/*
  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, -5, 4, 7, -6, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {29,29,2*w^5 - w^4 - 13*w^3 + 8*w^2 + 15*w - 6}>;

primesArray := [
[4, 2, -w^5 + 6*w^3 - w^2 - 6*w],
[19, 19, 3*w^5 - 2*w^4 - 19*w^3 + 14*w^2 + 18*w - 9],
[29, 29, -w^2 - w + 2],
[29, 29, 2*w^5 - w^4 - 13*w^3 + 7*w^2 + 14*w - 3],
[29, 29, -2*w^5 + w^4 + 13*w^3 - 8*w^2 - 15*w + 6],
[49, 7, -3*w^5 + 2*w^4 + 19*w^3 - 15*w^2 - 19*w + 9],
[49, 7, w^5 - w^4 - 7*w^3 + 6*w^2 + 9*w - 4],
[49, 7, w^2 - 3],
[59, 59, -3*w^5 + 2*w^4 + 18*w^3 - 15*w^2 - 14*w + 7],
[59, 59, 2*w^5 - 12*w^3 + 3*w^2 + 12*w - 3],
[59, 59, -2*w^5 + w^4 + 12*w^3 - 8*w^2 - 11*w + 3],
[61, 61, -4*w^5 + 2*w^4 + 24*w^3 - 16*w^2 - 19*w + 8],
[61, 61, 5*w^5 - 3*w^4 - 31*w^3 + 22*w^2 + 27*w - 13],
[61, 61, 3*w^5 - w^4 - 18*w^3 + 9*w^2 + 16*w - 4],
[71, 71, 2*w^5 - 12*w^3 + 2*w^2 + 11*w - 1],
[71, 71, 3*w^5 - 2*w^4 - 18*w^3 + 15*w^2 + 13*w - 9],
[71, 71, -w^5 + w^4 + 6*w^3 - 7*w^2 - 6*w + 5],
[79, 79, -w^5 + 7*w^3 - w^2 - 10*w],
[79, 79, 4*w^5 - 2*w^4 - 25*w^3 + 15*w^2 + 23*w - 9],
[79, 79, -w^5 + 6*w^3 - 2*w^2 - 7*w + 4],
[89, 89, -w^5 + w^4 + 6*w^3 - 6*w^2 - 4*w + 4],
[89, 89, w^5 - w^4 - 7*w^3 + 6*w^2 + 8*w - 1],
[89, 89, 4*w^5 - 2*w^4 - 25*w^3 + 16*w^2 + 24*w - 12],
[101, 101, -2*w^5 + 2*w^4 + 13*w^3 - 13*w^2 - 12*w + 6],
[101, 101, w^3 + w^2 - 4*w - 1],
[101, 101, -4*w^5 + 2*w^4 + 25*w^3 - 16*w^2 - 25*w + 10],
[109, 109, -w^3 + 5*w],
[109, 109, -2*w^5 + w^4 + 12*w^3 - 9*w^2 - 11*w + 8],
[109, 109, 4*w^5 - 2*w^4 - 25*w^3 + 15*w^2 + 24*w - 9],
[125, 5, 2*w^5 - w^4 - 12*w^3 + 8*w^2 + 10*w - 7],
[149, 149, 3*w^5 - 2*w^4 - 19*w^3 + 14*w^2 + 20*w - 9],
[149, 149, 5*w^5 - 2*w^4 - 31*w^3 + 16*w^2 + 30*w - 9],
[149, 149, -2*w^5 + 12*w^3 - 3*w^2 - 10*w + 1],
[199, 199, -3*w^5 + 3*w^4 + 20*w^3 - 19*w^2 - 20*w + 10],
[199, 199, 2*w^5 - w^4 - 13*w^3 + 7*w^2 + 15*w - 2],
[199, 199, 3*w^5 - w^4 - 17*w^3 + 10*w^2 + 10*w - 5],
[211, 211, 9*w^5 - 5*w^4 - 57*w^3 + 38*w^2 + 56*w - 21],
[211, 211, -5*w^5 + 4*w^4 + 31*w^3 - 27*w^2 - 27*w + 12],
[211, 211, -7*w^5 + 3*w^4 + 43*w^3 - 26*w^2 - 39*w + 15],
[241, 241, 5*w^5 - 3*w^4 - 30*w^3 + 22*w^2 + 24*w - 12],
[241, 241, -5*w^5 + 3*w^4 + 32*w^3 - 21*w^2 - 32*w + 12],
[241, 241, -3*w^5 + w^4 + 17*w^3 - 10*w^2 - 12*w + 7],
[251, 251, -3*w^5 + 17*w^3 - 4*w^2 - 12*w + 2],
[251, 251, -8*w^5 + 5*w^4 + 49*w^3 - 38*w^2 - 42*w + 23],
[251, 251, -3*w^5 + 2*w^4 + 18*w^3 - 14*w^2 - 16*w + 7],
[269, 269, -w^4 + 6*w^2 - w - 3],
[269, 269, -w^5 + 6*w^3 - w^2 - 4*w - 1],
[269, 269, w^5 - w^4 - 6*w^3 + 7*w^2 + 3*w - 5],
[271, 271, -6*w^5 + 4*w^4 + 37*w^3 - 28*w^2 - 33*w + 14],
[271, 271, -3*w^5 + 2*w^4 + 19*w^3 - 14*w^2 - 20*w + 8],
[271, 271, 8*w^5 - 4*w^4 - 49*w^3 + 33*w^2 + 44*w - 21],
[311, 311, 6*w^5 - 4*w^4 - 37*w^3 + 29*w^2 + 34*w - 17],
[311, 311, -3*w^5 + w^4 + 18*w^3 - 8*w^2 - 15*w + 1],
[311, 311, -5*w^5 + 3*w^4 + 31*w^3 - 23*w^2 - 30*w + 13],
[311, 311, -6*w^5 + 3*w^4 + 36*w^3 - 25*w^2 - 29*w + 16],
[311, 311, 4*w^5 - 3*w^4 - 25*w^3 + 21*w^2 + 22*w - 9],
[311, 311, w^5 - 7*w^3 + 2*w^2 + 9*w - 1],
[331, 331, 6*w^5 - 3*w^4 - 37*w^3 + 24*w^2 + 32*w - 13],
[331, 331, 8*w^5 - 4*w^4 - 49*w^3 + 31*w^2 + 44*w - 17],
[331, 331, 7*w^5 - 4*w^4 - 44*w^3 + 30*w^2 + 42*w - 19],
[331, 331, -7*w^5 + 4*w^4 + 43*w^3 - 31*w^2 - 37*w + 18],
[331, 331, 6*w^5 - 3*w^4 - 38*w^3 + 24*w^2 + 38*w - 14],
[331, 331, 5*w^5 - 2*w^4 - 30*w^3 + 17*w^2 + 24*w - 11],
[349, 349, 3*w^5 - 3*w^4 - 19*w^3 + 19*w^2 + 17*w - 10],
[349, 349, 5*w^5 - 2*w^4 - 31*w^3 + 18*w^2 + 29*w - 13],
[349, 349, -12*w^5 + 6*w^4 + 74*w^3 - 46*w^2 - 67*w + 23],
[349, 349, -w^5 + w^4 + 6*w^3 - 8*w^2 - 4*w + 4],
[349, 349, -w^5 + w^4 + 7*w^3 - 6*w^2 - 10*w + 1],
[349, 349, 2*w^5 - 2*w^4 - 13*w^3 + 14*w^2 + 14*w - 12],
[389, 389, 8*w^5 - 4*w^4 - 49*w^3 + 33*w^2 + 43*w - 21],
[389, 389, -5*w^5 + 4*w^4 + 31*w^3 - 27*w^2 - 28*w + 14],
[389, 389, -2*w^5 + 2*w^4 + 12*w^3 - 14*w^2 - 9*w + 6],
[401, 401, w^3 + w^2 - 2*w - 3],
[401, 401, -6*w^5 + 2*w^4 + 37*w^3 - 18*w^2 - 35*w + 12],
[401, 401, 4*w^5 - 3*w^4 - 24*w^3 + 21*w^2 + 17*w - 12],
[409, 409, 8*w^5 - 4*w^4 - 49*w^3 + 31*w^2 + 44*w - 16],
[409, 409, 4*w^5 - w^4 - 25*w^3 + 10*w^2 + 26*w - 6],
[409, 409, -2*w^5 + 12*w^3 - 3*w^2 - 11*w + 6],
[421, 421, -7*w^5 + 5*w^4 + 43*w^3 - 36*w^2 - 37*w + 20],
[421, 421, -2*w^5 + 2*w^4 + 13*w^3 - 13*w^2 - 11*w + 5],
[421, 421, w^5 - 2*w^4 - 6*w^3 + 12*w^2 + 3*w - 8],
[431, 431, -4*w^5 + 4*w^4 + 26*w^3 - 26*w^2 - 25*w + 14],
[431, 431, 7*w^5 - 4*w^4 - 44*w^3 + 31*w^2 + 41*w - 20],
[431, 431, -6*w^5 + 4*w^4 + 37*w^3 - 29*w^2 - 32*w + 14],
[439, 439, 3*w^5 - w^4 - 19*w^3 + 9*w^2 + 18*w - 7],
[439, 439, 4*w^5 - 2*w^4 - 26*w^3 + 15*w^2 + 28*w - 11],
[439, 439, w^5 - w^4 - 7*w^3 + 8*w^2 + 8*w - 7],
[461, 461, 5*w^5 - 3*w^4 - 31*w^3 + 22*w^2 + 30*w - 13],
[461, 461, -3*w^5 + 3*w^4 + 19*w^3 - 19*w^2 - 16*w + 7],
[461, 461, 5*w^5 - 3*w^4 - 30*w^3 + 24*w^2 + 24*w - 16],
[479, 479, 5*w^5 - 3*w^4 - 30*w^3 + 22*w^2 + 23*w - 11],
[479, 479, -4*w^5 + 3*w^4 + 25*w^3 - 21*w^2 - 21*w + 9],
[479, 479, -3*w^5 + w^4 + 19*w^3 - 9*w^2 - 18*w + 8],
[491, 491, 6*w^5 - 4*w^4 - 38*w^3 + 28*w^2 + 35*w - 14],
[491, 491, 5*w^5 - 2*w^4 - 31*w^3 + 18*w^2 + 28*w - 13],
[491, 491, w^5 + w^4 - 6*w^3 - 3*w^2 + 8*w],
[541, 541, -w^4 - w^3 + 6*w^2 + 2*w - 5],
[541, 541, -5*w^5 + 4*w^4 + 31*w^3 - 28*w^2 - 28*w + 16],
[541, 541, 5*w^5 - 2*w^4 - 30*w^3 + 18*w^2 + 24*w - 11],
[619, 619, w^3 - 6*w],
[619, 619, -w^5 + w^4 + 6*w^3 - 8*w^2 - 6*w + 8],
[619, 619, 3*w^5 - w^4 - 19*w^3 + 8*w^2 + 20*w - 5],
[619, 619, -2*w^5 + w^4 + 13*w^3 - 8*w^2 - 14*w + 2],
[619, 619, -4*w^5 + 3*w^4 + 26*w^3 - 21*w^2 - 25*w + 11],
[619, 619, -6*w^5 + 2*w^4 + 37*w^3 - 19*w^2 - 34*w + 14],
[631, 631, 6*w^5 - 4*w^4 - 38*w^3 + 29*w^2 + 35*w - 16],
[631, 631, -4*w^5 + w^4 + 25*w^3 - 11*w^2 - 24*w + 9],
[631, 631, -2*w^5 + 3*w^4 + 13*w^3 - 18*w^2 - 11*w + 10],
[641, 641, -3*w^5 + 2*w^4 + 20*w^3 - 14*w^2 - 23*w + 10],
[641, 641, -6*w^5 + 3*w^4 + 37*w^3 - 24*w^2 - 32*w + 14],
[641, 641, -4*w^5 + 2*w^4 + 25*w^3 - 17*w^2 - 25*w + 11],
[661, 661, 2*w^5 - w^4 - 13*w^3 + 9*w^2 + 15*w - 6],
[661, 661, -4*w^5 + 3*w^4 + 25*w^3 - 22*w^2 - 22*w + 12],
[661, 661, -5*w^5 + 3*w^4 + 32*w^3 - 22*w^2 - 33*w + 15],
[709, 709, 2*w^5 - 2*w^4 - 12*w^3 + 13*w^2 + 10*w - 5],
[709, 709, 5*w^5 - 3*w^4 - 31*w^3 + 23*w^2 + 26*w - 14],
[709, 709, 7*w^5 - 4*w^4 - 43*w^3 + 31*w^2 + 37*w - 17],
[719, 719, 6*w^5 - 4*w^4 - 38*w^3 + 29*w^2 + 38*w - 16],
[719, 719, -6*w^5 + 4*w^4 + 37*w^3 - 28*w^2 - 31*w + 14],
[719, 719, -6*w^5 + 2*w^4 + 37*w^3 - 19*w^2 - 36*w + 13],
[729, 3, -3],
[739, 739, 11*w^5 - 5*w^4 - 69*w^3 + 41*w^2 + 67*w - 24],
[739, 739, -9*w^5 + 5*w^4 + 56*w^3 - 38*w^2 - 51*w + 18],
[739, 739, 8*w^5 - 4*w^4 - 49*w^3 + 31*w^2 + 43*w - 17],
[751, 751, -7*w^5 + 4*w^4 + 44*w^3 - 31*w^2 - 43*w + 19],
[751, 751, -7*w^5 + 5*w^4 + 44*w^3 - 35*w^2 - 40*w + 18],
[751, 751, -7*w^5 + 5*w^4 + 44*w^3 - 36*w^2 - 41*w + 20],
[761, 761, -3*w^5 + 2*w^4 + 19*w^3 - 16*w^2 - 20*w + 11],
[761, 761, 7*w^5 - 4*w^4 - 42*w^3 + 31*w^2 + 33*w - 17],
[761, 761, 3*w^5 - 2*w^4 - 20*w^3 + 14*w^2 + 24*w - 10],
[761, 761, -6*w^5 + 2*w^4 + 36*w^3 - 18*w^2 - 31*w + 9],
[761, 761, -8*w^5 + 5*w^4 + 50*w^3 - 36*w^2 - 45*w + 21],
[761, 761, 3*w^5 - 2*w^4 - 20*w^3 + 15*w^2 + 23*w - 10],
[769, 769, 5*w^5 - 3*w^4 - 30*w^3 + 22*w^2 + 22*w - 10],
[769, 769, w^5 - w^4 - 5*w^3 + 8*w^2 + 2*w - 5],
[769, 769, -6*w^5 + 2*w^4 + 37*w^3 - 18*w^2 - 36*w + 10],
[811, 811, 5*w^5 - w^4 - 30*w^3 + 13*w^2 + 26*w - 10],
[811, 811, -3*w^5 + 2*w^4 + 20*w^3 - 15*w^2 - 21*w + 10],
[811, 811, -8*w^5 + 6*w^4 + 50*w^3 - 42*w^2 - 45*w + 24],
[821, 821, 8*w^5 - 4*w^4 - 50*w^3 + 32*w^2 + 47*w - 21],
[821, 821, -2*w^5 + 13*w^3 - 3*w^2 - 16*w + 2],
[821, 821, -6*w^5 + 3*w^4 + 38*w^3 - 23*w^2 - 37*w + 15],
[829, 829, -9*w^5 + 5*w^4 + 56*w^3 - 38*w^2 - 51*w + 24],
[829, 829, -5*w^5 + 3*w^4 + 30*w^3 - 22*w^2 - 23*w + 13],
[829, 829, -5*w^5 + 2*w^4 + 32*w^3 - 17*w^2 - 33*w + 13],
[829, 829, 6*w^5 - 4*w^4 - 38*w^3 + 28*w^2 + 35*w - 13],
[829, 829, -2*w^5 + w^4 + 11*w^3 - 9*w^2 - 7*w + 8],
[829, 829, 7*w^5 - 3*w^4 - 43*w^3 + 25*w^2 + 40*w - 17],
[859, 859, 3*w^5 - w^4 - 19*w^3 + 9*w^2 + 20*w - 9],
[859, 859, -11*w^5 + 6*w^4 + 68*w^3 - 45*w^2 - 61*w + 23],
[859, 859, -9*w^5 + 4*w^4 + 55*w^3 - 32*w^2 - 49*w + 15],
[919, 919, -2*w^5 + 2*w^4 + 11*w^3 - 14*w^2 - 5*w + 8],
[919, 919, -5*w^5 + 3*w^4 + 32*w^3 - 22*w^2 - 32*w + 16],
[919, 919, 4*w^5 - 3*w^4 - 24*w^3 + 21*w^2 + 19*w - 9],
[919, 919, -8*w^5 + 4*w^4 + 49*w^3 - 32*w^2 - 43*w + 17],
[919, 919, 3*w^5 - w^4 - 18*w^3 + 11*w^2 + 17*w - 11],
[919, 919, -7*w^5 + 4*w^4 + 44*w^3 - 29*w^2 - 42*w + 17],
[929, 929, 7*w^5 - 4*w^4 - 43*w^3 + 30*w^2 + 39*w - 15],
[929, 929, 6*w^5 - 3*w^4 - 36*w^3 + 25*w^2 + 30*w - 14],
[929, 929, -3*w^5 + w^4 + 17*w^3 - 9*w^2 - 11*w + 2],
[941, 941, 7*w^5 - 4*w^4 - 42*w^3 + 31*w^2 + 34*w - 19],
[941, 941, 6*w^5 - 3*w^4 - 36*w^3 + 24*w^2 + 31*w - 15],
[941, 941, 7*w^5 - 3*w^4 - 42*w^3 + 25*w^2 + 35*w - 15]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

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

heckeEigenvaluesArray := [-2, 4, -6, 10, 1, -2, -10, -2, -12, 0, 4, 2, -10, -14, -16, -4, 0, -4, 4, 0, -6, -6, 14, -10, -2, 14, -14, -10, -10, 18, -6, -22, -18, -16, -16, -8, 28, 20, -8, 14, -10, -2, -28, -12, 4, -30, 18, -6, -24, 0, -4, 4, 16, -8, -32, 20, 32, 28, -20, -12, -16, 24, 20, 30, -14, 14, 18, -18, 34, 18, 26, 26, -14, 18, 18, 38, -22, 2, -6, -6, -22, -24, -24, 16, -32, -20, 24, -26, 38, 14, -16, 32, -4, 20, -16, -4, -18, 38, -26, 16, 20, -20, -20, 20, -44, -16, 16, 32, 46, -34, -18, 22, -46, -42, -18, 30, -26, -48, -8, -48, -34, -28, 16, 12, 8, 16, 4, -38, -14, 10, 18, 34, 42, -14, 34, 18, 48, 20, 20, 2, 10, 38, -50, -14, 30, -30, -2, -6, -28, 8, -28, 24, 16, -32, 16, -20, 44, -18, -42, -30, -18, -2, 30];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {29,29,2*w^5 - w^4 - 13*w^3 + 8*w^2 + 15*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;