/*
  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![-3, -7, 0, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {11, 11, -w^2 + 5}>;

primesArray := [
[3, 3, w],
[3, 3, w + 1],
[3, 3, w + 2],
[8, 2, 2],
[11, 11, -w^2 + 5],
[13, 13, w^2 - w - 7],
[17, 17, -w^2 + w + 4],
[19, 19, -w^2 - w + 4],
[29, 29, 2*w^2 - 2*w - 11],
[31, 31, w^2 - 2*w - 4],
[37, 37, -2*w^2 + 3*w + 7],
[41, 41, w^2 - 2],
[59, 59, 2*w^2 - 13],
[61, 61, w^2 + w - 10],
[67, 67, 2*w^2 - w - 11],
[73, 73, -w^2 - 1],
[83, 83, w^2 - w - 10],
[89, 89, -w^2 + 4*w - 2],
[97, 97, w^2 - 2*w - 7],
[97, 97, -w^2 - 4*w - 5],
[97, 97, 2*w^2 + w - 8],
[101, 101, -4*w^2 + 2*w + 25],
[103, 103, w^2 + w - 7],
[107, 107, -3*w - 4],
[109, 109, 5*w^2 - 3*w - 31],
[121, 11, 2*w^2 - 3*w - 4],
[125, 5, -5],
[137, 137, -5*w^2 + 3*w + 34],
[139, 139, w^2 + 2*w - 4],
[151, 151, 3*w^2 - 23],
[151, 151, w^2 - 3*w - 5],
[151, 151, 3*w^2 - 3*w - 17],
[157, 157, 4*w^2 - 9*w - 8],
[157, 157, w^2 + 3*w - 2],
[157, 157, 2*w^2 - w - 2],
[167, 167, 2*w^2 + w - 11],
[167, 167, -w^2 + 11],
[167, 167, -3*w^2 + 19],
[169, 13, w^2 - 4*w - 4],
[173, 173, 2*w^2 - 3*w - 10],
[179, 179, -3*w + 7],
[191, 191, -w^2 + 5*w - 5],
[193, 193, -w^2 + w - 2],
[199, 199, 3*w - 2],
[211, 211, w^2 - 3*w - 11],
[223, 223, 3*w^2 - 20],
[223, 223, w^2 - 5*w - 4],
[223, 223, 2*w^2 - w - 8],
[229, 229, 4*w^2 - 3*w - 26],
[229, 229, 2*w^2 + w - 20],
[229, 229, 2*w^2 - 2*w - 5],
[239, 239, -6*w^2 + 3*w + 38],
[241, 241, -w^2 + 4*w - 5],
[257, 257, -2*w^2 + 5*w - 1],
[263, 263, -2*w^2 - w + 17],
[269, 269, -w^2 - w + 13],
[269, 269, 3*w - 4],
[269, 269, -4*w^2 + w + 31],
[271, 271, 2*w^2 - 7],
[271, 271, 3*w - 5],
[271, 271, w^2 - 3*w - 8],
[289, 17, 2*w^2 + w - 5],
[307, 307, 2*w^2 - w - 5],
[311, 311, 3*w^2 - 3*w - 19],
[311, 311, -7*w^2 + 13*w + 19],
[311, 311, w^2 - 2*w - 13],
[313, 313, w^2 + 2*w - 7],
[317, 317, -5*w^2 + w + 35],
[331, 331, -w^2 + 3*w - 4],
[337, 337, 3*w^2 - 3*w - 13],
[343, 7, -7],
[347, 347, 2*w^2 - 3*w - 13],
[349, 349, 3*w^2 - 3*w - 20],
[359, 359, -w^2 + 5*w - 2],
[361, 19, -2*w^2 + 7*w - 1],
[389, 389, -2*w^2 + w - 1],
[397, 397, w^2 - 6*w + 10],
[401, 401, 6*w^2 - 3*w - 44],
[419, 419, -3*w^2 + 6*w + 10],
[419, 419, 4*w^2 - 2*w - 31],
[419, 419, -4*w^2 + 4*w + 19],
[421, 421, 6*w^2 - 12*w - 13],
[431, 431, 4*w^2 - 7*w - 10],
[433, 433, w^2 - 4*w - 7],
[443, 443, 2*w^2 + 4*w - 5],
[443, 443, 5*w^2 - 11*w - 11],
[443, 443, 3*w^2 - 14],
[449, 449, -2*w^2 + 2*w - 1],
[457, 457, w^2 - 6*w - 5],
[457, 457, w^2 + 3*w - 5],
[457, 457, 6*w^2 - 3*w - 37],
[463, 463, w^2 + 5*w - 1],
[467, 467, w^2 - w - 13],
[479, 479, 5*w^2 - 9*w - 16],
[487, 487, -6*w - 5],
[499, 499, 3*w^2 - 3*w - 4],
[499, 499, 3*w^2 + 3*w - 1],
[499, 499, w^2 - 5*w + 8],
[503, 503, 4*w^2 - 5*w - 16],
[509, 509, 2*w^2 - 5*w - 8],
[521, 521, 4*w^2 - w - 22],
[523, 523, w^2 - 5*w - 16],
[547, 547, 5*w^2 - 2*w - 38],
[547, 547, 8*w^2 - 3*w - 58],
[547, 547, 4*w^2 - 29],
[557, 557, 7*w^2 - 4*w - 43],
[557, 557, 3*w - 11],
[557, 557, 2*w^2 - 4*w - 11],
[569, 569, 5*w^2 - 4*w - 32],
[569, 569, 3*w^2 - 3*w - 11],
[569, 569, w^2 + 2*w - 16],
[571, 571, w^2 - 4*w - 10],
[593, 593, 2*w^2 - 6*w - 7],
[601, 601, -w^2 - 3*w + 14],
[607, 607, 3*w^2 + 3*w - 7],
[619, 619, 3*w^2 - 6*w - 11],
[631, 631, -w^2 - w - 5],
[641, 641, 3*w^2 - 3*w - 5],
[647, 647, -w^2 + 6*w - 1],
[653, 653, -5*w^2 + 4*w + 35],
[661, 661, 3*w^2 - 3*w - 10],
[673, 673, w^2 + 4*w - 4],
[677, 677, 5*w^2 - 6*w - 25],
[677, 677, w^2 - 14],
[677, 677, 4*w^2 - 3*w - 20],
[691, 691, 5*w^2 - 2*w - 29],
[701, 701, 5*w^2 - 10*w - 14],
[719, 719, -3*w - 11],
[727, 727, -2*w^2 + 9*w - 8],
[733, 733, -w^2 - 4*w - 8],
[739, 739, w^2 + 3*w - 8],
[743, 743, 2*w^2 - 4*w - 17],
[757, 757, -w^2 + w - 5],
[769, 769, 3*w^2 - 3*w - 7],
[769, 769, -w^2 - 3*w - 7],
[769, 769, 5*w^2 - 3*w - 28],
[773, 773, 7*w^2 - 6*w - 38],
[787, 787, 2*w^2 - w - 20],
[797, 797, -5*w^2 + 14*w - 1],
[821, 821, -w^2 + 6*w - 4],
[823, 823, 7*w^2 - 6*w - 41],
[839, 839, 4*w^2 - 6*w - 11],
[841, 29, 5*w^2 - 4*w - 26],
[857, 857, 3*w^2 - 11],
[863, 863, 2*w^2 + 2*w - 17],
[881, 881, 5*w^2 - 5*w - 29],
[883, 883, -4*w^2 + 8*w + 13],
[907, 907, 4*w^2 - 9*w - 11],
[911, 911, 2*w^2 + 3*w - 10],
[947, 947, 2*w^2 + 5*w - 5],
[953, 953, 2*w^2 - 5*w - 11],
[961, 31, w^2 - 5*w - 13],
[967, 967, 2*w^2 - 3*w - 22],
[967, 967, 5*w^2 - 9*w - 10],
[967, 967, 3*w^2 - 10],
[977, 977, 5*w^2 - 37],
[983, 983, 7*w^2 - 5*w - 40],
[997, 997, -7*w^2 + 5*w + 49]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^4 - 3*x^3 - 3*x^2 + 11*x - 5;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, -e^3 + 2*e^2 + 5*e - 6, e, -e + 2, 1, 2*e^3 - 4*e^2 - 10*e + 12, -2*e^3 + 4*e^2 + 9*e - 9, -3*e^3 + 5*e^2 + 15*e - 16, e^3 - 2*e^2 - 5*e + 7, -e^3 + 3*e + 7, 2*e^3 - 4*e^2 - 7*e + 13, -e^3 + 3*e^2 + 6*e - 10, -5*e^3 + 10*e^2 + 24*e - 30, 4*e^3 - 9*e^2 - 21*e + 23, -e^3 + 2*e^2 + 5*e + 3, -2*e^2 + 3*e + 6, 5*e^3 - 10*e^2 - 21*e + 30, 4*e^3 - 9*e^2 - 22*e + 33, e^3 + e^2 - 9*e - 1, -5*e^3 + 7*e^2 + 23*e - 17, 4*e^3 - 7*e^2 - 20*e + 23, -4*e^3 + 8*e^2 + 17*e - 25, 2*e^3 - e^2 - 9*e - 5, -e^3 + 4*e^2 - e - 9, 2*e^3 - 3*e^2 - 10*e + 15, -e^2 + 4*e + 1, -2*e^3 + 6*e^2 + 8*e - 6, 6*e^3 - 11*e^2 - 27*e + 38, -3*e^3 + 6*e^2 + 19*e - 27, 5*e^3 - 11*e^2 - 20*e + 33, -3*e^3 + 7*e^2 + 13*e - 29, e^3 - 6*e^2 - e + 23, -7*e^3 + 10*e^2 + 32*e - 23, e^3 - 4*e^2 - 4*e + 9, -2*e^3 + 6*e^2 + 9*e - 22, -7*e^3 + 12*e^2 + 35*e - 35, 4*e^3 - 7*e^2 - 16*e + 22, -5*e^3 + 9*e^2 + 18*e - 14, 5*e^3 - 8*e^2 - 27*e + 20, -e^3 - 2*e^2 + 3*e + 16, -e^3 + 3*e^2 + 3*e - 25, -2*e^3 + 5*e^2 + 7*e - 30, e^3 + e^2 - 11*e - 7, 6*e^3 - 11*e^2 - 24*e + 33, e^3 - 5*e^2 - 2*e + 5, e^3 + 3*e^2 - 12*e - 16, -3*e^3 + 6*e^2 + 18*e - 16, 3*e^3 - 2*e^2 - 15*e + 3, 2*e^3 - 5*e^2 - 11*e + 6, -10*e^3 + 20*e^2 + 46*e - 52, e^2 + 16, 10*e^3 - 20*e^2 - 48*e + 57, -e^3 + 4*e^2 - 22, e^3 - 5*e^2 - e + 14, 6*e^3 - 13*e^2 - 28*e + 24, 7*e^3 - 8*e^2 - 38*e + 25, 9*e^3 - 19*e^2 - 35*e + 50, 2*e^3 - 6*e^2 - 15*e + 23, -2*e^3 + 13*e + 7, 7*e^3 - 16*e^2 - 35*e + 55, 6*e^3 - 10*e^2 - 33*e + 26, e^2 - 4*e + 15, -6*e^2 + 8*e + 9, -2*e^3 + 8*e^2 + 7*e - 24, -2*e^3 + 20*e - 3, 3*e^3 - 4*e^2 - 21*e + 4, 9*e^3 - 17*e^2 - 40*e + 60, 6*e^3 - 12*e^2 - 35*e + 46, 7*e^3 - 14*e^2 - 33*e + 31, 3*e^3 - 4*e^2 - 30*e + 11, -2*e^3 - 3*e^2 + 8*e + 26, 12*e^3 - 23*e^2 - 53*e + 72, -4*e^3 + 7*e^2 + 18*e - 18, e^3 - 3*e^2 - 14*e + 26, -5*e^3 + 13*e^2 + 24*e - 29, 11*e^3 - 19*e^2 - 55*e + 56, 6*e^3 - 7*e^2 - 27*e + 14, 4*e^3 - 8*e^2 - 14*e + 22, -3*e^3 + 9*e^2 + 17*e - 40, 6*e^3 - 12*e^2 - 32*e + 37, 11*e^3 - 24*e^2 - 42*e + 59, -3*e^3 + 3*e^2 + 21*e - 2, -2*e^3 - 2*e^2 + 3*e + 30, e^3 + 3*e^2 - 17*e - 6, 2*e^3 - 11*e^2 - 3*e + 31, 2*e^2 + 10*e - 13, -e^3 - 2*e^2 + 10*e + 5, -2*e^3 + 7*e^2 + 4*e - 23, -5*e^3 + 5*e^2 + 26*e - 22, e^3 - e^2 - 14*e + 26, 2*e^3 - 11*e^2 - 3*e + 23, 10*e^3 - 17*e^2 - 38*e + 39, e^3 - e^2 - 5*e, 5*e^3 - 5*e^2 - 28*e, 15*e^3 - 26*e^2 - 74*e + 86, -2*e^3 + e^2 + 15*e - 20, 9*e^3 - 22*e^2 - 27*e + 65, -7*e^3 + 17*e^2 + 31*e - 42, -6*e^3 + 4*e^2 + 28*e - 3, -10*e^3 + 27*e^2 + 41*e - 62, 14*e^3 - 29*e^2 - 56*e + 83, -6*e^3 + 22*e^2 + 21*e - 56, 9*e^3 - 19*e^2 - 38*e + 54, 9*e^3 - 17*e^2 - 50*e + 37, -3*e^3 + e^2 + 19*e - 1, -11*e^3 + 23*e^2 + 45*e - 67, -5*e^3 + 7*e^2 + 40*e - 32, 4*e^3 - 16*e^2 - 12*e + 56, -e^3 + 14*e^2 - 7*e - 46, -15*e^3 + 35*e^2 + 65*e - 93, 3*e^3 - 4*e^2 - 6*e + 4, -5*e^2 - 7*e + 20, 5*e + 2, -12*e^3 + 20*e^2 + 62*e - 67, -10*e^3 + 19*e^2 + 49*e - 33, -13*e^3 + 24*e^2 + 52*e - 64, 3*e^3 + 2*e^2 - 22*e - 10, 5*e^2 - 8*e + 11, -6*e^3 + 4*e^2 + 33*e + 11, -7*e^3 + 18*e^2 + 21*e - 40, 9*e^3 - 10*e^2 - 44*e + 15, 6*e^3 - 9*e^2 - 30*e + 4, -4*e^3 + 11*e^2 + 16*e - 32, -10*e^3 + 12*e^2 + 61*e - 42, 11*e^3 - 17*e^2 - 52*e + 69, -e^3 + 7*e^2 + e - 20, 7*e^3 - 11*e^2 - 25*e + 35, -9*e^3 + 21*e^2 + 43*e - 59, -8*e^3 + 11*e^2 + 35*e - 33, 4*e^3 - e^2 - 12*e - 8, 7*e^3 - 13*e^2 - 37*e + 54, -e^3 + 5*e^2 - 9, 4*e^3 - 11*e^2 - 26*e + 56, -9*e^3 + 14*e^2 + 58*e - 45, -21*e^3 + 43*e^2 + 91*e - 104, 5*e^3 - 14*e^2 - 19*e + 35, 11*e^3 - 24*e^2 - 63*e + 68, -12*e^3 + 21*e^2 + 58*e - 57, -e^3 - 8*e^2 + 9*e + 52, 11*e^3 - 20*e^2 - 54*e + 49, 12*e^3 - 24*e^2 - 56*e + 59, -3*e^3 + 7*e^2 + e + 3, -5*e^3 + e^2 + 41*e + 3, -6*e^3 + 8*e^2 + 39*e - 32, 5*e^3 - 5*e^2 - 32*e + 26, -e^2 + 5*e + 9, 6*e^3 + 5*e^2 - 44*e - 20, -11*e^3 + 19*e^2 + 56*e - 27, 2*e^3 - 11*e + 3, 5*e^3 + 3*e^2 - 37*e + 13, 2*e^3 - e^2 - 18*e + 12, -6*e^3 + 6*e^2 + 42*e - 7, 14*e^3 - 33*e^2 - 50*e + 85, -8*e^3 + 22*e^2 + 28*e - 51, 7*e^3 - 21*e^2 - 20*e + 78, -7*e^3 + 22*e^2 + 35*e - 78, -12*e^3 + 33*e^2 + 57*e - 93, -7*e^3 + 15*e^2 + 30*e - 47];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {11, 11, -w^2 + 5}>] := -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;