/*
  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![-28, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

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

primesArray := [
[2, 2, -w + 6],
[2, 2, w + 5],
[7, 7, 6*w - 35],
[7, 7, -6*w - 29],
[9, 3, 3],
[11, 11, 4*w + 19],
[11, 11, 4*w - 23],
[13, 13, -2*w + 11],
[13, 13, 2*w + 9],
[25, 5, -5],
[31, 31, 2*w - 13],
[31, 31, -2*w - 11],
[41, 41, -8*w - 39],
[41, 41, 8*w - 47],
[53, 53, -26*w - 125],
[53, 53, 26*w - 151],
[61, 61, -14*w + 81],
[61, 61, -14*w - 67],
[83, 83, 2*w - 15],
[83, 83, -2*w - 13],
[97, 97, 2*w - 5],
[97, 97, -2*w - 3],
[109, 109, 2*w - 3],
[109, 109, -2*w - 1],
[113, 113, 2*w - 1],
[127, 127, 8*w - 45],
[127, 127, 8*w + 37],
[131, 131, -50*w + 291],
[131, 131, 50*w + 241],
[139, 139, 6*w - 37],
[139, 139, -6*w - 31],
[149, 149, 20*w + 97],
[149, 149, 20*w - 117],
[157, 157, -12*w - 59],
[157, 157, 12*w - 71],
[163, 163, 4*w - 19],
[163, 163, 4*w + 15],
[173, 173, 4*w + 23],
[173, 173, 4*w - 27],
[211, 211, 2*w - 19],
[211, 211, -2*w - 17],
[227, 227, -4*w - 13],
[227, 227, 4*w - 17],
[233, 233, 6*w + 25],
[233, 233, -6*w + 31],
[239, 239, -14*w - 69],
[239, 239, 14*w - 83],
[241, 241, 46*w + 221],
[241, 241, -46*w + 267],
[251, 251, 22*w - 129],
[251, 251, 22*w + 107],
[257, 257, -34*w + 197],
[257, 257, -34*w - 163],
[277, 277, 4*w - 29],
[277, 277, -4*w - 25],
[283, 283, 4*w - 15],
[283, 283, -4*w - 11],
[289, 17, -17],
[307, 307, 68*w - 395],
[307, 307, 68*w + 327],
[311, 311, 10*w - 61],
[311, 311, -10*w - 51],
[313, 313, -32*w - 155],
[313, 313, 32*w - 187],
[317, 317, -18*w - 85],
[317, 317, 18*w - 103],
[331, 331, -4*w - 9],
[331, 331, 4*w - 13],
[337, 337, -16*w - 79],
[337, 337, 16*w - 95],
[347, 347, -12*w + 67],
[347, 347, 12*w + 55],
[353, 353, 14*w - 79],
[353, 353, 14*w + 65],
[361, 19, -19],
[367, 367, -32*w - 153],
[367, 367, -32*w + 185],
[383, 383, 56*w + 269],
[383, 383, 56*w - 325],
[389, 389, -4*w - 27],
[389, 389, 4*w - 31],
[401, 401, 8*w - 51],
[401, 401, -8*w - 43],
[421, 421, 12*w - 73],
[421, 421, -12*w - 61],
[439, 439, -8*w - 33],
[439, 439, 8*w - 41],
[443, 443, -4*w - 1],
[443, 443, 4*w - 5],
[461, 461, -30*w + 173],
[461, 461, 30*w + 143],
[463, 463, 2*w - 25],
[463, 463, -2*w - 23],
[467, 467, 34*w + 165],
[467, 467, 34*w - 199],
[503, 503, -26*w - 127],
[503, 503, 26*w - 153],
[509, 509, 4*w - 33],
[509, 509, -4*w - 29],
[521, 521, -10*w + 53],
[521, 521, 10*w + 43],
[529, 23, -23],
[547, 547, 14*w - 85],
[547, 547, -14*w - 71],
[557, 557, 66*w + 317],
[557, 557, 66*w - 383],
[563, 563, 2*w - 27],
[563, 563, -2*w - 25],
[569, 569, -64*w + 373],
[569, 569, 64*w + 309],
[587, 587, 12*w + 53],
[587, 587, -12*w + 65],
[593, 593, 8*w + 45],
[593, 593, 8*w - 53],
[601, 601, -26*w - 123],
[601, 601, 26*w - 149],
[617, 617, 6*w - 23],
[617, 617, -6*w - 17],
[647, 647, 24*w - 137],
[647, 647, -24*w - 113],
[653, 653, 28*w - 165],
[653, 653, -28*w - 137],
[677, 677, -22*w - 103],
[677, 677, 22*w - 125],
[691, 691, 20*w - 113],
[691, 691, 20*w + 93],
[709, 709, -10*w - 41],
[709, 709, 10*w - 51],
[719, 719, -8*w + 37],
[719, 719, -8*w - 29],
[727, 727, -22*w - 109],
[727, 727, 22*w - 131],
[739, 739, 46*w - 269],
[739, 739, -46*w - 223],
[761, 761, -6*w - 13],
[761, 761, 6*w - 19],
[769, 769, 110*w - 639],
[769, 769, 110*w + 529],
[773, 773, 4*w - 37],
[773, 773, -4*w - 33],
[787, 787, 2*w - 31],
[787, 787, -2*w - 29],
[809, 809, 56*w + 271],
[809, 809, -56*w + 327],
[821, 821, 6*w - 17],
[821, 821, -6*w - 11],
[823, 823, 38*w - 223],
[823, 823, -38*w - 185],
[827, 827, 132*w - 767],
[827, 827, 132*w + 635],
[841, 29, -29],
[853, 853, -76*w + 443],
[853, 853, 76*w + 367],
[863, 863, 14*w - 87],
[863, 863, -14*w - 73],
[911, 911, 2*w - 33],
[911, 911, -2*w - 31],
[919, 919, -6*w - 41],
[919, 919, 6*w - 47],
[929, 929, 50*w + 239],
[929, 929, 50*w - 289],
[953, 953, 6*w - 11],
[953, 953, -6*w - 5],
[967, 967, -8*w - 25],
[967, 967, 8*w - 33],
[991, 991, 16*w + 71],
[991, 991, -16*w + 87]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

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

heckeEigenvaluesArray := [e, e, -e^2 - e + 2, -e^2 - e + 2, e^3 - e^2 - 4*e + 4, e^3 - e^2 - 3*e + 3, e^3 - e^2 - 3*e + 3, -e^3 + e^2 + 5*e - 1, -e^3 + e^2 + 5*e - 1, -3*e^3 + 11*e + 2, e^3 + e^2 - 6*e - 4, e^3 + e^2 - 6*e - 4, 2*e^3 + 2*e^2 - 9*e - 3, 2*e^3 + 2*e^2 - 9*e - 3, -4*e^3 + 17*e + 3, -4*e^3 + 17*e + 3, -e^2 - 2*e + 5, -e^2 - 2*e + 5, -e^3 + 6*e^2 + 6*e - 15, -e^3 + 6*e^2 + 6*e - 15, 2*e^3 + e^2 - 5*e - 4, 2*e^3 + e^2 - 5*e - 4, e^3 - 4*e^2 - 7*e + 14, e^3 - 4*e^2 - 7*e + 14, -6*e^3 + 2*e^2 + 18*e - 12, 5*e^3 - 20*e - 1, 5*e^3 - 20*e - 1, 4*e^3 + 2*e^2 - 17*e + 3, 4*e^3 + 2*e^2 - 17*e + 3, -4*e^3 - 3*e^2 + 11*e + 8, -4*e^3 - 3*e^2 + 11*e + 8, -2*e^3 + 2*e^2 + 11*e - 9, -2*e^3 + 2*e^2 + 11*e - 9, -2*e^3 + 5*e^2 + 6*e - 13, -2*e^3 + 5*e^2 + 6*e - 13, -3*e^3 - 5*e^2 + 12*e + 20, -3*e^3 - 5*e^2 + 12*e + 20, -3*e^3 - 8*e^2 + 13*e + 18, -3*e^3 - 8*e^2 + 13*e + 18, -e^2 - 3*e - 4, -e^2 - 3*e - 4, 2*e^3 + 2*e^2 - 14*e - 18, 2*e^3 + 2*e^2 - 14*e - 18, 2*e^2 - 3*e + 9, 2*e^2 - 3*e + 9, e^2 + 2*e + 9, e^2 + 2*e + 9, -4*e^3 + 2*e^2 + 20*e + 2, -4*e^3 + 2*e^2 + 20*e + 2, -3*e^3 + 3*e^2 + 18*e - 18, -3*e^3 + 3*e^2 + 18*e - 18, 4*e^3 - e^2 - 20*e + 3, 4*e^3 - e^2 - 20*e + 3, -3*e^3 - 5*e^2 + 16*e + 14, -3*e^3 - 5*e^2 + 16*e + 14, 4*e^3 + 2*e^2 - 12*e + 14, 4*e^3 + 2*e^2 - 12*e + 14, e^3 + 4*e + 17, e^3 - 3*e^2 - 10*e - 4, e^3 - 3*e^2 - 10*e - 4, 6*e^3 + e^2 - 20*e - 15, 6*e^3 + e^2 - 20*e - 15, 6*e^3 - 2*e^2 - 20*e - 4, 6*e^3 - 2*e^2 - 20*e - 4, e^3 - 2*e^2 - 4*e - 3, e^3 - 2*e^2 - 4*e - 3, 5*e^3 + 7*e^2 - 26*e - 22, 5*e^3 + 7*e^2 - 26*e - 22, -e^3 - 3*e^2 + 9*e + 23, -e^3 - 3*e^2 + 9*e + 23, 8*e^3 + 6*e^2 - 23*e - 15, 8*e^3 + 6*e^2 - 23*e - 15, -5*e^3 + 5*e^2 + 10*e - 18, -5*e^3 + 5*e^2 + 10*e - 18, -e^3 - 4*e^2 - 10*e + 23, -11*e^3 - 2*e^2 + 42*e - 1, -11*e^3 - 2*e^2 + 42*e - 1, e^3 - 9*e^2 - 3*e + 15, e^3 - 9*e^2 - 3*e + 15, 2*e^3 + 5*e^2 - 9*e - 6, 2*e^3 + 5*e^2 - 9*e - 6, -3*e^3 + 4*e^2 + 12*e - 21, -3*e^3 + 4*e^2 + 12*e - 21, -11*e^3 + 39*e - 10, -11*e^3 + 39*e - 10, 3*e^3 + 10*e^2 - 9*e - 28, 3*e^3 + 10*e^2 - 9*e - 28, -10*e^3 + 3*e^2 + 50*e - 3, -10*e^3 + 3*e^2 + 50*e - 3, -e^3 + 15*e^2 + 6*e - 42, -e^3 + 15*e^2 + 6*e - 42, -10*e^3 + e^2 + 40*e - 7, -10*e^3 + e^2 + 40*e - 7, -4*e^3 - 7*e^2 + 11*e + 12, -4*e^3 - 7*e^2 + 11*e + 12, -3*e^3 - e^2 + 9*e + 27, -3*e^3 - e^2 + 9*e + 27, 8*e^3 - 8*e^2 - 30*e + 24, 8*e^3 - 8*e^2 - 30*e + 24, -7*e^3 - 8*e^2 + 37*e + 24, -7*e^3 - 8*e^2 + 37*e + 24, 7*e^3 - 9*e^2 - 25*e + 47, 2*e^3 - 8*e^2 - 19*e + 29, 2*e^3 - 8*e^2 - 19*e + 29, 2*e^3 + 7*e^2 - 12*e - 9, 2*e^3 + 7*e^2 - 12*e - 9, 4*e^3 + 9*e^2 - 12*e - 21, 4*e^3 + 9*e^2 - 12*e - 21, e^3 + 9*e^2 + 4*e - 36, e^3 + 9*e^2 + 4*e - 36, -2*e^3 - 12*e^2 + 5*e + 33, -2*e^3 - 12*e^2 + 5*e + 33, 10*e^3 + 8*e^2 - 35*e - 21, 10*e^3 + 8*e^2 - 35*e - 21, -e^2 + 4*e - 13, -e^2 + 4*e - 13, 6*e^3 - 13*e^2 - 20*e + 39, 6*e^3 - 13*e^2 - 20*e + 39, 13*e^3 - 4*e^2 - 41*e + 24, 13*e^3 - 4*e^2 - 41*e + 24, -4*e^3 + 2*e^2 + 13*e + 3, -4*e^3 + 2*e^2 + 13*e + 3, -16*e^3 - 7*e^2 + 62*e + 15, -16*e^3 - 7*e^2 + 62*e + 15, -8*e^3 + 39*e + 17, -8*e^3 + 39*e + 17, 2*e^3 - 3*e^2 - 11*e - 4, 2*e^3 - 3*e^2 - 11*e - 4, -11*e^3 + 2*e^2 + 41*e + 12, -11*e^3 + 2*e^2 + 41*e + 12, -3*e^3 + 4*e^2 + 8*e - 25, -3*e^3 + 4*e^2 + 8*e - 25, e^3 - 7*e^2 + 16*e + 26, e^3 - 7*e^2 + 16*e + 26, 10*e^3 + 7*e^2 - 31*e - 18, 10*e^3 + 7*e^2 - 31*e - 18, 8*e^3 + 13*e^2 - 29*e - 16, 8*e^3 + 13*e^2 - 29*e - 16, -10*e^3 + 7*e^2 + 31*e - 30, -10*e^3 + 7*e^2 + 31*e - 30, 3*e^3 - 3*e^2 - 7*e - 1, 3*e^3 - 3*e^2 - 7*e - 1, -12*e^3 - 7*e^2 + 46*e + 33, -12*e^3 - 7*e^2 + 46*e + 33, -12*e^3 - 8*e^2 + 42*e + 12, -12*e^3 - 8*e^2 + 42*e + 12, 2*e^2 - 3*e + 5, 2*e^2 - 3*e + 5, -3*e^3 + 9*e^2 - 3*e - 27, -3*e^3 + 9*e^2 - 3*e - 27, -e^3 - 5*e^2 + 18*e + 50, e^3 - 12*e^2 - 7*e + 38, e^3 - 12*e^2 - 7*e + 38, 9*e^3 - 17*e^2 - 45*e + 45, 9*e^3 - 17*e^2 - 45*e + 45, e^3 + 14*e^2 - 51, e^3 + 14*e^2 - 51, -e^3 - 9*e^2 - 2*e + 8, -e^3 - 9*e^2 - 2*e + 8, 7*e^3 - 33*e - 24, 7*e^3 - 33*e - 24, e^3 + 10*e^2 + 4*e - 39, e^3 + 10*e^2 + 4*e - 39, 3*e^3 - 7*e^2 - 3*e + 23, 3*e^3 - 7*e^2 - 3*e + 23, -7*e^3 + 8*e^2 + 25*e - 10, -7*e^3 + 8*e^2 + 25*e - 10];
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;