/*
  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![2, -4, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {29, 29, 2*w + 1}>;

primesArray := [
[2, 2, -w],
[2, 2, w - 1],
[11, 11, w^2 - w - 1],
[17, 17, -w^2 - w + 3],
[19, 19, w^2 - w + 1],
[23, 23, 2*w - 3],
[27, 3, 3],
[29, 29, 2*w + 1],
[31, 31, 2*w^2 - 2*w - 9],
[37, 37, 2*w^2 - 2*w - 5],
[41, 41, 2*w^2 - 9],
[43, 43, w^2 + w - 5],
[43, 43, -3*w^2 + w + 15],
[43, 43, -2*w^2 + 2*w + 11],
[53, 53, w^2 - w - 7],
[61, 61, 4*w^2 - 2*w - 15],
[67, 67, -5*w^2 + 3*w + 23],
[73, 73, 2*w^2 - 3],
[73, 73, -3*w^2 - w + 7],
[73, 73, -6*w^2 + 4*w + 25],
[79, 79, w^2 - 5*w + 1],
[79, 79, 3*w^2 - 3*w - 11],
[83, 83, 2*w^2 - 2*w - 3],
[109, 109, w^2 - 3*w - 3],
[113, 113, 3*w^2 - 3*w - 13],
[121, 11, 3*w^2 - w - 9],
[125, 5, -5],
[131, 131, -6*w^2 + 2*w + 25],
[137, 137, -w^2 + w - 3],
[149, 149, 2*w - 7],
[151, 151, w^2 - 3*w - 5],
[157, 157, 3*w^2 - 3*w - 7],
[163, 163, 2*w^2 - 13],
[167, 167, -4*w + 5],
[173, 173, -7*w^2 + 3*w + 27],
[179, 179, 4*w^2 - 4*w - 11],
[181, 181, 4*w^2 - 15],
[181, 181, w^2 - 3*w + 5],
[181, 181, w^2 + 3*w - 5],
[193, 193, 2*w^2 - 4*w - 5],
[197, 197, 2*w^2 + 2*w - 7],
[211, 211, 4*w^2 - 2*w - 13],
[211, 211, -7*w^2 + 5*w + 29],
[211, 211, w^2 - w - 9],
[223, 223, 5*w^2 - w - 23],
[227, 227, -3*w^2 + 3*w + 17],
[227, 227, 3*w^2 - w - 7],
[227, 227, 3*w^2 + w - 11],
[229, 229, 5*w^2 - w - 21],
[233, 233, 3*w^2 - 5*w - 7],
[239, 239, -w^2 + 7*w - 5],
[257, 257, -4*w - 3],
[257, 257, w^2 - 5*w - 1],
[257, 257, -3*w^2 + w + 17],
[263, 263, 3*w^2 - 3*w - 5],
[271, 271, 4*w^2 - 4*w - 17],
[271, 271, 3*w^2 - w - 3],
[271, 271, w^2 + 3*w - 7],
[283, 283, 6*w - 1],
[289, 17, w^2 + 3*w - 9],
[293, 293, 3*w^2 - w - 5],
[307, 307, 2*w^2 + 2*w - 9],
[307, 307, -2*w^2 + 2*w - 3],
[307, 307, 4*w^2 - 17],
[311, 311, w^2 + w - 11],
[311, 311, 2*w^2 - 4*w - 7],
[311, 311, -2*w^2 - 2*w + 13],
[313, 313, -2*w - 7],
[313, 313, 4*w^2 - 19],
[313, 313, -5*w^2 - w + 13],
[331, 331, 2*w^2 - 4*w - 11],
[331, 331, 6*w^2 - 4*w - 21],
[331, 331, -4*w^2 + 2*w + 21],
[343, 7, -7],
[347, 347, -2*w^2 + 8*w - 1],
[349, 349, w^2 - 7*w + 1],
[353, 353, -6*w^2 + 2*w + 21],
[361, 19, -5*w^2 + 3*w + 25],
[367, 367, 3*w^2 + w - 15],
[373, 373, 2*w^2 - 4*w - 9],
[383, 383, -9*w^2 + 3*w + 37],
[401, 401, 4*w^2 - 2*w - 11],
[409, 409, 4*w^2 - 4*w - 9],
[431, 431, -5*w^2 + 5*w + 19],
[439, 439, 2*w - 9],
[443, 443, w^2 - 5*w - 3],
[443, 443, 5*w^2 - 5*w - 23],
[443, 443, 3*w^2 - 5*w - 17],
[449, 449, -4*w^2 + 4*w + 1],
[461, 461, 5*w^2 - 7*w - 9],
[467, 467, 5*w^2 + w - 17],
[479, 479, 2*w^2 + 4*w - 7],
[487, 487, 5*w^2 - 5*w - 13],
[499, 499, -w^2 + 3*w - 7],
[509, 509, -7*w^2 + w + 27],
[521, 521, w^2 - w - 11],
[523, 523, 4*w^2 - 6*w - 11],
[529, 23, 4*w^2 + 2*w - 13],
[547, 547, -11*w^2 + 5*w + 43],
[563, 563, 4*w^2 - 7],
[569, 569, -6*w - 1],
[569, 569, 4*w^2 - 5],
[569, 569, w^2 - 5*w - 9],
[571, 571, 5*w^2 - 3*w - 15],
[577, 577, w^2 + 5*w - 7],
[599, 599, 4*w^2 - 4*w - 7],
[601, 601, -12*w^2 + 8*w + 51],
[613, 613, 4*w^2 + 1],
[643, 643, 6*w^2 - 2*w - 31],
[647, 647, 2*w^2 - 2*w - 15],
[647, 647, 4*w^2 - 4*w - 23],
[647, 647, w^2 - 5*w - 7],
[661, 661, -8*w^2 + 6*w + 39],
[673, 673, 3*w^2 - 5*w - 15],
[683, 683, -w^2 + 9*w - 7],
[701, 701, 4*w^2 - 4*w - 5],
[709, 709, 4*w^2 - 2*w - 7],
[719, 719, 3*w^2 - 5*w - 13],
[727, 727, w^2 - 7*w - 1],
[733, 733, 2*w^2 + 4*w - 9],
[733, 733, 3*w^2 + 3*w - 11],
[733, 733, w^2 + w - 13],
[743, 743, 2*w^2 - 6*w - 5],
[751, 751, 6*w^2 - 2*w - 19],
[761, 761, -12*w^2 + 6*w + 47],
[761, 761, 3*w^2 - 5*w - 21],
[761, 761, 2*w^2 + 6*w - 7],
[769, 769, -6*w^2 + 6*w + 23],
[773, 773, -2*w^2 + 2*w - 5],
[773, 773, -13*w^2 + 5*w + 57],
[773, 773, -11*w^2 + 7*w + 51],
[787, 787, 5*w^2 - w - 13],
[797, 797, -2*w^2 + 8*w - 11],
[811, 811, -5*w^2 + 5*w + 1],
[821, 821, -7*w^2 + 3*w + 35],
[821, 821, -4*w^2 + 2*w + 23],
[821, 821, 5*w^2 + w - 19],
[823, 823, w^2 + 5*w - 13],
[823, 823, -12*w^2 + 4*w + 49],
[823, 823, -7*w^2 + w + 29],
[829, 829, -w^2 + w - 7],
[839, 839, 4*w^2 + 2*w - 15],
[841, 29, 4*w^2 - 6*w - 13],
[853, 853, 7*w^2 - 7*w - 25],
[863, 863, 7*w^2 - 5*w - 35],
[877, 877, -6*w - 5],
[877, 877, w^2 + 5*w - 11],
[877, 877, 5*w^2 - 3*w - 13],
[881, 881, w^2 - 3*w - 13],
[887, 887, -9*w^2 + 5*w + 33],
[907, 907, w^2 - 9*w + 1],
[911, 911, -6*w^2 + 6*w + 25],
[919, 919, 5*w^2 - 3*w - 27],
[929, 929, 2*w - 11],
[937, 937, -7*w^2 + w + 31],
[941, 941, -6*w^2 + 4*w + 31],
[941, 941, -8*w^2 + 2*w + 27],
[941, 941, 3*w^2 + 5*w - 9],
[947, 947, -10*w + 7],
[947, 947, 2*w^2 + 8*w - 7],
[947, 947, 2*w^2 + 4*w - 11],
[961, 31, -14*w^2 + 8*w + 55],
[967, 967, -w^2 - w - 7],
[971, 971, -10*w^2 + 8*w + 39],
[977, 977, -10*w^2 + 6*w + 37],
[983, 983, 3*w^2 + 3*w - 13],
[983, 983, -4*w - 11],
[983, 983, -10*w^2 + 4*w + 37],
[997, 997, w^2 + 9*w - 7],
[997, 997, 5*w^2 - 7*w - 27],
[997, 997, 6*w^2 - 8*w - 11]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

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

heckeEigenvaluesArray := [e, e^2 + 2*e - 1, -3*e^2 - 4*e, e^2 + 5*e, -3*e - 5, 2*e^2 - e - 4, e^2 - e, -1, -e - 3, -e^2 - 3*e + 1, e^2 - e - 2, -e - 8, -2*e^2 - 2, -2*e^2 - 5*e - 5, 6*e + 3, 4*e^2 + 5*e - 8, 3*e^2 + 5*e - 3, -4*e^2 - 7*e - 4, -5*e^2 - 4*e + 8, 7*e^2 + 14*e - 1, 8*e^2 + 10*e - 8, 5*e^2 + 11*e - 9, 3*e^2 + 2*e, -8*e^2 - 12*e + 13, -e^2 + 2*e - 1, 6*e^2 + 7*e - 17, 3*e^2 + 11*e + 3, -11*e^2 - 11*e + 10, 3*e^2 - 4*e - 14, -9*e^2 - 14*e + 3, -e^2 + 3*e + 9, -3*e^2 - 10*e - 2, -5*e^2 - 2*e + 11, -9*e^2 - 20*e + 9, -3*e^2 + 2*e + 7, -6*e^2 - 12*e - 5, 3*e^2 + 9*e + 2, -3*e^2 - 11*e - 11, 5*e^2 + 10*e + 1, -7*e^2 - 24*e + 6, -6*e^2 - 16*e + 2, 7*e^2 + 27*e - 1, 13*e^2 + 15*e - 13, -e^2 + 6*e + 9, -5*e^2 + 3*e + 17, 18*e^2 + 36*e - 15, -6*e^2 - 10*e - 3, 9*e^2 + 10*e - 23, -15*e^2 - 25*e + 18, 17*e^2 + 21*e - 19, 3*e^2 - 17, -8*e^2 - 8*e + 7, 5*e^2 + 23*e - 2, 3*e^2 + 6*e + 7, 17*e^2 + 27*e - 13, 5*e^2 + 17*e - 8, -e^2 - 6*e - 17, 2*e^2 + 13*e + 4, -9*e^2 - 8*e + 2, -11*e + 1, 7*e^2 - 3*e - 35, -7*e^2 - 10*e + 14, e^2 + 13*e + 14, -7*e^2 + e + 30, 7*e^2 + 14*e - 16, 20*e^2 + 30*e - 15, -7*e^2 - 14*e + 13, -8*e^2 + 3*e + 23, -7*e^2 - 27*e + 8, 3*e^2 + 15*e + 16, -24*e^2 - 39*e + 15, -25*e^2 - 36*e + 23, -11*e^2 - 12*e + 4, -e^2 + 11*e + 6, 4*e^2 + 11*e + 15, -21*e^2 - 38*e + 15, 4*e^2 + 16*e + 1, -13*e^2 - 34*e + 12, 6*e - 11, -18*e^2 - 27*e + 5, -7*e^2 - 8*e, 8*e^2 + 15*e - 26, 5*e^2 - 5*e - 26, -19*e^2 - 35*e, 18*e^2 + 25*e - 8, 11*e^2 + 32*e + 7, 14*e^2 + 16*e - 14, -6*e^2 + 7*e + 24, 4*e^2 + 11*e + 3, -e^2 - 21*e - 13, 2*e^2 - 2*e - 8, 13*e^2 + 16*e - 10, 4*e^2 + 5*e + 3, -11*e^2 - 18*e - 2, 7*e^2 - 14*e - 31, 25*e^2 + 31*e - 19, 7*e^2 + 12*e - 30, -23*e^2 - 30*e + 9, e^2 - 17*e - 21, -30*e^2 - 44*e + 11, 3*e^2 + 7*e - 31, -e^2 + 3*e - 4, -15*e^2 - 30*e + 15, 7*e^2 + 22*e - 6, 22*e^2 + 30*e, e^2 + 20*e + 6, -e^2 - 26*e - 10, 20*e^2 + 39*e - 17, 24*e^2 + 48*e - 11, 7*e^2 + 17*e + 20, 3*e^2 - 10*e - 26, 11*e + 34, -12*e^2 - 37*e + 3, 18*e^2 + 12*e - 29, -8*e^2 - 15*e - 15, -18*e^2 - 20*e + 40, 10*e + 15, -4*e^2 - 4*e - 10, 12*e^2 + 35*e - 27, -26*e^2 - 31*e + 22, 22*e + 9, -19*e^2 - 44*e + 27, 14*e^2 + 13*e - 37, e^2 + 8*e + 21, 6*e^2 + 29*e + 17, 7*e^2 - 3*e - 16, -39*e^2 - 51*e + 33, -16*e^2 - 38*e - 7, 5*e^2 - 11*e - 40, 17*e^2 + 27*e - 11, -17*e^2 - 30*e + 30, 8*e^2 + 27*e - 5, -e^2 - 24*e - 30, -5*e^2 - 16*e + 25, -7*e^2 - 33*e - 20, e^2 - 22*e - 17, -8*e^2 + e + 3, 28*e^2 + 51*e - 34, 3*e + 2, 5*e^2 + 15*e + 4, 12*e^2 + 32*e - 2, -e^2 + 3*e + 24, -16*e^2 - 23*e + 34, -3*e^2 + 10*e + 41, -15*e^2 - 15*e + 18, e^2 + 11*e - 22, 18*e^2 + 18*e - 23, 7*e^2 + 14*e + 34, -2*e^2 - 25*e - 9, 11*e^2 - 31, -4*e^2 + e + 39, 31*e^2 + 43*e - 8, e^2 + 11*e - 34, 4*e^2 - 6*e - 26, 5*e^2 - 11*e - 31, 5*e^2 - 9*e - 14, -7*e^2 - 6*e + 38, 17*e^2 + 45*e - 7, 2*e^2 + 4*e - 7, -e^2 - 15*e + 20, -25*e^2 - 28*e + 33, 4*e^2 - 4*e - 56, 36*e^2 + 63*e - 23, -30*e^2 - 73*e + 17, -2*e^2 - 12*e + 20, 25*e^2 + 35*e - 43, -3*e^2 - 8*e + 22, -18*e^2 - 43*e - 12, 19*e^2 + 7*e - 23, -10*e^2 - 10*e + 9, -29*e^2 - 46*e + 3];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {29, 29, 2*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;