/*
  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![41, 0, -13, 0, 1];
F<w> := NumberField(g);
ZF := Integers(F);

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

primesArray := [
[4, 2, -w^3 + 2*w^2 + 6*w - 11],
[5, 5, -w - 2],
[5, 5, -w + 2],
[11, 11, -2*w^2 - w + 12],
[11, 11, 2*w^2 - w - 12],
[19, 19, -w^2 - w + 4],
[19, 19, w^2 - w - 4],
[29, 29, -w - 1],
[29, 29, w - 1],
[31, 31, -2*w^2 - w + 14],
[31, 31, w^3 - 3*w^2 - 6*w + 19],
[31, 31, -w^3 - 3*w^2 + 6*w + 19],
[31, 31, -2*w^2 + w + 14],
[41, 41, -w],
[71, 71, w^3 + 3*w^2 - 7*w - 19],
[71, 71, w^3 - 3*w^2 - 7*w + 19],
[79, 79, w^3 + 3*w^2 - 9*w - 25],
[79, 79, w^3 - 3*w^2 - 9*w + 25],
[81, 3, -3],
[89, 89, -w^3 - 4*w^2 + 6*w + 24],
[89, 89, w^3 - 4*w^2 - 6*w + 24],
[101, 101, w^3 - 4*w^2 - 7*w + 26],
[101, 101, -w^3 - 4*w^2 + 7*w + 26],
[109, 109, -4*w^2 + w + 27],
[109, 109, -4*w^2 - w + 27],
[121, 11, -3*w^2 + 19],
[149, 149, -4*w^2 - w + 25],
[149, 149, 4*w^2 - w - 25],
[151, 151, -2*w^2 - w + 17],
[151, 151, -2*w^2 + w + 17],
[179, 179, -w^3 + w^2 + 9*w - 13],
[179, 179, -w^3 - w^2 + 9*w + 13],
[181, 181, 4*w^2 + w - 26],
[181, 181, 4*w^2 - w - 26],
[191, 191, -w^2 - 2*w + 11],
[191, 191, w^2 - 2*w - 11],
[199, 199, -w^3 - 2*w^2 + 5*w + 13],
[199, 199, w^3 - 2*w^2 - 5*w + 13],
[211, 211, -w^3 + 2*w^2 + 7*w - 11],
[211, 211, w^3 + 2*w^2 - 7*w - 11],
[229, 229, 2*w^3 + 4*w^2 - 11*w - 18],
[229, 229, -2*w^3 - 5*w^2 + 10*w + 23],
[239, 239, -w^3 - 5*w^2 + 4*w + 27],
[239, 239, -w^3 + 5*w^2 + 4*w - 27],
[241, 241, w^3 - 7*w - 5],
[241, 241, w^2 - 2*w - 9],
[241, 241, w^2 + 2*w - 9],
[241, 241, -w^3 + 7*w - 5],
[251, 251, w^2 + 2*w - 12],
[251, 251, -w^3 - w^2 + 6*w + 11],
[251, 251, w^3 - w^2 - 6*w + 11],
[251, 251, w^2 - 2*w - 12],
[269, 269, -3*w^3 - 10*w^2 + 14*w + 52],
[269, 269, 2*w - 3],
[269, 269, 2*w + 3],
[269, 269, -2*w^2 - 2*w + 11],
[281, 281, -4*w^3 - 12*w^2 + 20*w + 63],
[281, 281, -w^3 - 4*w^2 + 4*w + 22],
[311, 311, -4*w^2 + w + 20],
[311, 311, -w^3 - 4*w^2 + 7*w + 24],
[331, 331, w^3 - 2*w^2 - 8*w + 12],
[331, 331, -w^3 - 2*w^2 + 8*w + 12],
[361, 19, 4*w^2 - 27],
[379, 379, 3*w^2 + 2*w - 22],
[379, 379, -3*w^3 - 5*w^2 + 17*w + 23],
[379, 379, 4*w^3 + 11*w^2 - 23*w - 64],
[379, 379, 3*w^2 - 2*w - 22],
[409, 409, w^3 - 5*w^2 - 6*w + 32],
[409, 409, -5*w^2 + w + 34],
[409, 409, 5*w^2 + w - 34],
[409, 409, -w^3 - 5*w^2 + 6*w + 32],
[419, 419, -5*w^2 + 2*w + 30],
[419, 419, w^3 - w^2 - 8*w + 4],
[419, 419, w^3 + w^2 - 8*w - 4],
[419, 419, -5*w^2 - 2*w + 30],
[421, 421, 2*w^2 + 2*w - 13],
[421, 421, 2*w^2 - 2*w - 13],
[431, 431, 3*w^3 + 7*w^2 - 19*w - 45],
[431, 431, 3*w^3 + 5*w^2 - 19*w - 27],
[431, 431, -3*w^2 - 3*w + 14],
[431, 431, w^3 - 3*w^2 - 10*w + 28],
[439, 439, 2*w^3 - 4*w^2 - 13*w + 23],
[439, 439, -2*w^3 - 4*w^2 + 13*w + 23],
[479, 479, -2*w^3 - 4*w^2 + 13*w + 28],
[479, 479, 2*w^3 - 4*w^2 - 13*w + 28],
[491, 491, 4*w^2 - 2*w - 23],
[491, 491, w^3 - 2*w^2 - 6*w + 16],
[491, 491, -w^3 - 2*w^2 + 6*w + 16],
[491, 491, 4*w^2 + 2*w - 23],
[499, 499, -w^3 - w^2 + 4*w + 1],
[499, 499, -w^3 + w^2 + 4*w - 1],
[509, 509, -2*w^3 + 7*w^2 + 14*w - 49],
[509, 509, 2*w^3 + 7*w^2 - 14*w - 49],
[521, 521, w^3 + 3*w^2 - 9*w - 21],
[521, 521, -w^3 + 3*w^2 + 9*w - 21],
[529, 23, -w^3 + 5*w^2 + 7*w - 33],
[529, 23, w^3 + 5*w^2 - 7*w - 33],
[541, 541, w^3 + w^2 - 9*w - 5],
[541, 541, 2*w^3 + 7*w^2 - 12*w - 43],
[541, 541, -7*w^3 - 18*w^2 + 38*w + 94],
[541, 541, -w^3 + w^2 + 9*w - 5],
[569, 569, -3*w^3 + 9*w^2 + 20*w - 58],
[569, 569, w^2 + 2*w - 7],
[569, 569, w^2 - 2*w - 7],
[569, 569, -3*w^3 - 9*w^2 + 20*w + 58],
[571, 571, -w^3 + 5*w - 6],
[571, 571, w^3 - 5*w - 6],
[601, 601, -w^3 + 4*w - 2],
[601, 601, w^3 - 4*w - 2],
[619, 619, -w^3 + 4*w^2 + 5*w - 26],
[619, 619, w^3 + 3*w^2 - 6*w - 22],
[619, 619, -w^3 + 3*w^2 + 6*w - 22],
[619, 619, w^3 + 4*w^2 - 5*w - 26],
[631, 631, w^3 + 4*w^2 - 7*w - 23],
[631, 631, -2*w^3 + 3*w^2 + 14*w - 18],
[631, 631, 2*w^3 + 3*w^2 - 14*w - 18],
[631, 631, -w^3 + 4*w^2 + 7*w - 23],
[641, 641, 2*w^3 + 6*w^2 - 8*w - 27],
[641, 641, 2*w^3 + 7*w^2 - 13*w - 44],
[659, 659, w^3 + 5*w^2 - 6*w - 35],
[659, 659, -w^3 + 5*w^2 + 6*w - 35],
[661, 661, 2*w^3 - 7*w^2 - 13*w + 46],
[661, 661, -3*w^3 + 4*w^2 + 21*w - 26],
[661, 661, 3*w^3 + 4*w^2 - 21*w - 26],
[661, 661, -2*w^3 - 7*w^2 + 13*w + 46],
[691, 691, 2*w^3 + 6*w^2 - 17*w - 48],
[691, 691, 2*w^3 - 6*w^2 - 17*w + 48],
[701, 701, -2*w^3 - 7*w^2 + 8*w + 34],
[701, 701, w^3 - 9*w + 2],
[701, 701, -w^3 + 9*w + 2],
[701, 701, -w^3 - 6*w^2 + 3*w + 34],
[709, 709, 5*w^2 + 3*w - 32],
[709, 709, -5*w^2 + 3*w + 32],
[719, 719, -2*w^2 - 4*w + 3],
[719, 719, 2*w^2 - 4*w - 3],
[739, 739, -w^3 + 9*w + 7],
[739, 739, -w^3 + 9*w + 6],
[739, 739, w^3 - 9*w + 6],
[739, 739, -w^3 + 9*w - 7],
[751, 751, w^3 + 3*w^2 - 8*w - 18],
[751, 751, -w^3 + 3*w^2 + 8*w - 18],
[761, 761, -2*w^3 + 5*w^2 + 17*w - 40],
[761, 761, 2*w^3 + w^2 - 13*w - 8],
[761, 761, -2*w^3 + w^2 + 13*w - 8],
[761, 761, -2*w^3 - 5*w^2 + 17*w + 40],
[809, 809, 2*w^3 + 5*w^2 - 9*w - 24],
[809, 809, 2*w^3 - 5*w^2 - 9*w + 24],
[811, 811, 2*w^3 - w^2 - 16*w + 11],
[811, 811, -2*w^3 - 3*w^2 + 10*w + 9],
[811, 811, -2*w^3 - 8*w^2 + 11*w + 50],
[811, 811, 2*w^3 + w^2 - 16*w - 11],
[839, 839, -3*w^3 - 9*w^2 + 18*w + 56],
[839, 839, -3*w^3 - 4*w^2 + 17*w + 15],
[841, 29, -5*w^2 + 31],
[881, 881, w^3 + 7*w^2 - 6*w - 48],
[881, 881, -5*w^3 - 12*w^2 + 30*w + 70],
[881, 881, -4*w^3 - 6*w^2 + 24*w + 29],
[881, 881, -w^3 + 7*w^2 + 6*w - 48],
[919, 919, 2*w^3 - 12*w - 5],
[919, 919, 2*w^3 - 12*w + 5],
[929, 929, w^2 + 4*w + 2],
[929, 929, -w^2 + 4*w - 2],
[941, 941, 2*w^3 + w^2 - 13*w - 6],
[941, 941, -w^3 - 2*w^2 + 4*w + 4],
[941, 941, 4*w^3 + 8*w^2 - 22*w - 37],
[941, 941, -2*w^3 + w^2 + 13*w - 6],
[971, 971, 4*w^2 - 2*w - 29],
[971, 971, 4*w^2 + 2*w - 29],
[991, 991, 2*w^3 - w^2 - 15*w + 12],
[991, 991, -w^3 + 4*w^2 + 7*w - 19]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^4 - 52*x^2 + 578;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [-1/7*e^2 + 33/7, -1/7*e^2 + 40/7, -1, -1/119*e^3 - 16/119*e, e, -13/119*e^3 + 387/119*e, 10/119*e^3 - 197/119*e, -4/7*e^2 + 104/7, 3/7*e^2 - 36/7, 9/119*e^3 - 213/119*e, 8/119*e^3 - 348/119*e, -2/17*e^3 + 53/17*e, 8/119*e^3 - 348/119*e, 4, -18/119*e^3 + 545/119*e, 4/119*e^3 - 174/119*e, 2/17*e^3 - 70/17*e, 2/17*e^3 - 70/17*e, 8/7*e^2 - 250/7, -2/7*e^2 + 94/7, 6/7*e^2 - 128/7, 2/7*e^2 - 52/7, 9/7*e^2 - 192/7, -2/7*e^2 + 66/7, 6/7*e^2 - 156/7, -2/7*e^2 - 18/7, -12/7*e^2 + 312/7, 11/7*e^2 - 272/7, 6/17*e^3 - 176/17*e, -24/119*e^3 + 925/119*e, 29/119*e^3 - 964/119*e, 31/119*e^3 - 694/119*e, -6/7*e^2 + 128/7, 2/7*e^2 - 94/7, -40/119*e^3 + 907/119*e, 4/17*e^3 - 140/17*e, 1/7*e^3 - 19/7*e, -6/119*e^3 + 261/119*e, -5/119*e^3 + 396/119*e, 1/7*e^3 - 19/7*e, -5/7*e^2 + 116/7, 8/7*e^2 - 82/7, -11/119*e^3 + 181/119*e, 33/119*e^3 - 1257/119*e, -2/7*e^2 + 38/7, 4/7*e^2 - 20/7, 4/7*e^2 - 20/7, 12/7*e^2 - 242/7, -47/119*e^3 + 1390/119*e, -10/119*e^3 + 435/119*e, 31/119*e^3 - 1408/119*e, 44/119*e^3 - 1081/119*e, 2/7*e^2 - 94/7, 2/7*e^2 + 60/7, -3/7*e^2 + 36/7, -1/7*e^2 + 152/7, 6/7*e^2 - 268/7, 2/7*e^2 + 60/7, 38/119*e^3 - 1296/119*e, 2/17*e^3 - 121/17*e, -3/17*e^3 + 20/17*e, -39/119*e^3 + 1399/119*e, 9/7*e^2 - 206/7, -32/119*e^3 + 1511/119*e, -1/7*e^3 + 68/7*e, -67/119*e^3 + 1784/119*e, 54/119*e^3 - 1635/119*e, -8/7*e^2 + 166/7, -2/7*e^2 - 18/7, 20/7*e^2 - 520/7, -8, -1/7*e^3 + 19/7*e, 1/17*e^3 - 1/17*e, 5/119*e^3 - 277/119*e, -15/119*e^3 + 593/119*e, 2/7*e^2 + 172/7, -3/7*e^2 + 148/7, -29/119*e^3 + 369/119*e, -62/119*e^3 + 1745/119*e, -38/119*e^3 + 1296/119*e, 64/119*e^3 - 1832/119*e, 61/119*e^3 - 1761/119*e, 36/119*e^3 - 1447/119*e, 8/119*e^3 - 824/119*e, -80/119*e^3 + 2052/119*e, 11/119*e^3 - 181/119*e, -78/119*e^3 + 2441/119*e, -79/119*e^3 + 2306/119*e, -1/17*e^3 + 154/17*e, 58/119*e^3 - 1571/119*e, -13/119*e^3 - 89/119*e, 3/7*e^2 - 176/7, -2/7*e^2 + 234/7, 12/7*e^2 - 340/7, -6/7*e^2 + 268/7, 6/7*e^2 - 240/7, 19/7*e^2 - 438/7, 8/7*e^2 - 306/7, 2, -22/7*e^2 + 516/7, 3/7*e^2 + 104/7, 2/7*e^2 - 178/7, 13/7*e^2 - 338/7, 6/7*e^2 - 198/7, -9/7*e^2 + 290/7, -10/119*e^3 + 1149/119*e, -61/119*e^3 + 1642/119*e, -3*e^2 + 74, 11/7*e^2 - 370/7, -57/119*e^3 + 1230/119*e, 1/7*e^3 - 75/7*e, -3/17*e^3 + 139/17*e, -13/119*e^3 - 208/119*e, 20/119*e^3 - 870/119*e, 46/119*e^3 - 930/119*e, -66/119*e^3 + 2395/119*e, -4/119*e^3 - 421/119*e, 26/7*e^2 - 718/7, -6/7*e^2 + 170/7, 52/119*e^3 - 1191/119*e, 27/119*e^3 - 877/119*e, -4/7*e^2 + 202/7, 2/7*e^2 - 192/7, 2/7*e^2 - 192/7, 9/7*e^2 + 4/7, -9/119*e^3 + 213/119*e, -33/119*e^3 + 662/119*e, -6/7*e^2 + 380/7, 23/7*e^2 - 612/7, 9/7*e^2 - 332/7, -2*e^2 + 24, -23/7*e^2 + 612/7, 10/7*e^2 - 358/7, -40/119*e^3 + 1740/119*e, -86/119*e^3 + 2908/119*e, -3/7*e^3 + 57/7*e, -27/119*e^3 + 1829/119*e, -8/17*e^3 + 229/17*e, 43/119*e^3 - 1097/119*e, 48/119*e^3 - 1017/119*e, -22/119*e^3 + 600/119*e, 6/7*e^2 + 68/7, -2*e^2 + 48, 4/7*e^2 - 272/7, -3/7*e^2 - 62/7, -2*e^2 + 64, -1/7*e^2 + 250/7, 30/119*e^3 - 829/119*e, 33/119*e^3 - 1852/119*e, 36/119*e^3 - 1447/119*e, -19/119*e^3 - 66/119*e, -2/17*e^3 + 155/17*e, -66/119*e^3 + 1443/119*e, -6/7*e^2 + 44/7, 15/7*e^2 - 278/7, -22/7*e^2 + 726/7, 12/7*e^2 - 326/7, -10/7*e^2 + 36/7, 44/119*e^3 - 724/119*e, -50/119*e^3 + 1342/119*e, -4*e^2 + 110, -8/7*e^2 + 432/7, 29/7*e^2 - 852/7, -18/7*e^2 + 496/7, -12/7*e^2 + 438/7, -26/7*e^2 + 620/7, 25/119*e^3 - 314/119*e, -45/119*e^3 + 1303/119*e, -19/119*e^3 + 1243/119*e, -26/119*e^3 + 298/119*e];
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;