/*
  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![-80, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

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

primesArray := [
[2, 2, w],
[2, 2, w + 1],
[3, 3, -2*w + 19],
[5, 5, w],
[5, 5, w + 4],
[13, 13, w + 1],
[13, 13, w + 11],
[17, 17, w + 3],
[17, 17, w + 13],
[19, 19, w + 6],
[19, 19, w + 12],
[37, 37, w + 2],
[37, 37, w + 34],
[49, 7, -7],
[59, 59, -4*w - 33],
[59, 59, 4*w - 37],
[61, 61, w + 28],
[61, 61, w + 32],
[71, 71, w + 22],
[71, 71, w + 48],
[79, 79, 2*w - 21],
[79, 79, 2*w + 19],
[107, 107, 12*w - 113],
[113, 113, w + 17],
[113, 113, w + 95],
[121, 11, -11],
[131, 131, w + 18],
[131, 131, w + 112],
[151, 151, w + 43],
[151, 151, w + 107],
[163, 163, 2*w - 23],
[163, 163, -2*w - 21],
[167, 167, w + 66],
[167, 167, w + 100],
[173, 173, w + 59],
[173, 173, w + 113],
[179, 179, w + 60],
[179, 179, w + 118],
[191, 191, w + 21],
[191, 191, w + 169],
[193, 193, 32*w - 303],
[193, 193, 8*w - 77],
[199, 199, w + 45],
[199, 199, w + 153],
[223, 223, w + 52],
[223, 223, w + 170],
[241, 241, w + 85],
[241, 241, w + 155],
[257, 257, 2*w - 9],
[257, 257, -2*w - 7],
[269, 269, w + 101],
[269, 269, w + 167],
[271, 271, w + 104],
[271, 271, w + 166],
[281, 281, -10*w + 93],
[281, 281, 38*w - 359],
[283, 283, w + 98],
[283, 283, w + 184],
[311, 311, w + 26],
[311, 311, w + 284],
[313, 313, w + 36],
[313, 313, w + 276],
[317, 317, 2*w - 3],
[317, 317, -2*w - 1],
[331, 331, w + 131],
[331, 331, w + 199],
[337, 337, w + 127],
[337, 337, w + 209],
[347, 347, w + 46],
[347, 347, w + 300],
[353, 353, 34*w - 321],
[353, 353, -14*w + 131],
[359, 359, w + 156],
[359, 359, w + 202],
[373, 373, w + 67],
[373, 373, w + 305],
[389, 389, 6*w - 53],
[389, 389, 6*w + 47],
[397, 397, 4*w + 39],
[397, 397, -4*w + 43],
[401, 401, -18*w + 169],
[401, 401, 30*w - 283],
[419, 419, w + 126],
[419, 419, w + 292],
[421, 421, w + 65],
[421, 421, w + 355],
[439, 439, 10*w - 97],
[439, 439, -10*w - 87],
[443, 443, 4*w - 31],
[443, 443, -4*w - 27],
[449, 449, w + 182],
[449, 449, w + 266],
[457, 457, w + 135],
[457, 457, w + 321],
[463, 463, 2*w - 29],
[463, 463, -2*w - 27],
[479, 479, w + 193],
[479, 479, w + 285],
[491, 491, w + 99],
[491, 491, w + 391],
[521, 521, w + 33],
[521, 521, w + 487],
[529, 23, -23],
[547, 547, w + 74],
[547, 547, w + 472],
[557, 557, w + 58],
[557, 557, w + 498],
[563, 563, w + 253],
[563, 563, w + 309],
[571, 571, w + 241],
[571, 571, w + 329],
[577, 577, w + 96],
[577, 577, w + 480],
[593, 593, w + 248],
[593, 593, w + 344],
[617, 617, w + 61],
[617, 617, w + 555],
[641, 641, w + 143],
[641, 641, w + 497],
[643, 643, w + 51],
[643, 643, w + 591],
[647, 647, 8*w - 71],
[647, 647, 8*w + 63],
[659, 659, 4*w - 27],
[659, 659, -4*w - 23],
[661, 661, 44*w - 417],
[661, 661, 20*w - 191],
[691, 691, w + 91],
[691, 691, w + 599],
[701, 701, w + 38],
[701, 701, w + 662],
[719, 719, w + 303],
[719, 719, w + 415],
[727, 727, w + 295],
[727, 727, w + 431],
[773, 773, -6*w - 43],
[773, 773, 6*w - 49],
[809, 809, w + 220],
[809, 809, w + 588],
[811, 811, -10*w - 89],
[811, 811, 10*w - 99],
[821, 821, w + 41],
[821, 821, w + 779],
[827, 827, w + 177],
[827, 827, w + 649],
[841, 29, -29],
[859, 859, w + 318],
[859, 859, w + 540],
[863, 863, w + 42],
[863, 863, w + 820],
[883, 883, w + 173],
[883, 883, w + 709],
[887, 887, w + 73],
[887, 887, w + 813],
[911, 911, 8*w - 69],
[911, 911, 8*w + 61],
[929, 929, w + 251],
[929, 929, w + 677],
[937, 937, w + 372],
[937, 937, w + 564],
[947, 947, 12*w + 97],
[947, 947, 12*w - 109],
[953, 953, 6*w - 47],
[953, 953, -6*w - 41],
[961, 31, -31],
[967, 967, w + 388],
[967, 967, w + 578],
[971, 971, w + 176],
[971, 971, w + 794],
[983, 983, w + 140],
[983, 983, w + 842],
[997, 997, w + 399],
[997, 997, w + 597]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^4 + x^3 + 4*x^2 - 3*x + 9;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, -1/4*e^3 - e + 3/4, -2, 1/4*e^3 + e^2 + e - 3/4, -1/4*e^3 - e^2 - e - 9/4, 1/12*e^3 + 1/3*e^2 - 5/3*e + 3/4, 5/12*e^3 - 1/3*e^2 + 5/3*e - 5/4, 3/4*e^3 + e^2 + 3*e - 9/4, -1/4*e^3 - e^2 - 3*e - 9/4, -7/12*e^3 - 1/3*e^2 - 7/3*e + 7/4, 1/12*e^3 + 1/3*e^2 + 7/3*e + 3/4, -7/12*e^3 - 7/3*e^2 - 7/3*e + 7/4, 7/12*e^3 + 7/3*e^2 + 7/3*e + 21/4, -1/2*e^3 + 19/2, -1/2*e^3 + 15/2, -1/2*e^3 + 15/2, -5/12*e^3 - 5/3*e^2 + 1/3*e - 15/4, -1/12*e^3 + 5/3*e^2 - 1/3*e + 1/4, -3/4*e^3 + e^2 - 3*e + 9/4, -1/4*e^3 - e^2 + 3*e - 9/4, 1/2*e^3 - 11/2, 1/2*e^3 - 11/2, 18, -1/4*e^3 - e^2 - e - 9/4, 1/4*e^3 + e^2 + e - 3/4, -3/2*e^3 + 37/2, 1/4*e^3 + 3*e^2 + e - 3/4, -3/4*e^3 - 3*e^2 - e - 27/4, -19/12*e^3 - 19/3*e^2 - 19/3*e + 19/4, 19/12*e^3 + 19/3*e^2 + 19/3*e + 57/4, -3/2*e^3 + 13/2, -3/2*e^3 + 13/2, -1/4*e^3 - e^2 - 7*e - 9/4, 7/4*e^3 + e^2 + 7*e - 21/4, -5/4*e^3 - 5*e^2 - 5*e - 45/4, 5/4*e^3 + 5*e^2 + 5*e - 15/4, -3/4*e^3 - e^2 - 3*e + 9/4, 1/4*e^3 + e^2 + 3*e + 9/4, -3/4*e^3 - 3*e^2 - 17*e - 27/4, 17/4*e^3 + 3*e^2 + 17*e - 51/4, -e^3 + 5, -e^3 + 5, -25/12*e^3 - 13/3*e^2 - 25/3*e + 25/4, 13/12*e^3 + 13/3*e^2 + 25/3*e + 39/4, -5/12*e^3 - 5/3*e^2 + 19/3*e - 15/4, -19/12*e^3 + 5/3*e^2 - 19/3*e + 19/4, -1/12*e^3 - 1/3*e^2 - 1/3*e + 1/4, 1/12*e^3 + 1/3*e^2 + 1/3*e + 3/4, e^3 + 3, e^3 + 3, -15/4*e^3 - 3*e^2 - 15*e + 45/4, 3/4*e^3 + 3*e^2 + 15*e + 27/4, -43/12*e^3 - 19/3*e^2 - 43/3*e + 43/4, 19/12*e^3 + 19/3*e^2 + 43/3*e + 57/4, 3*e^3 - 21, 3*e^3 - 21, -5/12*e^3 - 5/3*e^2 - 41/3*e - 15/4, 41/12*e^3 + 5/3*e^2 + 41/3*e - 41/4, -11/4*e^3 - 3*e^2 - 11*e + 33/4, 3/4*e^3 + 3*e^2 + 11*e + 27/4, 1/12*e^3 + 1/3*e^2 - 17/3*e + 3/4, 17/12*e^3 - 1/3*e^2 + 17/3*e - 17/4, e^3 - 3, e^3 - 3, -23/12*e^3 - 23/3*e^2 - 17/3*e - 69/4, 17/12*e^3 + 23/3*e^2 + 17/3*e - 17/4, -5/12*e^3 - 5/3*e^2 - 23/3*e - 15/4, 23/12*e^3 + 5/3*e^2 + 23/3*e - 23/4, -11/4*e^3 - 5*e^2 - 11*e + 33/4, 5/4*e^3 + 5*e^2 + 11*e + 45/4, 6, 6, 9/4*e^3 + 5*e^2 + 9*e - 27/4, -5/4*e^3 - 5*e^2 - 9*e - 45/4, 5/12*e^3 - 7/3*e^2 + 5/3*e - 5/4, 7/12*e^3 + 7/3*e^2 - 5/3*e + 21/4, 1/2*e^3 + 21/2, 1/2*e^3 + 21/2, 3/2*e^3 - 41/2, 3/2*e^3 - 41/2, -3/2*e^3 - 3/2, -3/2*e^3 - 3/2, -1/4*e^3 - e^2 - 13*e - 9/4, 13/4*e^3 + e^2 + 13*e - 39/4, -19/12*e^3 - 7/3*e^2 - 19/3*e + 19/4, 7/12*e^3 + 7/3*e^2 + 19/3*e + 21/4, -2*e^3 + 8, -2*e^3 + 8, 3*e^3 - 3, 3*e^3 - 3, 1/4*e^3 + e^2 - 9*e + 9/4, 9/4*e^3 - e^2 + 9*e - 27/4, -17/12*e^3 - 17/3*e^2 - 17/3*e - 51/4, 17/12*e^3 + 17/3*e^2 + 17/3*e - 17/4, -e^3 - 13, -e^3 - 13, 11/4*e^3 + 7*e^2 + 11*e - 33/4, -7/4*e^3 - 7*e^2 - 11*e - 63/4, 5/4*e^3 + 5*e^2 + 11*e + 45/4, -11/4*e^3 - 5*e^2 - 11*e + 33/4, -3/4*e^3 - 3*e^2 - 15*e - 27/4, 15/4*e^3 + 3*e^2 + 15*e - 45/4, 6*e^3 - 28, -5/12*e^3 - 5/3*e^2 - 17/3*e - 15/4, 17/12*e^3 + 5/3*e^2 + 17/3*e - 17/4, 15/4*e^3 + 9*e^2 + 15*e - 45/4, -9/4*e^3 - 9*e^2 - 15*e - 81/4, 7/4*e^3 - 3*e^2 + 7*e - 21/4, 3/4*e^3 + 3*e^2 - 7*e + 27/4, 23/12*e^3 - 1/3*e^2 + 23/3*e - 23/4, 1/12*e^3 + 1/3*e^2 - 23/3*e + 3/4, 83/12*e^3 + 17/3*e^2 + 83/3*e - 83/4, -17/12*e^3 - 17/3*e^2 - 83/3*e - 51/4, 9/4*e^3 + 3*e^2 + 9*e - 27/4, -3/4*e^3 - 3*e^2 - 9*e - 27/4, 5/4*e^3 + 5*e^2 - e + 45/4, 1/4*e^3 - 5*e^2 + e - 3/4, -13/4*e^3 - e^2 - 13*e + 39/4, 1/4*e^3 + e^2 + 13*e + 9/4, 41/12*e^3 - 7/3*e^2 + 41/3*e - 41/4, 7/12*e^3 + 7/3*e^2 - 41/3*e + 21/4, 7/2*e^3 - 45/2, 7/2*e^3 - 45/2, -2*e^3 + 24, -2*e^3 + 24, 2*e^3 - 40, 2*e^3 - 40, -73/12*e^3 - 31/3*e^2 - 73/3*e + 73/4, 31/12*e^3 + 31/3*e^2 + 73/3*e + 93/4, 9/4*e^3 + 9*e^2 + 9*e - 27/4, -9/4*e^3 - 9*e^2 - 9*e - 81/4, 11/4*e^3 + 11*e^2 + 15*e + 99/4, -15/4*e^3 - 11*e^2 - 15*e + 45/4, -11/12*e^3 - 11/3*e^2 + 37/3*e - 33/4, -37/12*e^3 + 11/3*e^2 - 37/3*e + 37/4, -5/2*e^3 + 63/2, -5/2*e^3 + 63/2, 3/4*e^3 + 3*e^2 + 3*e - 9/4, -3/4*e^3 - 3*e^2 - 3*e - 27/4, -16, -16, 1/4*e^3 + e^2 + 7*e + 9/4, -7/4*e^3 - e^2 - 7*e + 21/4, -3/4*e^3 + e^2 - 3*e + 9/4, -1/4*e^3 - e^2 + 3*e - 9/4, 32, 41/12*e^3 + 35/3*e^2 + 41/3*e - 41/4, -35/12*e^3 - 35/3*e^2 - 41/3*e - 105/4, -3/4*e^3 - 9*e^2 - 3*e + 9/4, 9/4*e^3 + 9*e^2 + 3*e + 81/4, -7/12*e^3 - 7/3*e^2 - 7/3*e + 7/4, 7/12*e^3 + 7/3*e^2 + 7/3*e + 21/4, 1/4*e^3 + e^2 - e + 9/4, 1/4*e^3 - e^2 + e - 3/4, 3/2*e^3 - 69/2, 3/2*e^3 - 69/2, -5/4*e^3 - 5*e^2 - 17*e - 45/4, 17/4*e^3 + 5*e^2 + 17*e - 51/4, 29/12*e^3 + 47/3*e^2 + 29/3*e - 29/4, -47/12*e^3 - 47/3*e^2 - 29/3*e - 141/4, -2*e^3 + 6, -2*e^3 + 6, 5/2*e^3 - 63/2, 5/2*e^3 - 63/2, -10, -49/12*e^3 - 7/3*e^2 - 49/3*e + 49/4, 7/12*e^3 + 7/3*e^2 + 49/3*e + 21/4, 3/4*e^3 + 3*e^2 - 15*e + 27/4, 15/4*e^3 - 3*e^2 + 15*e - 45/4, -31/4*e^3 - 11*e^2 - 31*e + 93/4, 11/4*e^3 + 11*e^2 + 31*e + 99/4, 1/12*e^3 + 1/3*e^2 - 53/3*e + 3/4, 53/12*e^3 - 1/3*e^2 + 53/3*e - 53/4];
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;