/*
  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![1, -2, -5, 0, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {35, 35, -w^3 + w^2 + 4*w - 4}>;

primesArray := [
[4, 2, -w^3 + 4*w + 1],
[5, 5, w - 1],
[7, 7, -w^2 + w + 2],
[13, 13, -w^3 + w^2 + 4*w],
[13, 13, -w^2 + 3],
[17, 17, -w^2 + 2],
[19, 19, -w^3 + 5*w],
[31, 31, -w^2 + 2*w + 3],
[37, 37, -2*w^3 + w^2 + 8*w - 1],
[43, 43, -w - 3],
[53, 53, -w^3 + 2*w^2 + 3*w - 2],
[53, 53, w^3 - 6*w - 2],
[59, 59, 2*w^3 - w^2 - 10*w - 2],
[61, 61, 2*w^3 - w^2 - 10*w],
[61, 61, 2*w^3 - w^2 - 8*w],
[71, 71, 2*w^3 - 9*w - 2],
[73, 73, -w^3 - w^2 + 6*w + 3],
[81, 3, -3],
[83, 83, -w^3 + 2*w^2 + 3*w - 3],
[101, 101, 2*w^3 - 8*w - 3],
[125, 5, w^3 + w^2 - 4*w - 6],
[127, 127, 2*w^3 - w^2 - 7*w - 2],
[127, 127, w^3 - 7*w - 7],
[131, 131, w^3 - 3*w - 4],
[131, 131, 3*w^3 - 13*w - 7],
[137, 137, w^2 + w - 4],
[137, 137, -w^3 + 2*w^2 + 4*w - 4],
[149, 149, 3*w + 5],
[157, 157, -2*w^3 - w^2 + 7*w + 2],
[157, 157, -2*w^3 + 2*w^2 + 9*w - 1],
[163, 163, -2*w^3 + w^2 + 7*w],
[163, 163, -3*w^3 + w^2 + 15*w + 3],
[163, 163, 2*w^3 - 7*w - 1],
[163, 163, 2*w^3 - 11*w - 5],
[167, 167, -2*w^3 + w^2 + 9*w - 4],
[167, 167, w^3 - w^2 - 7*w + 1],
[167, 167, w^3 - w^2 - 7*w - 1],
[167, 167, w^2 - 2*w - 5],
[169, 13, w^3 + w^2 - 6*w - 2],
[173, 173, -3*w^3 + 13*w + 3],
[181, 181, 2*w^2 + w - 4],
[191, 191, 2*w^3 - 10*w - 1],
[191, 191, -2*w^3 + 2*w^2 + 6*w - 3],
[193, 193, 3*w^3 - 13*w - 6],
[197, 197, -w^3 + w^2 + 2*w - 3],
[197, 197, w^3 + w^2 - 7*w - 7],
[199, 199, -w^2 + 2*w + 6],
[223, 223, -2*w^3 + w^2 + 11*w + 2],
[227, 227, -3*w^3 + 2*w^2 + 15*w - 1],
[229, 229, -w^3 + w^2 + 6*w - 5],
[233, 233, 2*w^3 - 11*w - 4],
[239, 239, -2*w^3 + 11*w + 3],
[251, 251, 2*w^3 - w^2 - 10*w - 4],
[257, 257, -2*w^3 + w^2 + 6*w + 1],
[257, 257, w^3 - 7*w - 2],
[263, 263, 2*w^3 - 9*w],
[269, 269, w^3 - 4*w - 6],
[269, 269, -2*w^3 + 8*w - 3],
[271, 271, 2*w^2 - w - 5],
[277, 277, -3*w^3 + 2*w^2 + 11*w - 2],
[277, 277, 3*w^3 - w^2 - 15*w - 5],
[281, 281, 2*w^2 - 5],
[283, 283, -3*w^3 + 2*w^2 + 12*w + 2],
[283, 283, -2*w^3 + w^2 + 6*w + 6],
[293, 293, -2*w^3 - w^2 + 12*w + 7],
[293, 293, -2*w^3 + 9*w - 1],
[311, 311, -3*w^3 + w^2 + 12*w + 4],
[313, 313, -w^3 + w^2 + 2*w - 4],
[313, 313, -4*w^3 + 2*w^2 + 16*w - 3],
[331, 331, 3*w^3 - w^2 - 16*w - 5],
[331, 331, 2*w^3 - 2*w^2 - 8*w - 1],
[343, 7, 2*w^3 - 3*w^2 - 9*w + 6],
[359, 359, -2*w^3 + 10*w - 1],
[367, 367, -w^3 + 2*w^2 + w - 4],
[367, 367, 3*w^3 - 11*w - 1],
[373, 373, -w^3 + w^2 + 2*w - 5],
[373, 373, 2*w^3 - 2*w^2 - 11*w - 1],
[389, 389, -2*w^3 + 3*w^2 + 8*w - 6],
[409, 409, 3*w^3 - 2*w^2 - 13*w - 1],
[419, 419, 3*w^3 - w^2 - 12*w - 1],
[419, 419, w^3 - w^2 - 4*w - 4],
[439, 439, w^2 + 2*w - 4],
[439, 439, 3*w^3 + w^2 - 11*w - 5],
[443, 443, 2*w^3 + 2*w^2 - 7*w - 8],
[449, 449, 4*w^3 - w^2 - 16*w - 8],
[457, 457, 2*w^3 + w^2 - 11*w - 4],
[461, 461, -5*w^3 + 2*w^2 + 24*w + 2],
[463, 463, -w^3 + 3*w^2 + 6*w - 5],
[467, 467, w^3 - 3*w^2 - 6*w + 4],
[467, 467, 3*w^3 - w^2 - 12*w - 2],
[479, 479, -w^3 + w^2 + 4*w - 6],
[479, 479, -w^3 + 2*w + 8],
[491, 491, -w^3 + 2*w^2 + 2*w - 6],
[491, 491, w - 5],
[499, 499, 2*w^3 + w^2 - 10*w - 4],
[509, 509, -4*w^3 + w^2 + 20*w + 5],
[521, 521, w^3 - w + 6],
[523, 523, -4*w^3 + 18*w + 5],
[523, 523, -2*w^3 - 2*w^2 + 9*w + 12],
[557, 557, w^2 - 3*w - 6],
[571, 571, 3*w^3 - 11*w - 8],
[577, 577, 2*w^3 + 2*w^2 - 11*w - 7],
[577, 577, -3*w^3 + 4*w^2 + 11*w - 6],
[587, 587, -w^3 - w^2 + 7*w + 11],
[593, 593, w^3 + w^2 - 8*w - 8],
[593, 593, -2*w^3 + 4*w^2 + 6*w - 9],
[599, 599, 4*w^3 - 17*w - 9],
[599, 599, w^3 + w^2 - 6*w - 10],
[601, 601, 3*w^2 - 10],
[607, 607, 4*w^3 - w^2 - 18*w - 8],
[619, 619, w^2 - w - 8],
[641, 641, -w^3 + 3*w^2 + 2*w - 10],
[643, 643, 3*w^3 - w^2 - 16*w + 1],
[653, 653, -2*w^3 + 2*w^2 + 9*w + 3],
[677, 677, w^3 + w^2 - 8*w - 11],
[677, 677, 2*w^3 - w^2 - 12*w - 3],
[683, 683, 3*w^3 - 2*w^2 - 13*w - 2],
[683, 683, 3*w^3 + 2*w^2 - 16*w - 10],
[691, 691, -2*w^3 - 2*w^2 + 12*w + 15],
[691, 691, 3*w^3 - 2*w^2 - 15*w - 2],
[691, 691, w^3 - 2*w^2 - 6*w + 6],
[691, 691, -4*w^3 + 3*w^2 + 15*w - 2],
[719, 719, 2*w^3 - 7*w - 8],
[719, 719, w^2 + 2*w - 5],
[727, 727, -w^3 + 3*w^2 + 4*w - 8],
[727, 727, 2*w^3 - 9*w - 9],
[733, 733, w^3 - 8*w - 2],
[739, 739, -w^3 - 2*w^2 + 4*w + 10],
[743, 743, w^3 - w^2 - 2*w + 8],
[743, 743, 4*w^3 - w^2 - 17*w - 4],
[751, 751, -2*w^3 + 2*w^2 + 6*w - 5],
[751, 751, w^3 - 5*w - 7],
[769, 769, 3*w^3 - 2*w^2 - 10*w],
[787, 787, -2*w^3 + 3*w^2 + 8*w],
[787, 787, 3*w^3 - w^2 - 10*w - 4],
[809, 809, 3*w^3 + w^2 - 16*w - 6],
[811, 811, 3*w^3 - 14*w - 2],
[811, 811, -w^3 + 3*w^2 + 5*w - 5],
[811, 811, w^3 - 2*w - 6],
[811, 811, 3*w^3 - w^2 - 11*w - 3],
[823, 823, w^3 + 2*w^2 - 3*w - 8],
[827, 827, -w^3 + 3*w^2 + 3*w - 7],
[829, 829, w^3 + w^2 - 3*w - 7],
[853, 853, 3*w^3 + w^2 - 14*w - 6],
[857, 857, 3*w^3 + w^2 - 13*w - 5],
[857, 857, 3*w^3 - 3*w^2 - 11*w - 1],
[859, 859, 4*w^3 - 2*w^2 - 18*w - 3],
[859, 859, 3*w^3 - 11*w - 3],
[863, 863, 3*w^3 + w^2 - 17*w - 9],
[877, 877, -w^3 - 2*w^2 + 9*w + 1],
[877, 877, 5*w^3 + w^2 - 23*w - 11],
[911, 911, w^2 - 4*w - 6],
[911, 911, -2*w^3 + 2*w^2 + 10*w + 3],
[919, 919, -4*w^3 + 3*w^2 + 18*w - 1],
[929, 929, 2*w^3 - 13*w - 12],
[929, 929, -w^3 + 2*w^2 + 7*w - 5],
[953, 953, -3*w^3 + 3*w^2 + 13*w - 1],
[967, 967, -3*w^3 + 4*w^2 + 13*w - 7],
[971, 971, 2*w^2 - 3*w - 12],
[971, 971, 3*w^3 - w^2 - 16*w - 3]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^4 - x^3 - 11*x^2 + 9*x + 10;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, 1, 1, -1/2*e^3 + 11/2*e - 1, 1/2*e^3 - 7/2*e - 1, -2, 1/2*e^3 - e^2 - 9/2*e + 5, 2*e + 2, -1/2*e^3 - e^2 + 9/2*e + 3, 1/2*e^3 - e^2 - 9/2*e + 9, -1/2*e^3 + 11/2*e - 1, 1/2*e^3 - e^2 - 13/2*e + 9, 0, 2, 2, -e^3 + e^2 + 6*e - 8, e^3 - 11*e + 4, -2*e^2 + 12, -2*e^2 + 14, -1/2*e^3 + e^2 + 13/2*e - 13, -1/2*e^3 + 3*e^2 + 5/2*e - 19, -2*e^3 + 16*e - 2, 2*e^2 - 2*e - 12, 1/2*e^3 + e^2 - 5/2*e - 3, -e^3 + 2*e^2 + 11*e - 8, -e^3 - 2*e^2 + 9*e + 8, e^2 - 3*e - 2, -3/2*e^3 + e^2 + 31/2*e - 5, -e^3 + 7*e + 8, 1/2*e^3 - 15/2*e + 3, 2*e^3 - 18*e + 4, -1/2*e^3 + e^2 + 9/2*e - 1, 1/2*e^3 - 11/2*e - 1, -e^3 + 5*e + 4, 2*e^3 - 18*e + 8, -e^3 - 2*e^2 + 11*e + 8, -e^3 + 2*e^2 + 9*e - 2, -e^3 - 2*e^2 + 7*e + 8, -3*e^2 - e + 20, e^3 - 11*e + 4, 2*e^3 - 18*e + 2, e^3 + 2*e^2 - 9*e - 18, -e^3 + 11*e + 2, -e^3 + 2*e^2 + 13*e - 16, -e^3 + 7*e + 8, -1/2*e^3 - e^2 - 3/2*e + 13, e^3 - 2*e^2 - 9*e + 10, e^3 - 3*e^2 - 8*e + 24, -1/2*e^3 + 3/2*e - 7, -1/2*e^3 + 19/2*e - 5, 2*e^3 - 16*e + 4, -2*e^2 + 2*e + 20, e^3 - 13*e + 12, -e^3 + 2*e^2 + 9*e - 12, 2*e^3 - 2*e^2 - 12*e + 18, -e^2 - 3*e - 6, 1/2*e^3 + 3*e^2 - 5/2*e - 15, 1/2*e^3 - 2*e^2 - 11/2*e + 15, -e^3 + e^2 + 14*e - 8, -e^3 - 4*e^2 + 7*e + 28, 1/2*e^3 + 4*e^2 - 15/2*e - 17, e^3 + 2*e^2 - 13*e - 8, -3*e^3 + 25*e - 6, -e^3 + 5*e + 4, 2*e^3 - 4*e^2 - 22*e + 14, -1/2*e^3 - e^2 + 5/2*e + 9, -e^3 + 5*e + 12, e^2 + 3*e - 16, -e^3 + 4*e^2 + 5*e - 26, 3/2*e^3 - e^2 - 31/2*e + 7, 1/2*e^3 - 3/2*e + 7, -2*e^3 + 18*e - 16, e^3 + 2*e^2 - 7*e, 3*e^2 + e - 22, -e^3 + 4*e^2 + 7*e - 22, 2*e^3 - 14*e - 6, -5/2*e^3 + 3*e^2 + 53/2*e - 21, -8*e - 10, 3*e^3 - e^2 - 32*e + 10, -1/2*e^3 - e^2 + 21/2*e - 5, -2*e^3 + 2*e^2 + 18*e - 10, -e^3 + 2*e^2 + 7*e, -2*e^2 - 6*e + 20, e^3 - 6*e^2 - 9*e + 34, e^3 + 2*e^2 - 7*e - 10, e^3 + 2*e^2 - 13*e + 8, 2*e^3 + 2*e^2 - 24*e + 2, e^3 + 2*e^2 - 5*e - 6, e^3 - 2*e^2 - 13*e + 18, 1/2*e^3 - 11/2*e - 17, 2*e^3 - 28*e + 10, e^3 + 2*e^2 - 13*e - 30, 1/2*e^3 + e^2 - 21/2*e - 23, 3/2*e^3 - 25/2*e - 3, 7/2*e^3 - 3*e^2 - 63/2*e + 15, 3/2*e^3 + 3*e^2 - 23/2*e - 15, -e^3 - e^2 + 22*e + 12, 7/2*e^3 - 5*e^2 - 75/2*e + 19, e^3 - 5*e - 16, 5/2*e^3 - 35/2*e + 3, 2*e^2 - 10*e - 8, -e^3 - 2*e^2 + 13*e + 8, -3*e^3 - 2*e^2 + 33*e - 2, -5/2*e^3 + 47/2*e + 13, e^3 + 6*e^2 - 13*e - 36, e^3 - 2*e^2 - 9*e + 24, e^3 - 11*e + 10, 3*e^3 + e^2 - 30*e, -2*e^2 - 12*e + 12, -e^3 + 2*e^2 + e - 2, 1/2*e^3 - 3*e^2 - 5/2*e + 5, 2*e^3 - 20*e - 8, 2*e^3 + 4*e^2 - 16*e - 26, 3*e^3 - 4*e^2 - 21*e + 24, -3/2*e^3 - e^2 + 31/2*e - 27, e^3 - 2*e^2 - 5*e + 28, -e^3 + 9*e + 4, -3*e^3 + 27*e + 4, -1/2*e^3 - 2*e^2 - 9/2*e + 37, 5/2*e^3 - 5*e^2 - 29/2*e + 37, 3/2*e^3 + 2*e^2 - 37/2*e - 23, -2*e^3 + 2*e^2 + 18*e - 18, 2*e^3 - 4*e^2 - 16*e + 10, -e^3 + 19*e + 10, e^3 + 6*e^2 - 9*e - 22, -2*e^3 + 2*e^2 + 24*e - 12, -3/2*e^3 - e^2 + 27/2*e + 19, -1/2*e^3 - 2*e^2 - 1/2*e + 5, 4*e^3 - 32*e + 4, -e^3 + 2*e^2 + 17*e - 26, -2*e^3 + 5*e^2 + 21*e - 38, -3*e^3 - 4*e^2 + 23*e + 32, -e^3 + 3*e, 4*e^3 - 2*e^2 - 34*e - 12, 4*e^3 - 4*e^2 - 36*e + 28, -3*e^3 + 4*e^2 + 29*e - 20, 2*e^3 - 4*e^2 - 14*e + 32, -3/2*e^3 + 6*e^2 + 37/2*e - 33, 1/2*e^3 + e^2 - 13/2*e + 17, -7/2*e^3 + 61/2*e - 33, -e^3 - 2*e^2 + 7*e + 4, -2*e^3 + 28*e - 2, -3/2*e^3 - 3*e^2 + 39/2*e - 5, e^3 - 4*e^2 - 15*e + 44, -e^3 + 8*e^2 + 11*e - 32, 2*e^2 + 4*e - 12, -5/2*e^3 + 2*e^2 + 75/2*e - 15, -2*e^3 + 22*e - 40, e^3 - 4*e^2 - 7*e + 34, -3*e^3 + 2*e^2 + 27*e - 12, -16*e - 2, 5*e^3 - 4*e^2 - 41*e + 32, 2*e^3 - 3*e^2 - 11*e + 22, e^3 + 3*e^2 - 2*e, 2*e^3 - 8*e^2 - 10*e + 50, -e^3 - 4*e^2 + 19*e + 20, -3*e^3 + 4*e^2 + 33*e - 36, e^3 + 2*e^2 - 13*e + 18, 2*e^2 + 12*e - 18, -2*e^3 + 6*e^2 + 22*e - 38];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {5, 5, w - 1}>] := -1;
ALEigenvalues[ideal<ZF | {7, 7, -w^2 + 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;