/*
  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, 7, 12, -1, -8, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {31, 31, -4*w^5 + 6*w^4 + 28*w^3 - 8*w^2 - 39*w - 11}>;

primesArray := [
[4, 2, -2*w^5 + 2*w^4 + 17*w^3 - w^2 - 27*w - 8],
[11, 11, -w^5 + 2*w^4 + 6*w^3 - 5*w^2 - 7*w + 1],
[11, 11, -w^5 + w^4 + 8*w^3 + w^2 - 12*w - 5],
[11, 11, w - 1],
[31, 31, -4*w^5 + 6*w^4 + 28*w^3 - 8*w^2 - 39*w - 11],
[41, 41, -4*w^5 + 5*w^4 + 31*w^3 - 4*w^2 - 49*w - 16],
[41, 41, w^4 - 3*w^3 - 3*w^2 + 8*w + 1],
[41, 41, -w^2 + w + 2],
[59, 59, w^2 - w - 4],
[59, 59, 4*w^5 - 5*w^4 - 31*w^3 + 4*w^2 + 49*w + 14],
[59, 59, -w^4 + 3*w^3 + 3*w^2 - 8*w - 3],
[71, 71, 5*w^5 - 7*w^4 - 37*w^3 + 9*w^2 + 55*w + 16],
[71, 71, -w^5 + 11*w^3 + 4*w^2 - 19*w - 7],
[71, 71, -w^4 + 2*w^3 + 5*w^2 - 4*w - 4],
[79, 79, 3*w^5 - 6*w^4 - 17*w^3 + 11*w^2 + 21*w + 7],
[79, 79, -5*w^5 + 8*w^4 + 35*w^3 - 14*w^2 - 52*w - 11],
[79, 79, -w^5 + 3*w^4 + 3*w^3 - 7*w^2 - 2*w + 2],
[101, 101, -2*w^5 + 3*w^4 + 14*w^3 - 3*w^2 - 21*w - 9],
[101, 101, -2*w^5 + 2*w^4 + 17*w^3 - 29*w - 9],
[101, 101, -w^5 + 2*w^4 + 6*w^3 - 5*w^2 - 9*w + 1],
[101, 101, -2*w^5 + 3*w^4 + 15*w^3 - 6*w^2 - 23*w - 4],
[101, 101, -3*w^5 + 3*w^4 + 26*w^3 - 2*w^2 - 43*w - 13],
[101, 101, -12*w^5 + 17*w^4 + 89*w^3 - 25*w^2 - 133*w - 30],
[125, 5, -2*w^5 + 3*w^4 + 14*w^3 - 4*w^2 - 20*w - 8],
[131, 131, -5*w^5 + 7*w^4 + 37*w^3 - 9*w^2 - 56*w - 17],
[131, 131, w^4 - 3*w^3 - 3*w^2 + 7*w],
[131, 131, w^5 - w^4 - 8*w^3 + 11*w + 5],
[149, 149, w^5 - w^4 - 9*w^3 + 2*w^2 + 16*w + 3],
[149, 149, -13*w^5 + 19*w^4 + 95*w^3 - 30*w^2 - 141*w - 28],
[149, 149, 2*w^5 - 2*w^4 - 17*w^3 + 30*w + 8],
[151, 151, -4*w^5 + 5*w^4 + 31*w^3 - 4*w^2 - 48*w - 14],
[151, 151, -w^5 + 11*w^3 + 4*w^2 - 20*w - 9],
[151, 151, 3*w^5 - 3*w^4 - 25*w^3 + 39*w + 12],
[151, 151, 3*w^5 - 4*w^4 - 23*w^3 + 5*w^2 + 35*w + 11],
[151, 151, 4*w^5 - 6*w^4 - 28*w^3 + 8*w^2 + 39*w + 10],
[151, 151, -2*w^5 + 4*w^4 + 12*w^3 - 9*w^2 - 16*w - 1],
[179, 179, w^5 - w^4 - 9*w^3 + w^2 + 18*w + 5],
[179, 179, 4*w^5 - 5*w^4 - 31*w^3 + 6*w^2 + 46*w + 11],
[179, 179, -7*w^5 + 10*w^4 + 51*w^3 - 12*w^2 - 78*w - 24],
[191, 191, -3*w^5 + 5*w^4 + 20*w^3 - 8*w^2 - 29*w - 11],
[191, 191, -w^5 + 3*w^4 + 4*w^3 - 10*w^2 - 4*w + 2],
[191, 191, 12*w^5 - 16*w^4 - 91*w^3 + 19*w^2 + 138*w + 36],
[211, 211, 5*w^5 - 7*w^4 - 37*w^3 + 10*w^2 + 55*w + 10],
[211, 211, -7*w^5 + 9*w^4 + 54*w^3 - 10*w^2 - 83*w - 24],
[211, 211, -w^5 + 11*w^3 + 4*w^2 - 21*w - 9],
[229, 229, 7*w^5 - 10*w^4 - 51*w^3 + 14*w^2 + 75*w + 19],
[229, 229, 6*w^5 - 7*w^4 - 48*w^3 + 5*w^2 + 75*w + 22],
[229, 229, 6*w^5 - 9*w^4 - 42*w^3 + 11*w^2 + 62*w + 20],
[239, 239, -12*w^5 + 16*w^4 + 91*w^3 - 19*w^2 - 139*w - 35],
[239, 239, -8*w^5 + 12*w^4 + 58*w^3 - 20*w^2 - 87*w - 16],
[239, 239, 5*w^5 - 7*w^4 - 37*w^3 + 10*w^2 + 55*w + 14],
[269, 269, -5*w^5 + 7*w^4 + 37*w^3 - 10*w^2 - 53*w - 12],
[269, 269, -7*w^5 + 9*w^4 + 54*w^3 - 10*w^2 - 82*w - 24],
[269, 269, 5*w^5 - 7*w^4 - 37*w^3 + 10*w^2 + 54*w + 15],
[331, 331, 7*w^5 - 10*w^4 - 51*w^3 + 12*w^2 + 77*w + 25],
[331, 331, 8*w^5 - 11*w^4 - 59*w^3 + 13*w^2 + 87*w + 24],
[331, 331, 3*w^5 - 3*w^4 - 25*w^3 + w^2 + 39*w + 12],
[359, 359, -w^5 + 2*w^4 + 7*w^3 - 7*w^2 - 11*w + 3],
[359, 359, -2*w^5 + 4*w^4 + 12*w^3 - 9*w^2 - 18*w - 3],
[359, 359, 5*w^5 - 7*w^4 - 37*w^3 + 10*w^2 + 53*w + 14],
[389, 389, -4*w^5 + 6*w^4 + 28*w^3 - 8*w^2 - 38*w - 13],
[389, 389, 6*w^5 - 7*w^4 - 48*w^3 + 6*w^2 + 74*w + 21],
[389, 389, -4*w^5 + 5*w^4 + 32*w^3 - 6*w^2 - 52*w - 13],
[421, 421, -4*w^5 + 6*w^4 + 29*w^3 - 11*w^2 - 42*w - 8],
[421, 421, -15*w^5 + 20*w^4 + 114*w^3 - 25*w^2 - 174*w - 43],
[421, 421, 8*w^5 - 11*w^4 - 60*w^3 + 14*w^2 + 92*w + 22],
[431, 431, 8*w^5 - 11*w^4 - 59*w^3 + 13*w^2 + 87*w + 23],
[431, 431, 3*w^5 - 5*w^4 - 19*w^3 + 7*w^2 + 24*w + 6],
[431, 431, 7*w^5 - 11*w^4 - 48*w^3 + 16*w^2 + 69*w + 18],
[439, 439, -6*w^5 + 7*w^4 + 48*w^3 - 6*w^2 - 74*w - 19],
[439, 439, 4*w^5 - 7*w^4 - 25*w^3 + 11*w^2 + 33*w + 8],
[439, 439, 5*w^5 - 6*w^4 - 39*w^3 + 5*w^2 + 58*w + 16],
[449, 449, -w^5 + 2*w^4 + 6*w^3 - 6*w^2 - 6*w + 1],
[449, 449, w^5 - 2*w^4 - 5*w^3 + 2*w^2 + 4*w + 6],
[449, 449, 4*w^5 - 5*w^4 - 31*w^3 + 4*w^2 + 50*w + 17],
[449, 449, 6*w^5 - 9*w^4 - 43*w^3 + 13*w^2 + 65*w + 17],
[449, 449, -3*w^5 + 3*w^4 + 25*w^3 - w^2 - 38*w - 10],
[449, 449, -3*w^5 + 3*w^4 + 26*w^3 - 2*w^2 - 43*w - 12],
[461, 461, 2*w^5 - 2*w^4 - 16*w^3 - 2*w^2 + 25*w + 9],
[461, 461, 2*w^5 - 2*w^4 - 17*w^3 + w^2 + 26*w + 5],
[461, 461, w^4 - 2*w^3 - 5*w^2 + 3*w + 6],
[461, 461, -4*w^5 + 6*w^4 + 29*w^3 - 10*w^2 - 43*w - 12],
[461, 461, w^5 - 3*w^4 - 4*w^3 + 9*w^2 + 6*w - 1],
[461, 461, 3*w^5 - 5*w^4 - 20*w^3 + 9*w^2 + 26*w + 3],
[479, 479, 15*w^5 - 21*w^4 - 111*w^3 + 28*w^2 + 167*w + 42],
[479, 479, -4*w^5 + 6*w^4 + 29*w^3 - 11*w^2 - 43*w - 7],
[479, 479, -9*w^5 + 12*w^4 + 68*w^3 - 14*w^2 - 104*w - 26],
[491, 491, -8*w^5 + 11*w^4 + 59*w^3 - 13*w^2 - 87*w - 25],
[491, 491, 7*w^5 - 11*w^4 - 48*w^3 + 16*w^2 + 69*w + 20],
[491, 491, 3*w^5 - 5*w^4 - 19*w^3 + 7*w^2 + 24*w + 8],
[499, 499, 3*w^5 - 4*w^4 - 23*w^3 + 5*w^2 + 35*w + 13],
[499, 499, -7*w^5 + 10*w^4 + 51*w^3 - 12*w^2 - 77*w - 23],
[499, 499, -2*w^5 + w^4 + 20*w^3 + 3*w^2 - 36*w - 12],
[509, 509, -2*w^5 + 3*w^4 + 15*w^3 - 7*w^2 - 21*w - 1],
[509, 509, -5*w^5 + 6*w^4 + 40*w^3 - 5*w^2 - 64*w - 19],
[509, 509, 4*w^5 - 7*w^4 - 26*w^3 + 12*w^2 + 37*w + 11],
[521, 521, 2*w^5 - w^4 - 19*w^3 - 4*w^2 + 30*w + 9],
[521, 521, 5*w^5 - 8*w^4 - 35*w^3 + 14*w^2 + 52*w + 10],
[521, 521, 5*w^5 - 6*w^4 - 40*w^3 + 6*w^2 + 62*w + 19],
[529, 23, -6*w^5 + 8*w^4 + 46*w^3 - 11*w^2 - 71*w - 18],
[529, 23, -4*w^5 + 5*w^4 + 30*w^3 - 3*w^2 - 44*w - 14],
[529, 23, -5*w^5 + 6*w^4 + 39*w^3 - 4*w^2 - 58*w - 17],
[541, 541, 2*w^5 - w^4 - 19*w^3 - 5*w^2 + 31*w + 15],
[541, 541, w^5 - 3*w^4 - 4*w^3 + 10*w^2 + 3*w - 2],
[541, 541, 5*w^5 - 7*w^4 - 37*w^3 + 9*w^2 + 54*w + 14],
[569, 569, -3*w^5 + 4*w^4 + 23*w^3 - 6*w^2 - 34*w - 10],
[569, 569, -2*w^5 + 3*w^4 + 15*w^3 - 6*w^2 - 24*w - 1],
[569, 569, 5*w^5 - 6*w^4 - 39*w^3 + 4*w^2 + 60*w + 16],
[571, 571, 5*w^5 - 7*w^4 - 37*w^3 + 9*w^2 + 54*w + 16],
[571, 571, 5*w^5 - 6*w^4 - 40*w^3 + 6*w^2 + 64*w + 17],
[571, 571, 2*w^5 - w^4 - 19*w^3 - 5*w^2 + 31*w + 13],
[599, 599, 5*w^5 - 6*w^4 - 41*w^3 + 8*w^2 + 66*w + 15],
[599, 599, 5*w^5 - 10*w^4 - 29*w^3 + 21*w^2 + 37*w + 5],
[599, 599, 12*w^5 - 16*w^4 - 90*w^3 + 17*w^2 + 134*w + 39],
[601, 601, 5*w^5 - 5*w^4 - 42*w^3 + w^2 + 67*w + 19],
[601, 601, 9*w^5 - 13*w^4 - 65*w^3 + 17*w^2 + 96*w + 26],
[601, 601, 5*w^5 - 6*w^4 - 40*w^3 + 5*w^2 + 64*w + 21],
[631, 631, -8*w^5 + 11*w^4 + 60*w^3 - 14*w^2 - 91*w - 25],
[631, 631, -4*w^5 + 6*w^4 + 28*w^3 - 7*w^2 - 40*w - 12],
[631, 631, w^5 - 11*w^3 - 5*w^2 + 21*w + 12],
[659, 659, -4*w^5 + 5*w^4 + 31*w^3 - 4*w^2 - 47*w - 18],
[659, 659, -6*w^5 + 7*w^4 + 48*w^3 - 5*w^2 - 76*w - 24],
[659, 659, 6*w^5 - 9*w^4 - 43*w^3 + 14*w^2 + 62*w + 17],
[659, 659, 5*w^5 - 7*w^4 - 37*w^3 + 10*w^2 + 53*w + 13],
[659, 659, -2*w^5 + 4*w^4 + 11*w^3 - 8*w^2 - 11*w],
[659, 659, 6*w^5 - 8*w^4 - 45*w^3 + 9*w^2 + 68*w + 17],
[691, 691, -4*w^5 + 7*w^4 + 26*w^3 - 12*w^2 - 38*w - 10],
[691, 691, 4*w^5 - 5*w^4 - 32*w^3 + 6*w^2 + 52*w + 14],
[691, 691, 6*w^5 - 7*w^4 - 48*w^3 + 6*w^2 + 74*w + 20],
[701, 701, w^3 - 3*w^2 - 2*w + 3],
[701, 701, -2*w^5 + 3*w^4 + 14*w^3 - 5*w^2 - 17*w - 2],
[701, 701, -4*w^5 + 6*w^4 + 29*w^3 - 10*w^2 - 45*w - 8],
[709, 709, 4*w^5 - 4*w^4 - 34*w^3 + w^2 + 56*w + 19],
[709, 709, 4*w^5 - 6*w^4 - 28*w^3 + 9*w^2 + 36*w + 8],
[709, 709, -5*w^5 + 7*w^4 + 37*w^3 - 8*w^2 - 57*w - 16],
[709, 709, 7*w^5 - 11*w^4 - 48*w^3 + 17*w^2 + 68*w + 15],
[709, 709, w^5 - 11*w^3 - 3*w^2 + 19*w + 3],
[709, 709, -7*w^5 + 11*w^4 + 50*w^3 - 20*w^2 - 75*w - 16],
[729, 3, -3],
[739, 739, 5*w^5 - 7*w^4 - 36*w^3 + 7*w^2 + 52*w + 14],
[739, 739, w^5 - w^4 - 9*w^3 + 18*w + 5],
[739, 739, -7*w^5 + 9*w^4 + 54*w^3 - 10*w^2 - 84*w - 24],
[769, 769, 5*w^5 - 6*w^4 - 40*w^3 + 6*w^2 + 63*w + 20],
[769, 769, 4*w^5 - 6*w^4 - 29*w^3 + 9*w^2 + 45*w + 9],
[769, 769, w^5 - 11*w^3 - 3*w^2 + 18*w + 2],
[811, 811, 4*w^5 - 5*w^4 - 32*w^3 + 6*w^2 + 51*w + 13],
[811, 811, 5*w^5 - 9*w^4 - 32*w^3 + 17*w^2 + 45*w + 11],
[811, 811, 7*w^5 - 8*w^4 - 56*w^3 + 5*w^2 + 86*w + 27],
[859, 859, -11*w^5 + 16*w^4 + 81*w^3 - 26*w^2 - 121*w - 23],
[859, 859, -12*w^5 + 17*w^4 + 88*w^3 - 23*w^2 - 131*w - 32],
[859, 859, 7*w^5 - 11*w^4 - 49*w^3 + 19*w^2 + 72*w + 13],
[911, 911, -2*w^5 + w^4 + 20*w^3 + 3*w^2 - 37*w - 12],
[911, 911, 2*w^5 - 2*w^4 - 17*w^3 + 2*w^2 + 27*w + 4],
[911, 911, 6*w^5 - 8*w^4 - 45*w^3 + 7*w^2 + 70*w + 23],
[919, 919, w^4 - 2*w^3 - 7*w^2 + 8*w + 6],
[919, 919, 16*w^5 - 22*w^4 - 120*w^3 + 29*w^2 + 182*w + 46],
[919, 919, -5*w^5 + 9*w^4 + 32*w^3 - 18*w^2 - 43*w - 11]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x;
K := Rationals(); e := 1;

heckeEigenvaluesArray := [-3, 4, 4, 4, -1, -6, -6, -6, 12, 12, 12, -8, -8, -8, 0, 0, 0, 6, 6, -10, 6, -10, -10, 22, 4, 4, 4, -10, -10, -10, 8, 8, 16, 16, 8, 16, 4, 4, 4, 0, 0, 0, -12, -12, -12, -26, -26, -26, 0, 0, 0, -10, -10, -10, -20, -20, -20, 24, 24, 24, 14, 14, 14, -10, -10, -10, 8, 8, 8, -24, -24, -24, -14, -14, -14, 42, 42, 42, 30, -18, -18, -18, 30, 30, 32, 32, 32, 28, 28, 28, 20, 20, 20, 30, 30, 30, 10, 10, 10, 10, 10, 10, -18, -18, -18, 18, 18, 18, 44, 44, 44, -16, -16, -16, 26, 26, 26, 8, 8, 8, -28, -20, -20, -20, -28, -28, 52, 52, 52, -50, -50, -50, 22, 6, 22, 6, 22, 6, -46, -52, -52, -52, 2, 2, 2, -4, -4, -4, -20, -20, -20, 24, 24, 24, 32, 32, 32];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {31, 31, -4*w^5 + 6*w^4 + 28*w^3 - 8*w^2 - 39*w - 11}>] := 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;