/*
  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![-34, 0, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {1, 1, 1}>;

primesArray := [
[2, 2, w - 6],
[3, 3, w + 1],
[3, 3, w + 2],
[5, 5, w + 2],
[5, 5, w + 3],
[11, 11, w + 1],
[11, 11, w + 10],
[17, 17, -3*w + 17],
[29, 29, w + 11],
[29, 29, w + 18],
[37, 37, w + 16],
[37, 37, w + 21],
[47, 47, -w - 9],
[47, 47, w - 9],
[49, 7, -7],
[61, 61, w + 20],
[61, 61, w + 41],
[89, 89, 2*w - 15],
[89, 89, -2*w - 15],
[103, 103, -14*w + 81],
[103, 103, 4*w - 21],
[107, 107, w + 26],
[107, 107, w + 81],
[109, 109, w + 19],
[109, 109, w + 90],
[127, 127, 2*w - 3],
[127, 127, -2*w - 3],
[131, 131, w + 54],
[131, 131, w + 77],
[137, 137, -3*w - 13],
[137, 137, 3*w - 13],
[139, 139, w + 27],
[139, 139, w + 112],
[151, 151, 8*w - 45],
[151, 151, -10*w + 57],
[163, 163, w + 69],
[163, 163, w + 94],
[169, 13, -13],
[173, 173, w + 42],
[173, 173, w + 131],
[181, 181, w + 45],
[181, 181, w + 136],
[191, 191, -w - 15],
[191, 191, w - 15],
[197, 197, w + 25],
[197, 197, w + 172],
[211, 211, w + 33],
[211, 211, w + 178],
[223, 223, -3*w - 23],
[223, 223, 3*w - 23],
[227, 227, w + 48],
[227, 227, w + 179],
[239, 239, -5*w + 33],
[239, 239, -23*w + 135],
[257, 257, -3*w - 7],
[257, 257, 3*w - 7],
[263, 263, -24*w + 139],
[263, 263, 6*w - 31],
[269, 269, w + 29],
[269, 269, w + 240],
[271, 271, -9*w + 55],
[271, 271, -15*w + 89],
[277, 277, w + 119],
[277, 277, w + 158],
[281, 281, 3*w - 5],
[281, 281, -3*w - 5],
[283, 283, w + 113],
[283, 283, w + 170],
[317, 317, w + 44],
[317, 317, w + 273],
[347, 347, w + 46],
[347, 347, w + 301],
[353, 353, 9*w - 49],
[353, 353, -21*w + 121],
[359, 359, -25*w + 147],
[359, 359, -7*w + 45],
[361, 19, -19],
[379, 379, w + 105],
[379, 379, w + 274],
[383, 383, 6*w - 29],
[383, 383, -36*w + 209],
[397, 397, w + 35],
[397, 397, w + 362],
[409, 409, -5*w - 21],
[409, 409, 5*w - 21],
[419, 419, w + 156],
[419, 419, w + 263],
[433, 433, -12*w + 73],
[433, 433, -18*w + 107],
[457, 457, 6*w + 41],
[457, 457, -6*w + 41],
[463, 463, -4*w - 9],
[463, 463, 4*w - 9],
[499, 499, w + 212],
[499, 499, w + 287],
[529, 23, -23],
[541, 541, w + 121],
[541, 541, w + 420],
[547, 547, w + 157],
[547, 547, w + 390],
[569, 569, -28*w + 165],
[569, 569, -10*w + 63],
[571, 571, w + 264],
[571, 571, w + 307],
[577, 577, -47*w + 273],
[577, 577, 7*w - 33],
[593, 593, 2*w - 27],
[593, 593, -2*w - 27],
[599, 599, -6*w - 25],
[599, 599, 6*w - 25],
[619, 619, w + 167],
[619, 619, w + 452],
[631, 631, -15*w + 91],
[631, 631, -21*w + 125],
[643, 643, w + 57],
[643, 643, w + 586],
[647, 647, -11*w + 69],
[647, 647, -29*w + 171],
[653, 653, w + 120],
[653, 653, w + 533],
[677, 677, w + 64],
[677, 677, w + 613],
[683, 683, w + 248],
[683, 683, w + 435],
[691, 691, w + 322],
[691, 691, w + 369],
[709, 709, w + 265],
[709, 709, w + 444],
[727, 727, -9*w + 59],
[727, 727, -39*w + 229],
[761, 761, -27*w + 155],
[761, 761, 15*w - 83],
[769, 769, 5*w - 9],
[769, 769, -5*w - 9],
[787, 787, w + 63],
[787, 787, w + 724],
[811, 811, w + 70],
[811, 811, w + 741],
[821, 821, w + 149],
[821, 821, w + 672],
[827, 827, w + 193],
[827, 827, w + 634],
[853, 853, w + 253],
[853, 853, w + 600],
[863, 863, -6*w - 19],
[863, 863, 6*w - 19],
[877, 877, w + 339],
[877, 877, w + 538],
[907, 907, w + 74],
[907, 907, w + 833],
[919, 919, -3*w - 35],
[919, 919, 3*w - 35],
[937, 937, 7*w - 27],
[937, 937, -7*w - 27],
[941, 941, w + 144],
[941, 941, w + 797],
[947, 947, w + 292],
[947, 947, w + 655],
[953, 953, 2*w - 33],
[953, 953, -2*w - 33],
[961, 31, -31],
[967, 967, 9*w - 61],
[967, 967, -9*w - 61],
[977, 977, -4*w - 39],
[977, 977, 4*w - 39],
[997, 997, w + 55],
[997, 997, w + 942]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^8 + 14*x^6 + 56*x^4 + 76*x^2 + 32;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [-1/4*e^6 - 7/2*e^4 - 13*e^2 - 11, -e, e, -1/4*e^7 - 3*e^5 - 8*e^3 - 3*e, 1/4*e^7 + 3*e^5 + 8*e^3 + 3*e, -e^7 - 13*e^5 - 43*e^3 - 33*e, e^7 + 13*e^5 + 43*e^3 + 33*e, -e^6 - 12*e^4 - 34*e^2 - 22, 3/4*e^7 + 10*e^5 + 34*e^3 + 25*e, -3/4*e^7 - 10*e^5 - 34*e^3 - 25*e, -5/4*e^7 - 16*e^5 - 52*e^3 - 41*e, 5/4*e^7 + 16*e^5 + 52*e^3 + 41*e, -1/2*e^6 - 6*e^4 - 18*e^2 - 16, -1/2*e^6 - 6*e^4 - 18*e^2 - 16, -e^6 - 13*e^4 - 43*e^2 - 26, -5/4*e^7 - 16*e^5 - 52*e^3 - 43*e, 5/4*e^7 + 16*e^5 + 52*e^3 + 43*e, 2*e^6 + 25*e^4 + 75*e^2 + 46, 2*e^6 + 25*e^4 + 75*e^2 + 46, 3*e^6 + 40*e^4 + 138*e^2 + 104, 3*e^6 + 40*e^4 + 138*e^2 + 104, e^5 + 10*e^3 + 15*e, -e^5 - 10*e^3 - 15*e, 9/4*e^7 + 29*e^5 + 94*e^3 + 69*e, -9/4*e^7 - 29*e^5 - 94*e^3 - 69*e, 2*e^6 + 24*e^4 + 66*e^2 + 32, 2*e^6 + 24*e^4 + 66*e^2 + 32, -e^7 - 14*e^5 - 52*e^3 - 43*e, e^7 + 14*e^5 + 52*e^3 + 43*e, 2*e^6 + 24*e^4 + 67*e^2 + 38, 2*e^6 + 24*e^4 + 67*e^2 + 38, -e^3 - 5*e, e^3 + 5*e, -2*e^6 - 26*e^4 - 82*e^2 - 56, -2*e^6 - 26*e^4 - 82*e^2 - 56, -e^7 - 12*e^5 - 31*e^3 - 7*e, e^7 + 12*e^5 + 31*e^3 + 7*e, 3*e^6 + 39*e^4 + 128*e^2 + 106, -11/4*e^7 - 36*e^5 - 122*e^3 - 99*e, 11/4*e^7 + 36*e^5 + 122*e^3 + 99*e, -7/4*e^7 - 23*e^5 - 76*e^3 - 47*e, 7/4*e^7 + 23*e^5 + 76*e^3 + 47*e, 1/2*e^6 + 8*e^4 + 36*e^2 + 32, 1/2*e^6 + 8*e^4 + 36*e^2 + 32, -15/4*e^7 - 49*e^5 - 164*e^3 - 135*e, 15/4*e^7 + 49*e^5 + 164*e^3 + 135*e, 2*e^7 + 25*e^5 + 74*e^3 + 39*e, -2*e^7 - 25*e^5 - 74*e^3 - 39*e, -1/2*e^6 - 8*e^4 - 40*e^2 - 48, -1/2*e^6 - 8*e^4 - 40*e^2 - 48, 3*e^7 + 38*e^5 + 117*e^3 + 73*e, -3*e^7 - 38*e^5 - 117*e^3 - 73*e, -11/2*e^6 - 70*e^4 - 220*e^2 - 160, -11/2*e^6 - 70*e^4 - 220*e^2 - 160, -7*e^6 - 90*e^4 - 289*e^2 - 210, -7*e^6 - 90*e^4 - 289*e^2 - 210, -3*e^6 - 40*e^4 - 138*e^2 - 104, -3*e^6 - 40*e^4 - 138*e^2 - 104, 15/4*e^7 + 48*e^5 + 154*e^3 + 121*e, -15/4*e^7 - 48*e^5 - 154*e^3 - 121*e, 1/2*e^6 + 6*e^4 + 22*e^2 + 32, 1/2*e^6 + 6*e^4 + 22*e^2 + 32, 9/4*e^7 + 29*e^5 + 94*e^3 + 65*e, -9/4*e^7 - 29*e^5 - 94*e^3 - 65*e, 2*e^6 + 27*e^4 + 98*e^2 + 82, 2*e^6 + 27*e^4 + 98*e^2 + 82, 2*e^7 + 25*e^5 + 77*e^3 + 55*e, -2*e^7 - 25*e^5 - 77*e^3 - 55*e, -1/4*e^7 - 4*e^5 - 18*e^3 - 11*e, 1/4*e^7 + 4*e^5 + 18*e^3 + 11*e, -e^7 - 11*e^5 - 23*e^3 + 3*e, e^7 + 11*e^5 + 23*e^3 - 3*e, e^6 + 13*e^4 + 38*e^2 + 10, e^6 + 13*e^4 + 38*e^2 + 10, 3/2*e^6 + 16*e^4 + 34*e^2 + 8, 3/2*e^6 + 16*e^4 + 34*e^2 + 8, -2*e^6 - 28*e^4 - 100*e^2 - 54, -2*e^7 - 25*e^5 - 74*e^3 - 35*e, 2*e^7 + 25*e^5 + 74*e^3 + 35*e, 3*e^6 + 38*e^4 + 122*e^2 + 80, 3*e^6 + 38*e^4 + 122*e^2 + 80, 1/4*e^7 + 2*e^5 - 4*e^3 - 31*e, -1/4*e^7 - 2*e^5 + 4*e^3 + 31*e, -2*e^6 - 28*e^4 - 106*e^2 - 94, -2*e^6 - 28*e^4 - 106*e^2 - 94, 4*e^7 + 52*e^5 + 172*e^3 + 133*e, -4*e^7 - 52*e^5 - 172*e^3 - 133*e, -e^6 - 13*e^4 - 45*e^2 - 42, -e^6 - 13*e^4 - 45*e^2 - 42, -e^6 - 13*e^4 - 47*e^2 - 50, -e^6 - 13*e^4 - 47*e^2 - 50, -5*e^6 - 64*e^4 - 208*e^2 - 160, -5*e^6 - 64*e^4 - 208*e^2 - 160, -3*e^7 - 42*e^5 - 160*e^3 - 153*e, 3*e^7 + 42*e^5 + 160*e^3 + 153*e, -4*e^6 - 47*e^4 - 127*e^2 - 58, 3/4*e^7 + 11*e^5 + 46*e^3 + 63*e, -3/4*e^7 - 11*e^5 - 46*e^3 - 63*e, e^7 + 12*e^5 + 36*e^3 + 35*e, -e^7 - 12*e^5 - 36*e^3 - 35*e, 4*e^6 + 49*e^4 + 144*e^2 + 90, 4*e^6 + 49*e^4 + 144*e^2 + 90, 2*e^7 + 23*e^5 + 54*e^3 - e, -2*e^7 - 23*e^5 - 54*e^3 + e, -2*e^6 - 24*e^4 - 69*e^2 - 50, -2*e^6 - 24*e^4 - 69*e^2 - 50, -7*e^6 - 89*e^4 - 280*e^2 - 206, -7*e^6 - 89*e^4 - 280*e^2 - 206, 5*e^6 + 64*e^4 + 204*e^2 + 136, 5*e^6 + 64*e^4 + 204*e^2 + 136, e^7 + 15*e^5 + 65*e^3 + 71*e, -e^7 - 15*e^5 - 65*e^3 - 71*e, -7/2*e^6 - 42*e^4 - 120*e^2 - 88, -7/2*e^6 - 42*e^4 - 120*e^2 - 88, 4*e^7 + 50*e^5 + 156*e^3 + 123*e, -4*e^7 - 50*e^5 - 156*e^3 - 123*e, 15/2*e^6 + 96*e^4 + 302*e^2 + 216, 15/2*e^6 + 96*e^4 + 302*e^2 + 216, -15/4*e^7 - 45*e^5 - 124*e^3 - 65*e, 15/4*e^7 + 45*e^5 + 124*e^3 + 65*e, -9/4*e^7 - 28*e^5 - 84*e^3 - 37*e, 9/4*e^7 + 28*e^5 + 84*e^3 + 37*e, 2*e^7 + 25*e^5 + 74*e^3 + 45*e, -2*e^7 - 25*e^5 - 74*e^3 - 45*e, 3*e^7 + 40*e^5 + 141*e^3 + 129*e, -3*e^7 - 40*e^5 - 141*e^3 - 129*e, -3/4*e^7 - 11*e^5 - 44*e^3 - 45*e, 3/4*e^7 + 11*e^5 + 44*e^3 + 45*e, -3/2*e^6 - 16*e^4 - 32*e^2 + 8, -3/2*e^6 - 16*e^4 - 32*e^2 + 8, 3*e^6 + 38*e^4 + 123*e^2 + 86, 3*e^6 + 38*e^4 + 123*e^2 + 86, e^6 + 18*e^4 + 85*e^2 + 78, e^6 + 18*e^4 + 85*e^2 + 78, 8*e^7 + 105*e^5 + 353*e^3 + 279*e, -8*e^7 - 105*e^5 - 353*e^3 - 279*e, 6*e^7 + 76*e^5 + 235*e^3 + 159*e, -6*e^7 - 76*e^5 - 235*e^3 - 159*e, -5/4*e^7 - 16*e^5 - 54*e^3 - 61*e, 5/4*e^7 + 16*e^5 + 54*e^3 + 61*e, -e^7 - 11*e^5 - 26*e^3 - 17*e, e^7 + 11*e^5 + 26*e^3 + 17*e, 1/4*e^7 + 3*e^5 + 10*e^3 + 7*e, -1/4*e^7 - 3*e^5 - 10*e^3 - 7*e, 2*e^4 + 20*e^2 + 48, 2*e^4 + 20*e^2 + 48, 19/4*e^7 + 63*e^5 + 216*e^3 + 159*e, -19/4*e^7 - 63*e^5 - 216*e^3 - 159*e, 3*e^7 + 36*e^5 + 99*e^3 + 59*e, -3*e^7 - 36*e^5 - 99*e^3 - 59*e, -11/2*e^6 - 70*e^4 - 214*e^2 - 120, -11/2*e^6 - 70*e^4 - 214*e^2 - 120, e^6 + 16*e^4 + 82*e^2 + 98, e^6 + 16*e^4 + 82*e^2 + 98, -1/4*e^7 + 24*e^3 + 67*e, 1/4*e^7 - 24*e^3 - 67*e, -8*e^7 - 103*e^5 - 335*e^3 - 253*e, 8*e^7 + 103*e^5 + 335*e^3 + 253*e, 4*e^6 + 49*e^4 + 149*e^2 + 94, 4*e^6 + 49*e^4 + 149*e^2 + 94, 14*e^6 + 177*e^4 + 553*e^2 + 406, -7/2*e^6 - 42*e^4 - 116*e^2 - 56, -7/2*e^6 - 42*e^4 - 116*e^2 - 56, 3*e^4 + 36*e^2 + 50, 3*e^4 + 36*e^2 + 50, 9/4*e^7 + 29*e^5 + 96*e^3 + 93*e, -9/4*e^7 - 29*e^5 - 96*e^3 - 93*e];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();

// 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;