4*w^3 + w^2 - 1/2*w - 35/4],
[389, 389, 3/4*w^3 + 2*w^2 - 9/2*w - 53/4],
[389, 389, 1/4*w^3 - 2*w^2 - 3/2*w + 33/4],
[389, 389, w^3 + w^2 - 6*w - 9],
[401, 401, 3/4*w^3 + w^2 - 7/2*w - 29/4],
[401, 401, -3/4*w^3 + w^2 + 11/2*w - 15/4],
[401, 401, 3/4*w^3 + 2*w^2 - 13/2*w - 33/4],
[401, 401, 1/4*w^3 + 2*w^2 - 7/2*w - 51/4],
[409, 409, w^3 - 5*w - 1],
[409, 409, -7/4*w^3 - w^2 + 15/2*w + 21/4],
[409, 409, 5/4*w^3 - 17/2*w - 3/4],
[409, 409, 9/4*w^3 - w^2 - 33/2*w + 25/4],
[419, 419, 3/4*w^3 + w^2 - 11/2*w - 9/4],
[419, 419, -3/4*w^3 - 2*w^2 + 11/2*w + 21/4],
[419, 419, -1/4*w^3 - 4*w^2 + 7/2*w + 47/4],
[419, 419, -1/4*w^3 + w^2 + 1/2*w - 33/4],
[421, 421, -1/2*w^3 - 4*w^2 + 4*w + 59/2],
[421, 421, -7/4*w^3 + 25/2*w + 1/4],
[431, 431, -5/4*w^3 - w^2 + 19/2*w + 23/4],
[431, 431, 1/2*w^3 + 2*w^2 - 5*w - 13/2],
[431, 431, -1/2*w^3 + w^2 + w - 9/2],
[431, 431, -1/2*w^3 - 2*w^2 + 5*w + 29/2],
[439, 439, -3/2*w^3 + 10*w - 3/2],
[439, 439, 5/4*w^3 - 13/2*w + 5/4],
[449, 449, -3/4*w^3 + 13/2*w - 15/4],
[449, 449, 1/4*w^3 + 1/2*w + 21/4],
[449, 449, 5/4*w^3 + 2*w^2 - 19/2*w - 43/4],
[449, 449, 5/4*w^3 + 2*w^2 - 17/2*w - 59/4],
[461, 461, -5/4*w^3 + 19/2*w + 15/4],
[461, 461, -1/4*w^3 - 3*w^2 + 5/2*w + 31/4],
[461, 461, 3/2*w^3 + w^2 - 9*w - 21/2],
[461, 461, 3/4*w^3 - 5/2*w - 17/4],
[479, 479, 3/2*w^3 + 2*w^2 - 10*w - 29/2],
[479, 479, 3/4*w^3 - 2*w^2 - 7/2*w + 27/4],
[491, 491, -5/4*w^3 + w^2 + 7/2*w + 7/4],
[491, 491, 1/4*w^3 + w^2 - 1/2*w - 39/4],
[491, 491, -5/2*w^3 - w^2 + 19*w + 23/2],
[491, 491, -1/4*w^3 + w^2 + 5/2*w - 5/4],
[499, 499, -9/4*w^3 + w^2 + 35/2*w - 21/4],
[499, 499, -1/4*w^3 - 4*w^2 + 9/2*w + 47/4],
[509, 509, w^3 + w^2 - 4*w - 7],
[509, 509, -5/4*w^3 + w^2 + 19/2*w - 17/4],
[521, 521, -3/4*w^3 + w^2 + 7/2*w - 27/4],
[521, 521, 5/4*w^3 + w^2 - 17/2*w - 15/4],
[529, 23, -9/4*w^3 + w^2 + 33/2*w - 21/4],
[529, 23, 7/4*w^3 + w^2 - 15/2*w - 25/4],
[541, 541, -7/4*w^3 + 13/2*w + 13/4],
[541, 541, 1/4*w^3 + 2*w^2 - 5/2*w - 39/4],
[541, 541, 1/2*w^3 + 2*w^2 - 4*w - 23/2],
[541, 541, 3*w^2 - 2*w - 10],
[571, 571, -5/4*w^3 + w^2 + 9/2*w - 5/4],
[571, 571, 1/2*w^3 + 4*w^2 - 5*w - 25/2],
[601, 601, -3/4*w^3 - 2*w^2 + 9/2*w + 17/4],
[601, 601, -1/4*w^3 + 2*w^2 + 3/2*w - 69/4],
[641, 641, 5/4*w^3 + 2*w^2 - 15/2*w - 51/4],
[641, 641, -3/4*w^3 + 2*w^2 + 9/2*w - 35/4],
[659, 659, 5/4*w^3 + 2*w^2 - 17/2*w - 35/4],
[659, 659, -1/2*w^3 + 2*w^2 + 2*w - 25/2],
[661, 661, -11/4*w^3 - w^2 + 41/2*w + 41/4],
[661, 661, 11/4*w^3 + 5*w^2 - 43/2*w - 157/4],
[661, 661, 1/4*w^3 - 5*w^2 + 7/2*w + 53/4],
[661, 661, 3/2*w^3 - w^2 - 5*w - 1/2],
[691, 691, 5/4*w^3 + w^2 - 7/2*w - 31/4],
[691, 691, -2*w^3 + w^2 + 16*w - 4],
[709, 709, 3/4*w^3 + 3*w^2 - 11/2*w - 45/4],
[709, 709, 1/4*w^3 + 3*w^2 - 5/2*w - 83/4],
[719, 719, 1/4*w^3 + 2*w^2 - 1/2*w - 55/4],
[719, 719, 3/4*w^3 - w^2 - 9/2*w - 1/4],
[751, 751, 2*w^3 + 2*w^2 - 13*w - 12],
[751, 751, 3/4*w^3 + 3*w^2 - 9/2*w - 73/4],
[809, 809, 5/4*w^3 + 2*w^2 - 11/2*w - 43/4],
[809, 809, 1/2*w^3 + 2*w^2 - 3*w - 33/2],
[821, 821, -5/4*w^3 + 19/2*w - 9/4],
[821, 821, 3/4*w^3 + 2*w^2 - 13/2*w - 29/4],
[821, 821, -1/4*w^3 - 3/2*w + 31/4],
[821, 821, 1/4*w^3 + 2*w^2 - 9/2*w - 15/4],
[839, 839, -5/4*w^3 - w^2 + 13/2*w + 35/4],
[839, 839, 5/4*w^3 - w^2 - 17/2*w + 9/4],
[841, 29, -5/4*w^3 + 15/2*w - 1/4],
[859, 859, 1/4*w^3 + w^2 - 7/2*w - 51/4],
[859, 859, -3/2*w^3 - 3*w^2 + 11*w + 41/2],
[859, 859, 2*w^3 + 2*w^2 - 12*w - 9],
[859, 859, 5/4*w^3 + 3*w^2 - 15/2*w - 63/4],
[919, 919, 5/4*w^3 - 2*w^2 - 11/2*w + 29/4],
[919, 919, 9/4*w^3 + 2*w^2 - 31/2*w - 55/4],
[929, 929, -3/2*w^3 - 3*w^2 + 11*w + 49/2],
[929, 929, -5/4*w^3 - w^2 + 9/2*w + 31/4],
[961, 31, 5/4*w^3 - 15/2*w - 7/4],
[961, 31, 1/2*w^3 - 3*w - 13/2],
[971, 971, 1/4*w^3 + 4*w^2 - 11/2*w - 43/4],
[971, 971, 5/4*w^3 - w^2 - 5/2*w - 11/4],
[991, 991, 5/4*w^3 - w^2 - 15/2*w + 17/4],
[991, 991, 3/2*w^3 + w^2 - 9*w - 13/2]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^4 - 4*x^3 - 13*x^2 + 64*x - 44;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, 0, 0, 0, 0, 2*e^3 - e^2 - 30*e + 29, -3/2*e^3 + 43/2*e - 10, e^3 - e^2 - 13*e + 19, 0, 0, 0, 0, 0, 0, 0, -7/2*e^3 + 103/2*e - 26, e^3 - e^2 - 17*e + 17, 0, 0, 0, 0, -3*e^3 + 41*e - 16, -e^3 + 19*e - 8, -7/2*e^3 + 4*e^2 + 99/2*e - 66, -2*e^3 + 4*e^2 + 30*e - 48, 6*e^3 - 4*e^2 - 90*e + 100, -13/2*e^3 + 4*e^2 + 197/2*e - 106, -2*e^3 + 3*e^2 + 30*e - 39, 0, 0, -2*e^3 + 4*e^2 + 30*e - 48, 6*e^3 - 4*e^2 - 90*e + 100, 0, 0, -5/2*e^3 + 65/2*e - 2, -3/2*e^3 + 2*e^2 + 47/2*e - 48, 0, 0, -5/2*e^3 + 69/2*e - 22, -5*e^3 + e^2 + 69*e - 33, 5*e^3 - 5*e^2 - 77*e + 97, 9/2*e^3 - 2*e^2 - 137/2*e + 68, 3/2*e^3 - 39/2*e + 14, -15/2*e^3 + 6*e^2 + 219/2*e - 124, 0, 0, -5/2*e^3 + 65/2*e - 2, 5/2*e^3 + 2*e^2 - 73/2*e + 8, -3/2*e^3 + 4*e^2 + 39/2*e - 50, -5/2*e^3 + 2*e^2 + 73/2*e - 60, 0, 0, 0, 0, 3*e^3 - 3*e^2 - 51*e + 59, 3*e^2 + 4*e - 27, -3/2*e^3 + 51/2*e - 10, -3/2*e^3 - 2*e^2 + 43/2*e - 4, 0, 0, 0, 0, 4*e^3 - 4*e^2 - 52*e + 76, -8*e^3 + 4*e^2 + 112*e - 88, 3*e^3 - 45*e + 8, 0, 0, 0, 0, 0, 0, 0, 0, 15/2*e^3 - 6*e^2 - 219/2*e + 116, -e^3 - 3*e^2 + 13*e + 13, 21/2*e^3 - 6*e^2 - 297/2*e + 128, 3*e^3 - e^2 - 47*e + 15, 0, 0, 0, 0, -2*e^3 - e^2 + 22*e + 11, 5*e^3 - e^2 - 73*e + 27, 0, 0, 0, 0, 9*e^3 - 7*e^2 - 137*e + 159, 1/2*e^3 - 4*e^2 - 33/2*e + 50, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -3*e^3 + 3*e^2 + 51*e - 79, -9/2*e^3 + 6*e^2 + 153/2*e - 112, 0, 0, 0, 0, -8*e^3 + 9*e^2 + 116*e - 143, -10*e^3 + 7*e^2 + 142*e - 137, 3/2*e^3 + 2*e^2 - 63/2*e - 20, -e^3 - 3*e^2 + 21*e + 29, 10*e^3 - 5*e^2 - 150*e + 147, -11/2*e^3 + 159/2*e - 46, 9/2*e^3 - 6*e^2 - 137/2*e + 80, 5*e^3 - 5*e^2 - 85*e + 105, 21/2*e^3 - 2*e^2 - 297/2*e + 76, 7/2*e^3 - 2*e^2 - 83/2*e + 48, 0, 0, 0, 0, -12*e^3 + 5*e^2 + 168*e - 119, 17/2*e^3 - 261/2*e + 66, -7/2*e^3 - 2*e^2 + 107/2*e - 28, 5*e^3 - 3*e^2 - 81*e + 65, 9/2*e^3 - 6*e^2 - 121/2*e + 96, 11*e^3 - e^2 - 163*e + 93, -5*e^3 + 8*e^2 + 83*e - 128, 7*e^3 - 8*e^2 - 113*e + 152, 0, 0, 2*e^3 - 4*e^2 - 22*e + 60, -10*e^3 + 4*e^2 + 142*e - 104, 0, 0, 0, 0, 0, 0, 0, 0, -15*e^3 + 9*e^2 + 219*e - 211, 14*e^3 - 4*e^2 - 210*e + 164, 3*e^3 - 9*e^2 - 51*e + 101, 9/2*e^3 - 6*e^2 - 153/2*e + 104, 6*e^3 + 4*e^2 - 90*e + 16, -3/2*e^3 + 6*e^2 + 27/2*e - 76, -6*e^3 - 3*e^2 + 90*e - 13, 0, 0, -9*e^3 + 135*e - 88, 9*e^3 - 135*e + 56, 0, 0, 14*e^3 - 4*e^2 - 210*e + 164, 6*e^3 + 4*e^2 - 90*e + 16];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {4,2,-1/2*w^3 + 4*w + 1/2}>] := 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;