/*
  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![-55, 0, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {6, 6, -w - 7}>;

primesArray := [
[2, 2, w + 1],
[3, 3, w + 1],
[3, 3, w + 2],
[5, 5, -2*w + 15],
[11, 11, 3*w - 22],
[13, 13, w + 4],
[13, 13, w + 9],
[17, 17, w + 2],
[17, 17, w + 15],
[19, 19, -w - 6],
[19, 19, w - 6],
[23, 23, w + 3],
[23, 23, w + 20],
[47, 47, w + 14],
[47, 47, w + 33],
[49, 7, -7],
[67, 67, w + 16],
[67, 67, w + 51],
[73, 73, w + 36],
[73, 73, w + 37],
[79, 79, 5*w + 36],
[79, 79, 5*w - 36],
[89, 89, -w - 12],
[89, 89, w - 12],
[103, 103, w + 40],
[103, 103, w + 63],
[131, 131, -6*w - 43],
[131, 131, -6*w + 43],
[139, 139, 2*w - 9],
[139, 139, -2*w - 9],
[151, 151, 4*w - 27],
[151, 151, 4*w + 27],
[163, 163, w + 50],
[163, 163, w + 113],
[173, 173, w + 48],
[173, 173, w + 125],
[181, 181, -3*w - 26],
[181, 181, 3*w - 26],
[193, 193, w + 21],
[193, 193, w + 172],
[197, 197, w + 45],
[197, 197, w + 152],
[211, 211, 2*w - 3],
[211, 211, -2*w - 3],
[223, 223, w + 72],
[223, 223, w + 151],
[229, 229, 6*w + 47],
[229, 229, 6*w - 47],
[233, 233, w + 88],
[233, 233, w + 145],
[239, 239, -3*w - 16],
[239, 239, 3*w - 16],
[269, 269, -w - 18],
[269, 269, w - 18],
[271, 271, -8*w - 57],
[271, 271, -8*w + 57],
[277, 277, w + 71],
[277, 277, w + 206],
[293, 293, w + 73],
[293, 293, w + 220],
[337, 337, w + 27],
[337, 337, w + 310],
[359, 359, 9*w + 64],
[359, 359, 9*w - 64],
[367, 367, w + 34],
[367, 367, w + 333],
[373, 373, w + 146],
[373, 373, w + 227],
[383, 383, w + 173],
[383, 383, w + 210],
[389, 389, -5*w - 42],
[389, 389, 5*w - 42],
[401, 401, 29*w - 216],
[401, 401, 11*w - 84],
[421, 421, -6*w + 49],
[421, 421, -6*w - 49],
[431, 431, 3*w - 8],
[431, 431, -3*w - 8],
[439, 439, -4*w - 21],
[439, 439, 4*w - 21],
[443, 443, w + 99],
[443, 443, w + 344],
[449, 449, -8*w - 63],
[449, 449, -8*w + 63],
[457, 457, w + 91],
[457, 457, w + 366],
[463, 463, w + 38],
[463, 463, w + 425],
[467, 467, w + 191],
[467, 467, w + 276],
[479, 479, -3*w - 4],
[479, 479, 3*w - 4],
[487, 487, w + 223],
[487, 487, w + 264],
[491, 491, 3*w - 2],
[491, 491, -3*w - 2],
[509, 509, 2*w - 27],
[509, 509, -2*w - 27],
[521, 521, -w - 24],
[521, 521, w - 24],
[557, 557, w + 75],
[557, 557, w + 482],
[571, 571, 11*w + 78],
[571, 571, 11*w - 78],
[587, 587, w + 246],
[587, 587, w + 341],
[593, 593, w + 184],
[593, 593, w + 409],
[613, 613, w + 235],
[613, 613, w + 378],
[641, 641, -4*w - 39],
[641, 641, 4*w - 39],
[643, 643, w + 174],
[643, 643, w + 469],
[647, 647, w + 257],
[647, 647, w + 390],
[659, 659, -18*w + 131],
[659, 659, 30*w - 221],
[661, 661, 3*w - 34],
[661, 661, -3*w - 34],
[673, 673, w + 196],
[673, 673, w + 477],
[677, 677, w + 240],
[677, 677, w + 437],
[683, 683, w + 136],
[683, 683, w + 547],
[709, 709, 21*w - 158],
[709, 709, 27*w - 202],
[727, 727, w + 339],
[727, 727, w + 388],
[733, 733, w + 39],
[733, 733, w + 694],
[739, 739, 10*w - 69],
[739, 739, 10*w + 69],
[811, 811, 35*w - 258],
[811, 811, -19*w + 138],
[823, 823, w + 177],
[823, 823, w + 646],
[829, 829, -6*w - 53],
[829, 829, 6*w - 53],
[841, 29, -29],
[853, 853, w + 316],
[853, 853, w + 537],
[857, 857, w + 270],
[857, 857, w + 587],
[863, 863, w + 405],
[863, 863, w + 458],
[877, 877, w + 377],
[877, 877, w + 500],
[881, 881, -19*w + 144],
[881, 881, -37*w + 276],
[883, 883, w + 52],
[883, 883, w + 831],
[907, 907, w + 350],
[907, 907, w + 557],
[919, 919, -8*w - 51],
[919, 919, 8*w - 51],
[929, 929, 5*w - 48],
[929, 929, -5*w - 48],
[937, 937, w + 176],
[937, 937, w + 761],
[947, 947, w + 252],
[947, 947, w + 695],
[953, 953, w + 334],
[953, 953, w + 619],
[961, 31, -31],
[983, 983, w + 277],
[983, 983, w + 706],
[997, 997, w + 371],
[997, 997, w + 626]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

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

heckeEigenvaluesArray := [1, 1, e, -e^3 - e^2 + 4*e + 1, -e^2 + 5, -e^2 + e + 3, e^3 + e^2 - 5*e, 2*e^3 - e^2 - 7*e + 3, -e^3 + e^2 + 6*e - 4, e^2 + e, -4*e^3 - 2*e^2 + 13*e + 3, 3*e^3 + 4*e^2 - 10*e - 8, e^3 - 2*e^2 - 3*e + 4, 2*e^3 + e^2 - 7*e + 1, -2*e^3 - 5*e^2 + 3*e + 14, 2*e^3 + 5*e^2 - 8*e - 11, 3*e^3 + 10*e^2 - 8*e - 22, 3*e^3 + 2*e^2 - 10*e - 1, 7*e^3 + 5*e^2 - 19*e - 10, e^3 + 7*e^2 - 3*e - 21, e^2 + 4*e + 2, 5*e^3 + 7*e^2 - 16*e - 12, 3*e^3 - 8*e - 1, -6*e^3 - 7*e^2 + 16*e + 16, -6*e^3 - 5*e^2 + 20*e + 5, -6*e^3 - e^2 + 17*e - 1, -e^2 - 2*e + 18, e^2 - 5*e - 2, -5*e^3 - 13*e^2 + 9*e + 34, 3*e^3 - 14*e + 4, 5*e^3 - 2*e^2 - 18*e + 5, -2*e^3 - e^2 + 7*e + 15, -9*e^2 + e + 28, -4*e^3 - 6*e^2 + 5*e + 13, -12*e^3 - 10*e^2 + 36*e + 15, 2*e^3 + 4*e^2 - 7*e - 13, 7*e^3 + 10*e^2 - 19*e - 28, 2*e^2 - 6*e - 10, 6*e^3 + 5*e^2 - 21*e - 2, -e^2 - 7*e + 8, -4*e^3 + 2*e^2 + 14*e - 2, e^3 + 5*e^2 - 4*e - 22, 3*e^3 + 15*e^2 - 11*e - 38, -5*e^3 - 11*e^2 + 8*e + 32, 3*e^3 - 2*e^2 - 5*e + 15, e^3 - 9*e^2 - 5*e + 25, -5*e^3 - 10*e^2 + 15*e + 29, e^3 + 11*e^2 - 5*e - 36, -7*e^3 - 14*e^2 + 24*e + 28, 4*e^3 - 8*e^2 - 18*e + 17, -2*e^3 + 6*e - 1, e^3 + 5*e^2 + 3*e - 8, -6*e^3 + e^2 + 22*e - 2, 2*e^3 + 9*e^2 - 7*e - 20, 10*e^3 + 3*e^2 - 25*e + 6, -2*e^3 + 3*e^2 + 9*e - 6, -7*e^3 - 7*e^2 + 24*e + 6, 11*e^3 + 11*e^2 - 32*e - 14, -5*e^3 - 18*e^2 + 8*e + 42, -e^3 - 3*e^2 + 7*e + 10, e^3 + 18*e^2 + e - 41, -11*e^3 - 18*e^2 + 30*e + 37, -5*e^3 + 17*e + 14, -4*e^2 + 6*e + 9, 4*e^3 - e^2 - 19*e + 20, 5*e^3 + 15*e^2 - 3*e - 42, -4*e^3 - 2*e^2 + 3*e + 6, -4*e^3 - 11*e^2 + e + 43, -13*e^3 - e^2 + 43*e + 2, 3*e^3 + 7*e^2 - 15*e - 21, 3*e^3 + 11*e^2 - 11*e - 26, 3*e^3 + 4*e^2 - 20*e - 16, -3*e^3 - 7*e^2 + 10*e + 22, e^3 + 2*e^2 - 14*e - 6, -3*e^3 + 8*e^2 + 20*e - 8, -e^3 - 11*e^2 - 5*e + 38, 9*e^3 + 8*e^2 - 38*e - 5, -4*e^3 + 2*e^2 + 18*e - 14, -5*e^3 + 3*e^2 + 15*e - 15, -4*e^2 + 8*e + 12, -e^3 - e^2 + 9*e + 5, 4*e^3 + 13*e^2 - e - 39, 7*e^2 - e - 3, -9*e^3 - 17*e^2 + 30*e + 28, 5*e^3 - 3*e^2 - 14*e + 12, 8*e^2 + 5*e - 12, 5*e^3 + 3*e^2 - 20*e - 8, -12*e^3 - 7*e^2 + 44*e - 5, -9*e^2 + 5*e + 17, -3*e^3 - 4*e^2 - e + 15, 6*e^3 + 3*e^2 - 32*e - 8, 9*e^3 + 8*e^2 - 35*e - 15, -8*e^3 + e^2 + 22*e - 24, -4*e^3 - 11*e^2 + 27*e + 30, -e^3 - 12*e^2 - 12*e + 37, 10*e^3 + 14*e^2 - 34*e - 29, 10*e^3 + 3*e^2 - 28*e - 20, 3*e^3 - 6*e^2 - 15*e + 14, -6*e^3 - 15*e^2 + 8*e + 56, -3*e^3 - e^2 + 20*e + 6, -10*e^3 - 17*e^2 + 32*e + 31, -7*e^3 - 5*e^2 + 20*e - 6, -6*e^3 - 4*e^2 + 28*e + 11, -9*e^3 - 17*e^2 + 20*e + 52, -7*e^3 - 7*e^2 + 27*e + 5, 15*e^3 + 16*e^2 - 50*e - 37, 2*e^3 + 16*e^2 - 4*e - 50, -5*e^3 - 3*e^2 + 25*e - 12, -15*e^3 - 23*e^2 + 39*e + 61, -3*e^3 + 7*e^2 + 8*e - 28, 2*e^2 + 13*e - 17, 3*e^3 + 3*e + 24, e^2 - 7*e - 34, 16*e^3 + 10*e^2 - 53*e - 29, 4*e^3 + 7*e^2 - 23*e + 4, -9*e^3 - 7*e^2 + 24*e + 21, 10*e^3 - 28*e - 6, 6*e^3 + 27*e^2 - 14*e - 54, -e^3 + 12*e^2 + 12*e - 20, -4*e^3 + 5*e^2 + 27*e - 29, -4*e^3 + 9*e^2 + 18*e - 22, e^3 + 10*e^2 - 17*e - 34, 6*e^3 + 11*e^2 - 24*e - 16, -15*e^3 - 7*e^2 + 49*e - 10, 12*e^3 + 11*e^2 - 30*e - 25, -5*e^3 - 9*e^2 + 8*e + 12, -20*e^3 - 9*e^2 + 63*e + 19, 8*e^3 + 5*e^2 - 17*e + 2, 6*e^3 + e^2 - 12*e + 8, -16*e^3 - 21*e^2 + 56*e + 24, -14*e^3 - 9*e^2 + 41*e, -3*e^3 - 9*e^2 - 4*e + 51, 3*e^3 + e^2 - 30*e + 8, -3*e^3 + 2*e^2 + 20*e - 8, 4*e^2 - 4*e + 5, -14*e^3 - 9*e^2 + 48*e + 4, 9*e^3 + 2*e^2 - 35*e + 11, -8*e^3 - 10*e^2 + 19*e + 40, e^3 + 11*e^2 + 13*e - 14, 14*e^3 + 20*e^2 - 37*e - 54, 22*e^3 + 8*e^2 - 73*e - 11, 5*e^3 + 7*e^2 - 27*e - 22, 20*e^3 + 16*e^2 - 47*e - 34, -3*e^3 - 16*e^2 + 8*e + 32, 4*e^3 - 14*e^2 - 21*e + 13, 16*e^3 - 55*e - 9, 3*e^3 - 18, -9*e^3 - 4*e^2 + 26*e + 34, 4*e^2 - 8*e - 28, 2*e^3 - 4*e^2 - 15*e + 4, -2*e^3 + 11*e^2 + 29*e - 41, -e^3 + 23*e^2 + 17*e - 40, e^3 + 11*e^2 + 5*e - 20, -20*e^3 - 19*e^2 + 65*e + 24, 18*e^3 + 26*e^2 - 44*e - 70, -3*e^3 + 11*e^2 + 34*e - 28, 7*e^3 + 21*e^2 - 21*e - 33, 7*e^3 - 16*e^2 - 19*e + 55, -18*e^3 - 16*e^2 + 59*e + 32, 8*e^3 + 8*e^2 - 30*e + 4, -10*e^3 - 12*e^2 + 23*e + 22, -8*e^3 + 46*e - 12, -11*e^3 - 28*e^2 + 32*e + 40, -7*e - 30, 6*e^3 + 11*e^2 - 29*e - 42, -19*e^3 - 10*e^2 + 60*e + 5, -14*e^3 - 22*e^2 + 48*e + 16, 8*e^3 + 23*e^2 - 32*e - 44, -8*e^3 - 26*e^2 + 15*e + 72, 4*e^3 - 3*e^2 + 6*e + 7];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

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