/*
  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![-49, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {36, 6, 6}>;

primesArray := [
[4, 2, 2],
[7, 7, w - 7],
[7, 7, w + 6],
[9, 3, 3],
[19, 19, w + 5],
[19, 19, w - 6],
[23, 23, w + 8],
[23, 23, -w + 9],
[25, 5, 5],
[29, 29, -w - 4],
[29, 29, w - 5],
[37, 37, -w - 3],
[37, 37, w - 4],
[41, 41, -w - 9],
[41, 41, w - 10],
[43, 43, -w - 2],
[43, 43, w - 3],
[47, 47, -w - 1],
[47, 47, w - 2],
[53, 53, 2*w - 13],
[53, 53, -2*w - 11],
[59, 59, -4*w - 25],
[59, 59, -4*w + 29],
[61, 61, -w - 10],
[61, 61, w - 11],
[83, 83, -w - 11],
[83, 83, w - 12],
[97, 97, 2*w - 11],
[97, 97, -2*w - 9],
[101, 101, 3*w - 20],
[101, 101, -3*w - 17],
[107, 107, -w - 12],
[107, 107, w - 13],
[109, 109, -5*w - 31],
[109, 109, -5*w + 36],
[121, 11, -11],
[127, 127, 2*w - 19],
[127, 127, -2*w - 17],
[137, 137, -3*w - 16],
[137, 137, 3*w - 19],
[157, 157, -3*w - 23],
[157, 157, 3*w - 26],
[163, 163, -4*w + 27],
[163, 163, 4*w + 23],
[169, 13, -13],
[173, 173, -6*w - 37],
[173, 173, -6*w + 43],
[181, 181, 2*w - 5],
[181, 181, -2*w - 3],
[191, 191, -w - 15],
[191, 191, w - 16],
[193, 193, 2*w - 3],
[193, 193, -2*w - 1],
[197, 197, 2*w - 1],
[223, 223, -w - 16],
[223, 223, w - 17],
[233, 233, -3*w - 13],
[233, 233, 3*w - 16],
[239, 239, -5*w + 34],
[239, 239, -5*w - 29],
[251, 251, 5*w - 41],
[251, 251, 5*w + 36],
[257, 257, -w - 17],
[257, 257, w - 18],
[289, 17, -17],
[293, 293, -w - 18],
[293, 293, w - 19],
[311, 311, -3*w - 10],
[311, 311, 3*w - 13],
[313, 313, -3*w - 26],
[313, 313, 3*w - 29],
[331, 331, -w - 19],
[331, 331, w - 20],
[347, 347, 4*w - 23],
[347, 347, -4*w - 19],
[353, 353, 3*w - 11],
[353, 353, -3*w - 8],
[379, 379, 2*w - 25],
[379, 379, -2*w - 23],
[401, 401, 3*w - 8],
[401, 401, -3*w - 5],
[409, 409, 5*w - 43],
[409, 409, 5*w + 38],
[419, 419, -5*w - 26],
[419, 419, 5*w - 31],
[431, 431, 3*w - 5],
[431, 431, -3*w - 2],
[433, 433, -7*w + 48],
[433, 433, -7*w - 41],
[443, 443, 3*w - 2],
[443, 443, 3*w - 1],
[449, 449, -9*w - 55],
[449, 449, -9*w + 64],
[457, 457, -w - 22],
[457, 457, w - 23],
[479, 479, 2*w - 27],
[479, 479, -2*w - 25],
[487, 487, -3*w - 29],
[487, 487, 3*w - 32],
[491, 491, -5*w - 39],
[491, 491, 5*w - 44],
[499, 499, 4*w - 19],
[499, 499, -4*w - 15],
[503, 503, -w - 23],
[503, 503, w - 24],
[521, 521, 11*w - 86],
[521, 521, -7*w + 47],
[557, 557, 7*w + 51],
[557, 557, 7*w - 58],
[563, 563, -4*w - 13],
[563, 563, 4*w - 17],
[569, 569, -10*w - 61],
[569, 569, -10*w + 71],
[587, 587, 2*w - 29],
[587, 587, -2*w - 27],
[601, 601, -w - 25],
[601, 601, w - 26],
[607, 607, -7*w - 39],
[607, 607, 7*w - 46],
[613, 613, 3*w - 34],
[613, 613, -3*w - 31],
[617, 617, -6*w - 31],
[617, 617, 6*w - 37],
[619, 619, 4*w - 15],
[619, 619, -4*w - 11],
[631, 631, 5*w - 27],
[631, 631, -5*w - 22],
[653, 653, -w - 26],
[653, 653, w - 27],
[661, 661, 5*w - 46],
[661, 661, -5*w - 41],
[683, 683, -9*w + 62],
[683, 683, -9*w - 53],
[691, 691, 7*w - 45],
[691, 691, -7*w - 38],
[727, 727, -6*w - 47],
[727, 727, 6*w - 53],
[733, 733, 4*w - 41],
[733, 733, -4*w - 37],
[739, 739, -4*w - 5],
[739, 739, 4*w - 9],
[751, 751, 8*w - 53],
[751, 751, -8*w - 45],
[769, 769, 5*w - 24],
[769, 769, -5*w - 19],
[773, 773, 7*w - 44],
[773, 773, -7*w - 37],
[787, 787, 4*w - 3],
[787, 787, 4*w - 1],
[797, 797, -9*w - 52],
[797, 797, -9*w + 61],
[811, 811, -5*w - 18],
[811, 811, 5*w - 23],
[821, 821, -w - 29],
[821, 821, w - 30],
[827, 827, 2*w - 33],
[827, 827, -2*w - 31],
[829, 829, -10*w + 69],
[829, 829, -10*w - 59],
[839, 839, 5*w - 48],
[839, 839, -5*w - 43],
[853, 853, -7*w - 36],
[853, 853, 7*w - 43],
[881, 881, -w - 30],
[881, 881, w - 31],
[961, 31, -31],
[991, 991, -5*w - 13],
[991, 991, 5*w - 18]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^4 + 8*x^3 + 14*x^2 - 8*x - 4;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [-1, -e - 4, e, 1, e^3 + 11/2*e^2 + 5*e - 3, -e^3 - 13/2*e^2 - 9*e + 1, 1/2*e^3 + 4*e^2 + 7*e - 2, -1/2*e^3 - 2*e^2 + e + 2, 3/2*e^2 + 6*e - 5, -1/2*e^3 - 3*e^2 + 6, 1/2*e^3 + 3*e^2 - 10, 1/2*e^3 + 4*e^2 + 5*e - 6, -1/2*e^3 - 2*e^2 + 3*e + 6, -1/2*e^3 - 7/2*e^2 - 6*e + 4, 1/2*e^3 + 5/2*e^2 + 2*e + 4, -1/2*e^3 - 5/2*e^2 + e + 6, 1/2*e^3 + 7/2*e^2 + 3*e - 6, 1/2*e^3 + e^2 - 6*e, -1/2*e^3 - 5*e^2 - 10*e + 8, -e^2 - 5*e + 2, -e^2 - 3*e + 6, 1/2*e^2 - e - 6, 1/2*e^2 + 5*e + 6, 1/2*e^3 + e^2 - 6*e - 6, -1/2*e^3 - 5*e^2 - 10*e + 2, 1/2*e^3 + 7/2*e^2 + 4*e - 5, -1/2*e^3 - 5/2*e^2 + 3, e^3 + 9/2*e^2 - 4*e - 14, -e^3 - 15/2*e^2 - 8*e + 10, -e^3 - 4*e^2 + 2*e + 8, e^3 + 8*e^2 + 14*e, e^3 + 13/2*e^2 + 9*e + 4, -e^3 - 11/2*e^2 - 5*e + 8, -2*e^2 - 7*e + 2, -2*e^2 - 9*e - 2, -7*e^2 - 28*e + 10, 1/2*e^3 - 8*e + 2, -1/2*e^3 - 6*e^2 - 16*e + 2, 1/2*e^3 + 5/2*e^2 + e + 13, -1/2*e^3 - 7/2*e^2 - 5*e + 17, -3/2*e^3 - 5*e^2 + 7*e + 4, 3/2*e^3 + 13*e^2 + 25*e - 8, 2*e^3 + 27/2*e^2 + 19*e - 11, -2*e^3 - 21/2*e^2 - 7*e + 1, 5/2*e^2 + 10*e, -e^3 - 3*e^2 + 9*e, e^3 + 9*e^2 + 15*e - 20, -1/2*e^3 + e^2 + 13*e - 10, 1/2*e^3 + 7*e^2 + 19*e - 14, e^3 + 10*e^2 + 19*e - 12, -e^3 - 2*e^2 + 13*e + 8, 3*e^3 + 39/2*e^2 + 24*e - 11, -3*e^3 - 33/2*e^2 - 12*e + 13, -10*e^2 - 40*e + 8, -5/2*e^3 - 19*e^2 - 31*e + 6, 5/2*e^3 + 11*e^2 - e - 14, 7/2*e^3 + 35/2*e^2 + 8*e - 14, -7/2*e^3 - 49/2*e^2 - 36*e + 10, -3/2*e^3 - 14*e^2 - 34*e + 6, 3/2*e^3 + 4*e^2 - 6*e + 14, 3/2*e^2 + 5*e - 4, 3/2*e^2 + 7*e, -5/2*e^3 - 29/2*e^2 - 17*e - 5, 5/2*e^3 + 31/2*e^2 + 21*e - 9, 5*e^2 + 20*e - 22, -1/2*e^3 + e^2 + 7*e - 8, 1/2*e^3 + 7*e^2 + 25*e + 12, -e^3 - 10*e^2 - 22*e + 18, e^3 + 2*e^2 - 10*e + 10, 5*e^3 + 65/2*e^2 + 40*e - 23, -5*e^3 - 55/2*e^2 - 20*e + 17, -5/2*e^3 - 29/2*e^2 - 11*e + 6, 5/2*e^3 + 31/2*e^2 + 15*e - 22, 3/2*e^3 + 21/2*e^2 + 14*e - 7, -3/2*e^3 - 15/2*e^2 - 2*e + 9, 1/2*e^3 - 5/2*e^2 - 20*e + 16, -1/2*e^3 - 17/2*e^2 - 24*e + 24, -e^3 + 3/2*e^2 + 21*e - 9, e^3 + 27/2*e^2 + 39*e - 5, 1/2*e^3 + 9/2*e^2 + 13*e - 1, -1/2*e^3 - 3/2*e^2 - e - 13, -3*e^3 - 41/2*e^2 - 30*e + 7, 3*e^3 + 31/2*e^2 + 10*e - 9, e^3 + 18*e^2 + 50*e - 18, -e^3 + 6*e^2 + 46*e + 6, -e^3 + 3*e^2 + 33*e - 4, e^3 + 15*e^2 + 39*e - 24, e^3 + 3/2*e^2 - 10*e - 8, -e^3 - 21/2*e^2 - 26*e - 8, -4*e^3 - 26*e^2 - 30*e + 6, 4*e^3 + 22*e^2 + 14*e - 34, -1/2*e^3 - 9/2*e^2 - 4*e + 36, 1/2*e^3 + 3/2*e^2 - 8*e + 12, 4*e^3 + 61/2*e^2 + 56*e - 14, -4*e^3 - 35/2*e^2 - 4*e - 6, 3/2*e^3 + 7*e^2 + 3*e + 8, -3/2*e^3 - 11*e^2 - 19*e + 12, 3/2*e^3 + 10*e^2 + 6*e - 32, -3/2*e^3 - 8*e^2 + 2*e + 8, -1/2*e^3 - 11/2*e^2 - 8*e + 27, 1/2*e^3 + 1/2*e^2 - 12*e + 3, -7/2*e^3 - 43/2*e^2 - 15*e + 20, 7/2*e^3 + 41/2*e^2 + 11*e - 40, 2*e^3 + 16*e^2 + 37*e + 8, -2*e^3 - 8*e^2 - 5*e - 12, 1/2*e^3 - 3/2*e^2 - 23*e - 29, -1/2*e^3 - 15/2*e^2 - 13*e + 7, 1/2*e^3 - 2*e^2 - 20*e + 4, -1/2*e^3 - 8*e^2 - 20*e + 20, -4*e^3 - 22*e^2 - 26*e - 8, 4*e^3 + 26*e^2 + 42*e, -9/2*e^3 - 69/2*e^2 - 61*e + 23, 9/2*e^3 + 39/2*e^2 + e + 3, -e^3 - 13/2*e^2 - 7*e - 6, e^3 + 11/2*e^2 + 3*e - 18, -4*e^3 - 41/2*e^2 - 6*e + 9, 4*e^3 + 55/2*e^2 + 34*e - 39, e^3 + 11*e^2 + 29*e - 16, -e^3 - e^2 + 11*e - 20, e^3 + 15*e^2 + 41*e - 26, -e^3 + 3*e^2 + 31*e - 14, -11/2*e^3 - 65/2*e^2 - 32*e + 26, 11/2*e^3 + 67/2*e^2 + 36*e - 14, 3*e^3 + 9*e^2 - 14*e + 14, -3*e^3 - 27*e^2 - 58*e + 22, -5/2*e^3 - 21*e^2 - 42*e + 2, 5/2*e^3 + 9*e^2 - 6*e - 6, -3/2*e^3 - 7*e^2 - 5*e - 24, 3/2*e^3 + 11*e^2 + 21*e - 20, 5*e^3 + 32*e^2 + 43*e - 30, -5*e^3 - 28*e^2 - 27*e - 10, e^3 + 10*e^2 + 24*e - 8, -e^3 - 2*e^2 + 8*e - 8, 5/2*e^3 + 13/2*e^2 - 17*e, -5/2*e^3 - 47/2*e^2 - 51*e + 12, -4*e^3 - 21*e^2 - 16*e + 16, 4*e^3 + 27*e^2 + 40*e, -4*e^2 - 21*e - 24, -4*e^2 - 11*e - 4, 1/2*e^3 + 7/2*e^2 - 9*e - 34, -1/2*e^3 - 5/2*e^2 + 13*e + 26, -5/2*e^3 - 17*e^2 - 23*e + 36, 5/2*e^3 + 13*e^2 + 7*e + 16, 2*e^3 + 15/2*e^2 - 10*e - 17, -2*e^3 - 33/2*e^2 - 26*e + 15, -4*e^3 - 31*e^2 - 48*e + 20, 4*e^3 + 17*e^2 - 8*e - 28, -3*e^3 - 31/2*e^2 - 11*e - 1, 3*e^3 + 41/2*e^2 + 31*e - 13, -3*e^3 - 14*e^2 + 3*e + 26, 3*e^3 + 22*e^2 + 29*e - 18, -e^3 - 25/2*e^2 - 39*e - 11, e^3 - 1/2*e^2 - 13*e + 9, 3*e^3 + 19*e^2 + 26*e - 38, -3*e^3 - 17*e^2 - 18*e - 30, -7/2*e^3 - 65/2*e^2 - 60*e + 33, 7/2*e^3 + 19/2*e^2 - 32*e - 23, -7/2*e^3 - 35*e^2 - 76*e + 28, 7/2*e^3 + 7*e^2 - 36*e - 4, -1/2*e^3 - e^2 + 9*e + 8, 1/2*e^3 + 5*e^2 + 7*e - 12, -19/2*e^3 - 55*e^2 - 49*e + 46, 19/2*e^3 + 59*e^2 + 65*e - 30, -5/2*e^3 - 19/2*e^2 + 15*e + 25, 5/2*e^3 + 41/2*e^2 + 29*e - 27, 15/2*e^2 + 30*e + 16, -9/2*e^3 - 28*e^2 - 19*e + 36, 9/2*e^3 + 26*e^2 + 11*e - 48];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {4, 2, 2}>] := 1;
ALEigenvalues[ideal<ZF | {9, 3, 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;