/*
  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![-39, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {9, 9, -w + 6}>;

primesArray := [
[3, 3, w + 6],
[3, 3, -w + 7],
[4, 2, 2],
[11, 11, -3*w - 17],
[11, 11, 3*w - 20],
[13, 13, 2*w - 13],
[13, 13, 2*w + 11],
[17, 17, w + 7],
[17, 17, -w + 8],
[19, 19, -w - 4],
[19, 19, -w + 5],
[25, 5, 5],
[31, 31, -6*w - 35],
[31, 31, -6*w + 41],
[37, 37, -w - 1],
[37, 37, w - 2],
[47, 47, 3*w + 16],
[47, 47, -3*w + 19],
[49, 7, -7],
[67, 67, 3*w - 22],
[67, 67, -3*w - 19],
[71, 71, -w - 10],
[71, 71, w - 11],
[89, 89, -8*w - 47],
[89, 89, 8*w - 55],
[101, 101, 4*w - 29],
[101, 101, -4*w - 25],
[109, 109, -3*w - 20],
[109, 109, 3*w - 23],
[113, 113, 3*w - 17],
[113, 113, -3*w - 14],
[127, 127, -9*w - 53],
[127, 127, 9*w - 62],
[157, 157, 2*w - 1],
[167, 167, 2*w - 19],
[167, 167, -2*w - 17],
[173, 173, 13*w - 89],
[173, 173, -13*w - 76],
[193, 193, 11*w - 73],
[193, 193, -11*w - 62],
[197, 197, 3*w - 14],
[197, 197, -3*w - 11],
[199, 199, 3*w - 25],
[199, 199, -3*w - 22],
[233, 233, -w - 16],
[233, 233, w - 17],
[239, 239, 7*w - 50],
[239, 239, -7*w - 43],
[257, 257, 6*w + 31],
[257, 257, -6*w + 37],
[263, 263, 3*w - 11],
[263, 263, -3*w - 8],
[277, 277, -12*w - 71],
[277, 277, 12*w - 83],
[281, 281, -3*w - 7],
[281, 281, 3*w - 10],
[283, 283, -7*w + 44],
[283, 283, 7*w + 37],
[311, 311, 3*w - 8],
[311, 311, -3*w - 5],
[313, 313, -13*w - 73],
[313, 313, 13*w - 86],
[317, 317, -9*w + 58],
[317, 317, 9*w + 49],
[331, 331, -5*w - 23],
[331, 331, 5*w - 28],
[347, 347, -3*w - 1],
[347, 347, 3*w - 4],
[349, 349, 3*w - 28],
[349, 349, -3*w - 25],
[353, 353, 3*w - 2],
[353, 353, 3*w - 1],
[389, 389, 6*w - 35],
[389, 389, 6*w + 29],
[419, 419, 2*w - 25],
[419, 419, -2*w - 23],
[431, 431, 10*w - 71],
[431, 431, -10*w - 61],
[457, 457, 27*w + 157],
[457, 457, 27*w - 184],
[461, 461, -4*w - 31],
[461, 461, 4*w - 35],
[467, 467, -w - 22],
[467, 467, w - 23],
[487, 487, 8*w + 41],
[487, 487, -8*w + 49],
[523, 523, -6*w - 41],
[523, 523, 6*w - 47],
[529, 23, -23],
[547, 547, 4*w - 11],
[547, 547, -4*w - 7],
[557, 557, 28*w - 191],
[557, 557, -28*w - 163],
[571, 571, 20*w - 133],
[571, 571, -20*w - 113],
[577, 577, -3*w - 29],
[577, 577, 3*w - 32],
[593, 593, 19*w - 131],
[593, 593, -19*w - 112],
[601, 601, 5*w - 22],
[601, 601, -5*w - 17],
[617, 617, -18*w - 101],
[617, 617, 18*w - 119],
[619, 619, -4*w - 1],
[619, 619, 4*w - 5],
[631, 631, 17*w - 112],
[631, 631, -17*w - 95],
[641, 641, 15*w + 83],
[641, 641, 15*w - 98],
[647, 647, -17*w - 101],
[647, 647, 17*w - 118],
[653, 653, -11*w - 68],
[653, 653, 11*w - 79],
[659, 659, 5*w - 43],
[659, 659, -5*w - 38],
[661, 661, -25*w - 142],
[661, 661, 25*w - 167],
[677, 677, 13*w - 92],
[677, 677, -13*w - 79],
[709, 709, 5*w - 19],
[709, 709, -5*w - 14],
[727, 727, -9*w - 58],
[727, 727, 9*w - 67],
[733, 733, 7*w - 38],
[733, 733, -7*w - 31],
[739, 739, -18*w - 107],
[739, 739, 18*w - 125],
[743, 743, 2*w - 31],
[743, 743, -2*w - 29],
[769, 769, -3*w - 32],
[769, 769, 3*w - 35],
[773, 773, -w - 28],
[773, 773, w - 29],
[797, 797, 21*w - 139],
[797, 797, -21*w - 118],
[821, 821, -15*w + 97],
[821, 821, 15*w + 82],
[827, 827, 9*w - 53],
[827, 827, -9*w - 44],
[829, 829, -19*w - 106],
[829, 829, 19*w - 125],
[841, 29, -29],
[853, 853, 9*w - 68],
[853, 853, -9*w - 59],
[907, 907, 3*w - 37],
[907, 907, -3*w - 34],
[911, 911, 5*w - 46],
[911, 911, -5*w - 41],
[929, 929, -6*w - 19],
[929, 929, 6*w - 25],
[941, 941, -20*w - 119],
[941, 941, 20*w - 139],
[953, 953, -w - 31],
[953, 953, w - 32],
[967, 967, 11*w + 56],
[967, 967, -11*w + 67],
[977, 977, 16*w - 113],
[977, 977, -16*w - 97],
[991, 991, -8*w - 35],
[991, 991, 8*w - 43]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^5 + x^4 - 9*x^3 - x^2 + 12*x - 5;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [0, e, -5/3*e^4 - 3*e^3 + 13*e^2 + 35/3*e - 41/3, -e^4 - 2*e^3 + 7*e^2 + 8*e - 6, e^4 + 2*e^3 - 7*e^2 - 8*e + 6, e^4 + 2*e^3 - 8*e^2 - 9*e + 9, e^4 + 2*e^3 - 8*e^2 - 9*e + 9, -3*e^4 - 5*e^3 + 23*e^2 + 18*e - 22, 3*e^4 + 5*e^3 - 23*e^2 - 18*e + 22, -1/3*e^4 + 4*e^2 - 5/3*e - 19/3, -1/3*e^4 + 4*e^2 - 5/3*e - 19/3, 8/3*e^4 + 4*e^3 - 21*e^2 - 41/3*e + 41/3, 7/3*e^4 + 3*e^3 - 20*e^2 - 25/3*e + 58/3, 7/3*e^4 + 3*e^3 - 20*e^2 - 25/3*e + 58/3, 1/3*e^4 - 4*e^2 - 4/3*e + 10/3, 1/3*e^4 - 4*e^2 - 4/3*e + 10/3, -4*e^4 - 7*e^3 + 31*e^2 + 24*e - 28, 4*e^4 + 7*e^3 - 31*e^2 - 24*e + 28, -3*e^4 - 5*e^3 + 24*e^2 + 21*e - 18, 3*e^4 + 6*e^3 - 22*e^2 - 26*e + 17, 3*e^4 + 6*e^3 - 22*e^2 - 26*e + 17, -2*e^4 - 4*e^3 + 17*e^2 + 19*e - 24, 2*e^4 + 4*e^3 - 17*e^2 - 19*e + 24, -4*e^4 - 6*e^3 + 33*e^2 + 23*e - 33, 4*e^4 + 6*e^3 - 33*e^2 - 23*e + 33, 4*e^2 + 5*e - 14, -4*e^2 - 5*e + 14, -19/3*e^4 - 10*e^3 + 49*e^2 + 112/3*e - 130/3, -19/3*e^4 - 10*e^3 + 49*e^2 + 112/3*e - 130/3, 7*e^4 + 10*e^3 - 57*e^2 - 34*e + 50, -7*e^4 - 10*e^3 + 57*e^2 + 34*e - 50, -25/3*e^4 - 14*e^3 + 66*e^2 + 163/3*e - 190/3, -25/3*e^4 - 14*e^3 + 66*e^2 + 163/3*e - 190/3, -12*e^4 - 20*e^3 + 92*e^2 + 70*e - 80, 6*e^4 + 9*e^3 - 47*e^2 - 29*e + 42, -6*e^4 - 9*e^3 + 47*e^2 + 29*e - 42, 8*e^4 + 14*e^3 - 64*e^2 - 56*e + 65, -8*e^4 - 14*e^3 + 64*e^2 + 56*e - 65, -e^3 + 3*e^2 + 12*e - 10, -e^3 + 3*e^2 + 12*e - 10, 3*e^4 + 6*e^3 - 21*e^2 - 19*e + 8, -3*e^4 - 6*e^3 + 21*e^2 + 19*e - 8, 9*e^4 + 15*e^3 - 70*e^2 - 50*e + 62, 9*e^4 + 15*e^3 - 70*e^2 - 50*e + 62, -6*e^4 - 9*e^3 + 47*e^2 + 24*e - 50, 6*e^4 + 9*e^3 - 47*e^2 - 24*e + 50, e^4 - e^3 - 10*e^2 + 6*e + 5, -e^4 + e^3 + 10*e^2 - 6*e - 5, 3*e^4 + 7*e^3 - 22*e^2 - 28*e + 22, -3*e^4 - 7*e^3 + 22*e^2 + 28*e - 22, -12*e^4 - 22*e^3 + 89*e^2 + 84*e - 81, 12*e^4 + 22*e^3 - 89*e^2 - 84*e + 81, 26/3*e^4 + 17*e^3 - 67*e^2 - 209/3*e + 221/3, 26/3*e^4 + 17*e^3 - 67*e^2 - 209/3*e + 221/3, 6*e^4 + 13*e^3 - 43*e^2 - 60*e + 41, -6*e^4 - 13*e^3 + 43*e^2 + 60*e - 41, 10/3*e^4 + 5*e^3 - 22*e^2 - 43/3*e - 5/3, 10/3*e^4 + 5*e^3 - 22*e^2 - 43/3*e - 5/3, -7*e^4 - 11*e^3 + 59*e^2 + 49*e - 74, 7*e^4 + 11*e^3 - 59*e^2 - 49*e + 74, -46/3*e^4 - 27*e^3 + 119*e^2 + 304/3*e - 370/3, -46/3*e^4 - 27*e^3 + 119*e^2 + 304/3*e - 370/3, -e^4 + 9*e^2 - 7*e - 15, e^4 - 9*e^2 + 7*e + 15, -9*e^4 - 15*e^3 + 66*e^2 + 48*e - 60, -9*e^4 - 15*e^3 + 66*e^2 + 48*e - 60, 2*e^3 + e^2 - 18*e + 10, -2*e^3 - e^2 + 18*e - 10, 3*e^4 + 6*e^3 - 19*e^2 - 20*e + 12, 3*e^4 + 6*e^3 - 19*e^2 - 20*e + 12, -6*e^4 - 13*e^3 + 47*e^2 + 59*e - 49, 6*e^4 + 13*e^3 - 47*e^2 - 59*e + 49, -4*e^4 - 9*e^3 + 30*e^2 + 33*e - 47, 4*e^4 + 9*e^3 - 30*e^2 - 33*e + 47, -8*e^4 - 14*e^3 + 60*e^2 + 43*e - 47, 8*e^4 + 14*e^3 - 60*e^2 - 43*e + 47, -8*e^4 - 14*e^3 + 59*e^2 + 47*e - 39, 8*e^4 + 14*e^3 - 59*e^2 - 47*e + 39, 13*e^4 + 22*e^3 - 98*e^2 - 72*e + 72, 13*e^4 + 22*e^3 - 98*e^2 - 72*e + 72, -e^4 - 4*e^3 + 3*e^2 + 26*e + 5, e^4 + 4*e^3 - 3*e^2 - 26*e - 5, e^3 + 8*e^2 - 25, -e^3 - 8*e^2 + 25, e^4 + e^3 - 2*e^2 + 12*e - 20, e^4 + e^3 - 2*e^2 + 12*e - 20, -25/3*e^4 - 14*e^3 + 61*e^2 + 151/3*e - 187/3, -25/3*e^4 - 14*e^3 + 61*e^2 + 151/3*e - 187/3, 10/3*e^4 + 4*e^3 - 27*e^2 - 22/3*e - 8/3, -47/3*e^4 - 23*e^3 + 124*e^2 + 242/3*e - 329/3, -47/3*e^4 - 23*e^3 + 124*e^2 + 242/3*e - 329/3, -12*e^4 - 22*e^3 + 91*e^2 + 80*e - 90, 12*e^4 + 22*e^3 - 91*e^2 - 80*e + 90, -12*e^4 - 20*e^3 + 89*e^2 + 67*e - 79, -12*e^4 - 20*e^3 + 89*e^2 + 67*e - 79, 35/3*e^4 + 19*e^3 - 94*e^2 - 215/3*e + 275/3, 35/3*e^4 + 19*e^3 - 94*e^2 - 215/3*e + 275/3, -10*e^4 - 22*e^3 + 69*e^2 + 92*e - 76, 10*e^4 + 22*e^3 - 69*e^2 - 92*e + 76, 4*e^2 + 2*e - 29, 4*e^2 + 2*e - 29, 4*e^4 + 11*e^3 - 27*e^2 - 55*e + 15, -4*e^4 - 11*e^3 + 27*e^2 + 55*e - 15, 11/3*e^4 + 5*e^3 - 30*e^2 - 29/3*e + 134/3, 11/3*e^4 + 5*e^3 - 30*e^2 - 29/3*e + 134/3, -3*e^4 - 10*e^3 + 21*e^2 + 48*e - 36, -3*e^4 - 10*e^3 + 21*e^2 + 48*e - 36, -5*e^4 - 6*e^3 + 40*e^2 + 25*e - 25, 5*e^4 + 6*e^3 - 40*e^2 - 25*e + 25, -4*e^4 - 8*e^3 + 27*e^2 + 26*e - 35, 4*e^4 + 8*e^3 - 27*e^2 - 26*e + 35, e^4 - 15*e^2 + 4*e + 45, -e^4 + 15*e^2 - 4*e - 45, 11*e^4 + 20*e^3 - 91*e^2 - 90*e + 100, -11*e^4 - 20*e^3 + 91*e^2 + 90*e - 100, 35/3*e^4 + 21*e^3 - 87*e^2 - 233/3*e + 278/3, 35/3*e^4 + 21*e^3 - 87*e^2 - 233/3*e + 278/3, 11*e^4 + 23*e^3 - 81*e^2 - 101*e + 88, -11*e^4 - 23*e^3 + 81*e^2 + 101*e - 88, 9*e^4 + 11*e^3 - 77*e^2 - 28*e + 78, 9*e^4 + 11*e^3 - 77*e^2 - 28*e + 78, -14*e^4 - 21*e^3 + 113*e^2 + 65*e - 110, -14*e^4 - 21*e^3 + 113*e^2 + 65*e - 110, -31/3*e^4 - 16*e^3 + 81*e^2 + 124/3*e - 265/3, -31/3*e^4 - 16*e^3 + 81*e^2 + 124/3*e - 265/3, -56/3*e^4 - 34*e^3 + 145*e^2 + 401/3*e - 455/3, -56/3*e^4 - 34*e^3 + 145*e^2 + 401/3*e - 455/3, 5*e^4 + 5*e^3 - 44*e^2 - 6*e + 44, -5*e^4 - 5*e^3 + 44*e^2 + 6*e - 44, -14/3*e^4 - 10*e^3 + 33*e^2 + 155/3*e - 128/3, -14/3*e^4 - 10*e^3 + 33*e^2 + 155/3*e - 128/3, -7*e^4 - 15*e^3 + 47*e^2 + 68*e - 35, 7*e^4 + 15*e^3 - 47*e^2 - 68*e + 35, e^4 - 2*e^3 - 12*e^2 + 17*e + 20, -e^4 + 2*e^3 + 12*e^2 - 17*e - 20, -20*e^4 - 32*e^3 + 161*e^2 + 126*e - 164, 20*e^4 + 32*e^3 - 161*e^2 - 126*e + 164, e^4 - 8*e^2 + 2*e - 18, -e^4 + 8*e^2 - 2*e + 18, -8/3*e^4 - 2*e^3 + 17*e^2 - 13/3*e - 5/3, -8/3*e^4 - 2*e^3 + 17*e^2 - 13/3*e - 5/3, -11/3*e^4 - 4*e^3 + 29*e^2 + 8/3*e - 194/3, 34/3*e^4 + 14*e^3 - 94*e^2 - 130/3*e + 280/3, 34/3*e^4 + 14*e^3 - 94*e^2 - 130/3*e + 280/3, -5*e^4 - 11*e^3 + 30*e^2 + 46*e - 8, -5*e^4 - 11*e^3 + 30*e^2 + 46*e - 8, 23*e^4 + 40*e^3 - 179*e^2 - 137*e + 176, -23*e^4 - 40*e^3 + 179*e^2 + 137*e - 176, 20*e^4 + 32*e^3 - 162*e^2 - 114*e + 153, -20*e^4 - 32*e^3 + 162*e^2 + 114*e - 153, -e^4 - e^3 + 10*e^2 + 7*e - 36, e^4 + e^3 - 10*e^2 - 7*e + 36, -9*e^4 - 13*e^3 + 75*e^2 + 45*e - 85, 9*e^4 + 13*e^3 - 75*e^2 - 45*e + 85, -11*e^4 - 20*e^3 + 80*e^2 + 68*e - 57, -11*e^4 - 20*e^3 + 80*e^2 + 68*e - 57, e^4 + 3*e^3 - 4*e^2 - e - 25, -e^4 - 3*e^3 + 4*e^2 + e + 25, 10*e^4 + 18*e^3 - 71*e^2 - 66*e + 56, 10*e^4 + 18*e^3 - 71*e^2 - 66*e + 56];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {3, 3, w + 6}>] := -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;