/*
  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 | {5, 5, w + 2}>;

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^5 - x^4 - 5*x^3 + 3*x^2 + 4*x - 1;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, e^3 - e^2 - 3*e + 1, -e^4 + 5*e^2 + e - 3, 1, e^4 - e^3 - 5*e^2 + 3*e + 2, -e^4 + 2*e^3 + 4*e^2 - 7*e - 3, 2*e^4 - 2*e^3 - 9*e^2 + 3*e + 5, e^4 - e^3 - 4*e^2 + e, -e^4 + 3*e^3 + 3*e^2 - 9*e - 3, -2*e^4 + 2*e^3 + 9*e^2 - 5*e - 7, e^3 + 3*e^2 - 5*e - 11, e^4 - 2*e^3 - 3*e^2 + 5*e - 4, 2*e^4 + e^3 - 13*e^2 - 3*e + 11, e^4 - 3*e^3 - 2*e^2 + 9*e, -e^4 - e^3 + 6*e^2 + e - 3, 3*e^4 - e^3 - 17*e^2 + 2*e + 10, -4*e^4 + e^3 + 20*e^2 + 5*e - 16, 2*e^4 - 4*e^3 - 10*e^2 + 7*e + 14, -e^4 + 2*e^3 + 3*e^2 - 6*e + 4, -4*e^4 + 2*e^3 + 23*e^2 + e - 18, -e^4 - 3*e^3 + 6*e^2 + 13*e - 9, 4*e^4 - 8*e^3 - 16*e^2 + 24*e + 13, 6*e^4 - 4*e^3 - 27*e^2 + 9*e + 10, e^4 - 5*e^3 - e^2 + 19*e - 1, -2*e^4 + e^3 + 13*e^2 + 2*e - 20, -2*e^4 + 3*e^3 + 13*e^2 - 9*e - 7, 4*e^4 - 2*e^3 - 16*e^2 + 3*e - 3, 3*e^3 - e^2 - 20*e + 1, 4*e^4 - 2*e^3 - 18*e^2 + 5*e - 6, -3*e^4 + 4*e^3 + 15*e^2 - 11*e - 14, -4*e^4 + 7*e^3 + 10*e^2 - 16*e + 7, -3*e^4 - 3*e^3 + 19*e^2 + 14*e - 18, -e^4 + e^3 + 7*e^2 - 13*e - 10, 2*e^4 - 4*e^3 - 9*e^2 + 13*e + 5, -6*e^4 + 4*e^3 + 28*e^2 - 14*e - 10, 2*e^4 - 2*e^3 - 9*e^2 + 8*e + 10, -7*e^4 + 7*e^3 + 35*e^2 - 14*e - 21, -3*e^4 + 2*e^3 + 12*e^2 + 3*e - 5, e^4 - 4*e^3 + 3*e^2 + 4*e - 23, 2*e^4 - 9*e^2 + 4*e - 4, 3*e^4 + 2*e^3 - 15*e^2 - 15*e + 10, -4*e^4 + 8*e^3 + 12*e^2 - 25*e, 2*e^4 - 8*e^2 + 5*e - 4, e^4 - 5*e^3 - 3*e^2 + 11*e + 6, -2*e^4 - 3*e^3 + 11*e^2 + 10*e - 8, 2*e^4 - 18*e^2 + 18, -2*e^4 + 5*e^3 + 12*e^2 - 18*e - 11, -4*e^4 + 2*e^3 + 19*e^2 - 4*e - 9, -4*e^4 + 7*e^3 + 16*e^2 - 16*e - 10, 2*e^4 - 4*e^3 - 4*e^2 + 16*e - 11, 6*e^4 - 6*e^3 - 23*e^2 + 17*e + 2, 5*e^4 - 4*e^3 - 23*e^2 + 17*e + 14, 3*e^4 + 3*e^3 - 20*e^2 - 21*e + 16, 7*e^4 - 4*e^3 - 29*e^2 + 6*e + 11, -2*e^4 + 2*e^3 + 3*e^2 + e + 15, e^4 + e^3 - 2*e^2 - 2*e - 17, -2*e^4 + 8*e^3 + 3*e^2 - 24*e, 7*e^4 - 9*e^3 - 30*e^2 + 21*e + 9, 3*e^4 - 3*e^3 - 5*e^2 + 3*e - 21, e^4 + e^3 - 6*e^2 - e + 13, -3*e^4 - 2*e^3 + 19*e^2 + 13*e - 18, -4*e^4 + 5*e^3 + 13*e^2 - 10*e + 10, -3*e^4 - 3*e^3 + 18*e^2 + 21*e - 20, e^4 + 3*e^3 - 7*e^2 - 26*e + 7, -e^4 + 2*e^3 + 6*e^2 - 2*e - 4, 11*e^4 - 11*e^3 - 50*e^2 + 18*e + 27, 2*e^4 - 8*e^3 - 3*e^2 + 29*e, -6*e^4 + 3*e^3 + 20*e^2 - e + 1, -8*e^4 + 6*e^3 + 41*e^2 - 32, -4*e^4 - 2*e^3 + 18*e^2 + 14*e - 15, -2*e^4 - 7*e^3 + 10*e^2 + 23*e + 9, -3*e^4 - 3*e^3 + 13*e^2 + 16*e - 9, -7*e^4 + 5*e^3 + 27*e^2 - 2*e - 16, -7*e^4 + 14*e^3 + 21*e^2 - 39*e - 3, -5*e^4 + 10*e^3 + 13*e^2 - 36*e + 5, -4*e^4 + 3*e^3 + 16*e^2 - 9*e + 5, -7*e^4 + 8*e^3 + 30*e^2 - 27*e - 22, -8*e^4 + 5*e^3 + 31*e^2 - 7*e, -5*e^4 + 5*e^3 + 19*e^2 - 15*e - 4, 4*e^4 - 2*e^3 - 22*e^2 + 2*e - 7, 8*e^4 - 3*e^3 - 43*e^2 + 13*e + 21, 5*e^4 + e^3 - 30*e^2 - 11*e + 18, 15*e^4 - 11*e^3 - 69*e^2 + 16*e + 28, 9*e^4 - 6*e^3 - 40*e^2 + 16*e + 24, -e^4 + 8*e^3 - 10*e^2 - 25*e + 23, -9*e^4 + 3*e^3 + 49*e^2 - 2*e - 31, -6*e^4 - 3*e^3 + 26*e^2 + 20*e - 3, 4*e^4 + e^3 - 31*e^2 - e + 40, -9*e^4 + 14*e^3 + 28*e^2 - 38*e - 3, -5*e^4 + 7*e^3 + 27*e^2 - 19*e - 6, 2*e^4 + 5*e^3 - 6*e^2 - 22*e - 18, -6*e^4 + 8*e^3 + 30*e^2 - 23*e - 18, 4*e^4 + e^3 - 26*e^2 - 6*e + 35, 2*e^4 - 4*e^3 - e^2 + 5*e - 15, 4*e^4 - 8*e^3 - 14*e^2 + 10*e + 8, 11*e^4 - 3*e^3 - 50*e^2 + 14, 3*e^4 - 11*e^2 - 12*e + 5, -e^4 + 4*e^3 - 7*e^2 - 6*e + 19, -3*e^4 + 15*e^2 + 3*e - 18, e^4 - e^3 - 3*e^2 + 10*e - 18, -16*e^4 + 13*e^3 + 71*e^2 - 39*e - 34, e^4 + 8*e^3 - 12*e^2 - 32*e + 22, -10*e^4 + 3*e^3 + 56*e^2 + 3*e - 34, 4*e^4 + e^3 - 27*e^2 - 9*e + 26, 7*e^4 - 16*e^3 - 27*e^2 + 49*e + 7, -10*e^4 + 18*e^3 + 47*e^2 - 53*e - 24, 4*e^4 - 8*e^3 - 7*e^2 + 19*e - 15, -13*e^4 + 10*e^3 + 48*e^2 - 21*e - 6, 4*e^4 + 4*e^3 - 19*e^2 - 28*e + 16, 8*e^4 - 5*e^3 - 38*e^2 + 19*e + 30, -5*e^4 + 11*e^3 + 27*e^2 - 45*e - 9, -12*e^4 + 15*e^3 + 56*e^2 - 28*e - 39, -8*e^4 + 3*e^3 + 39*e^2 + 13*e - 10, -4*e^3 + 9*e^2 + 10*e - 21, 2*e^4 - 6*e^3 + 5*e^2 + 25*e - 30, -3*e^4 + 6*e^3 + 7*e^2 - 2*e + 12, 12*e^4 - 17*e^3 - 42*e^2 + 50*e + 12, 16*e^4 - 11*e^3 - 83*e^2 + 29*e + 36, 2*e^4 - 7*e - 27, -6*e^4 + 14*e^3 + 17*e^2 - 41*e - 10, 7*e^4 - 7*e^3 - 33*e^2 + 12*e - 13, 11*e^4 - 9*e^3 - 50*e^2 + 24*e + 16, 5*e^4 - e^3 - 32*e^2 - 2*e + 50, -3*e^4 + 3*e^3 - 3*e^2 + 2*e + 36, -9*e^4 + 5*e^3 + 33*e^2 - 10, 3*e^4 - 2*e^3 - 25*e^2 + 3*e + 35, 4*e^4 - 6*e^3 - 20*e^2 + 31*e + 20, -4*e^4 + 3*e^3 + 12*e^2 - 9*e + 15, e^4 + e^3 - 8*e^2 - 5*e - 34, 13*e^4 - 9*e^3 - 57*e^2 + 13*e + 41, 15*e^4 - 14*e^3 - 70*e^2 + 43*e + 33, -e^4 - 5*e^3 - e^2 + 28*e + 16, 15*e^4 - 69*e^2 - 25*e + 50, 10*e^4 - 13*e^3 - 48*e^2 + 36*e + 44, -4*e^4 + 8*e^3 + 7*e^2 - 27*e + 18, e^4 + e^2 - 6*e + 10, e^4 - 7*e^3 - 11*e^2 + 30*e + 32, e^4 + 2*e^3 - 3*e^2 - 8*e - 12, 3*e^4 + 7*e^3 - 30*e^2 - 28*e + 21, 8*e^4 - 19*e^3 - 35*e^2 + 56*e + 17, 9*e^4 - e^3 - 37*e^2 - 12*e + 14, -7*e^4 - e^3 + 39*e^2 + 22*e - 27, -14*e^4 + 6*e^3 + 75*e^2 + e - 35, -7*e^4 + 11*e^3 + 25*e^2 - 39*e - 3, -5*e^4 + 4*e^3 + 13*e^2 - 13*e + 18, 4*e^4 + e^3 - 21*e^2 - 17*e + 26, -7*e^4 - 2*e^3 + 32*e^2 + 19*e - 11, 3*e^4 + 3*e^3 - 15*e^2 - 19*e + 25, 4*e^4 + e^3 - 2*e^2 - 26*e - 28, e^4 - 9*e^2 - 3*e + 42, -4*e^4 + 8*e^3 + 17*e^2 - 25*e - 38, 6*e^4 - 12*e^3 - 27*e^2 + 26*e + 26, -13*e^4 + 19*e^3 + 53*e^2 - 61*e - 29, -2*e^4 - 4*e^3 + 9*e^2 + 16*e - 17, 6*e^4 + 2*e^3 - 38*e^2 - 14*e + 19, -2*e^4 - 2*e^3 + 10*e^2 + 20*e - 18, 10*e^3 - 18*e^2 - 26*e + 24, -3*e^4 + 8*e^3 - 6*e^2 - 23*e + 31, 6*e^4 - 11*e^3 - 19*e^2 + 35*e + 12, -2*e^4 - 3*e^3 + 21*e^2 + 4*e - 31, 10*e^4 - 8*e^3 - 48*e^2 + 27*e + 4, -17*e^4 + 20*e^3 + 68*e^2 - 57*e - 40, -4*e^4 + 12*e^3 + 14*e^2 - 32*e - 11, e^4 - 17*e^3 + 4*e^2 + 48*e - 7, -14*e^4 + 18*e^3 + 51*e^2 - 39*e - 3, 8*e^4 - 5*e^3 - 40*e^2 + 3*e + 19, 7*e^4 - e^3 - 32*e^2 - 25*e + 32];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {5, 5, w + 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;