/*
  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![7, 0, -6, 0, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {17,17,w^3 - 3*w - 1}>;

primesArray := [
[2, 2, w + 1],
[7, 7, w],
[9, 3, w^2 + w - 1],
[9, 3, w^2 - w - 1],
[17, 17, -w^3 + 3*w - 1],
[17, 17, w^3 - 3*w - 1],
[23, 23, -w^3 - w^2 + 3*w + 4],
[23, 23, w^3 - w^2 - 3*w + 4],
[41, 41, w^3 - w^2 - 2*w + 3],
[41, 41, -w^3 - w^2 + 2*w + 3],
[49, 7, w^2 - 6],
[71, 71, w^3 + 3*w^2 - 3*w - 6],
[71, 71, -2*w^2 - 2*w + 1],
[73, 73, w^2 - 2*w - 2],
[73, 73, w^2 + 2*w - 2],
[79, 79, w^3 - w^2 - 5*w + 2],
[79, 79, -w^3 - w^2 + 5*w + 2],
[89, 89, 2*w - 1],
[89, 89, -2*w - 1],
[97, 97, -w^3 + w^2 + 4*w - 1],
[97, 97, w^3 + w^2 - 4*w - 1],
[103, 103, -2*w^3 + w^2 + 8*w - 2],
[103, 103, w^3 - w^2 - 3*w + 6],
[103, 103, -w^3 - w^2 + 3*w + 6],
[103, 103, 2*w^3 + w^2 - 8*w - 2],
[127, 127, w^3 + 3*w^2 - 4*w - 11],
[127, 127, -w^3 + 3*w^2 + 4*w - 11],
[137, 137, -w^3 - 3*w^2 + 2*w + 9],
[137, 137, -2*w^2 + w + 10],
[137, 137, -2*w^2 - w + 10],
[137, 137, w^3 - 3*w^2 - 2*w + 9],
[151, 151, 3*w^2 - 2*w - 10],
[151, 151, w^3 + 3*w^2 - 4*w - 9],
[167, 167, w^3 + 4*w^2 - 2*w - 8],
[167, 167, 2*w^3 - 8*w - 3],
[167, 167, 2*w^3 - 8*w + 3],
[167, 167, 2*w^3 + 5*w^2 - 7*w - 13],
[191, 191, -w^3 + 6*w - 2],
[191, 191, w^3 - 6*w - 2],
[193, 193, -2*w^3 + 2*w^2 + 6*w - 5],
[193, 193, w^3 + 2*w^2 - 4*w - 4],
[193, 193, -w^3 + 2*w^2 + 4*w - 4],
[193, 193, -2*w^3 - 2*w^2 + 6*w + 5],
[199, 199, 2*w^2 - 3*w - 6],
[199, 199, w^3 + w^2 + w + 2],
[199, 199, 2*w^3 + 2*w^2 - 4*w - 5],
[199, 199, 2*w^2 + 3*w - 6],
[223, 223, 2*w^3 + w^2 - 6*w - 6],
[223, 223, w^3 - w^2 - 5*w - 2],
[223, 223, 2*w^3 + 4*w^2 - 6*w - 9],
[223, 223, -2*w^3 + w^2 + 6*w - 6],
[239, 239, -2*w^3 - 5*w^2 + 5*w + 11],
[239, 239, w^3 + 4*w^2 - 4*w - 10],
[241, 241, -3*w^3 - 4*w^2 + 9*w + 11],
[241, 241, -2*w^3 - w^2 + 6*w + 4],
[257, 257, 2*w^3 + 5*w^2 - 6*w - 12],
[257, 257, -w^3 - 4*w^2 + 3*w + 9],
[263, 263, -w^3 - w^2 + 6*w + 3],
[263, 263, w^3 - w^2 - 6*w + 3],
[271, 271, -2*w^3 - w^2 + 7*w + 3],
[271, 271, -w^3 + 2*w^2 + 3*w - 9],
[271, 271, w^3 + 2*w^2 - 3*w - 9],
[271, 271, 2*w^3 - w^2 - 7*w + 3],
[289, 17, 3*w^2 - 8],
[313, 313, -w^3 + 2*w^2 - 6],
[313, 313, w^3 + 2*w^2 - 6],
[353, 353, -w^3 + 3*w - 5],
[353, 353, w^3 - 3*w - 5],
[359, 359, w^3 - w + 3],
[359, 359, 4*w^3 + 5*w^2 - 14*w - 18],
[361, 19, -w^3 + 2*w^2 + 4*w - 2],
[361, 19, w^3 + 2*w^2 - 4*w - 2],
[367, 367, -2*w^3 - 3*w^2 + 6*w + 10],
[367, 367, -3*w - 2],
[367, 367, 3*w - 2],
[367, 367, w^3 + 2*w^2 - 7*w - 11],
[401, 401, -2*w^3 + 6*w - 1],
[401, 401, -3*w^3 + 3*w^2 + 11*w - 10],
[401, 401, 3*w^3 + 3*w^2 - 11*w - 10],
[401, 401, 2*w^3 - 6*w - 1],
[409, 409, w^3 - 3*w^2 - 6*w + 13],
[409, 409, -w^3 - 3*w^2 + 6*w + 13],
[431, 431, w^3 - 4*w^2 - 2*w + 10],
[431, 431, -w^3 - 4*w^2 + 2*w + 10],
[433, 433, w^3 - w^2 - 6*w - 3],
[433, 433, 2*w^3 + 4*w^2 - 7*w - 10],
[463, 463, w^2 + 3*w - 3],
[463, 463, w^2 - 3*w - 3],
[487, 487, 2*w^2 + 2*w - 9],
[487, 487, 2*w^2 - 2*w - 9],
[521, 521, -w^3 - 4*w^2 + 5*w + 13],
[521, 521, w^3 - 4*w^2 - 5*w + 13],
[529, 23, -w^2 - 2],
[577, 577, w^3 + 2*w^2 - 2*w - 8],
[577, 577, -w^3 + 2*w^2 + 2*w - 8],
[593, 593, -2*w^3 - 3*w^2 + 5*w + 9],
[593, 593, w^3 + 2*w^2 - 8*w - 12],
[599, 599, w^3 + 2*w^2 - 5*w - 3],
[599, 599, -w^3 + 2*w^2 + 5*w - 3],
[601, 601, -w^3 + 4*w^2 + 4*w - 8],
[601, 601, w^3 + 4*w^2 - 4*w - 8],
[607, 607, -w^3 + 8*w + 6],
[607, 607, -3*w^3 - 5*w^2 + 7*w + 12],
[607, 607, 2*w^2 - 4*w - 9],
[607, 607, -2*w^3 - 3*w^2 + 11*w + 13],
[617, 617, 3*w^3 + 2*w^2 - 11*w - 9],
[617, 617, -2*w^3 + w^2 + 9*w - 1],
[617, 617, 2*w^3 + w^2 - 9*w - 1],
[617, 617, 3*w^3 - 12*w - 4],
[625, 5, -5],
[631, 631, w^3 - w^2 - 7*w - 4],
[631, 631, 2*w^3 + 4*w^2 - 8*w - 11],
[641, 641, -2*w^3 + 3*w^2 + 9*w - 9],
[641, 641, w^3 - 2*w^2 - 7*w + 5],
[641, 641, w^3 + 2*w^2 - 7*w - 5],
[641, 641, 2*w^3 + 3*w^2 - 9*w - 9],
[647, 647, -w^3 + 3*w^2 - 1],
[647, 647, 2*w^3 - 3*w^2 - 8*w + 8],
[647, 647, -2*w^3 - 3*w^2 + 8*w + 8],
[647, 647, w^3 + 3*w^2 - 1],
[727, 727, -3*w^3 - 2*w^2 + 11*w + 3],
[727, 727, -2*w^3 + 5*w^2 + 9*w - 17],
[727, 727, 2*w^3 + 5*w^2 - 9*w - 17],
[727, 727, 3*w^3 - 2*w^2 - 11*w + 3],
[743, 743, 3*w^2 - 2*w - 12],
[743, 743, 3*w^2 + 2*w - 12],
[751, 751, 2*w^3 - 2*w^2 - 6*w + 9],
[751, 751, -2*w^3 - 2*w^2 + 6*w + 9],
[761, 761, -3*w^3 + 5*w^2 + 6*w - 11],
[761, 761, 3*w^3 + 5*w^2 - 6*w - 11],
[769, 769, w^3 - 7*w - 3],
[769, 769, -w^3 + 7*w - 3],
[823, 823, w^3 + 2*w^2 - 5*w - 1],
[823, 823, w^3 - 2*w^2 - 5*w + 1],
[839, 839, 4*w^2 + w - 12],
[839, 839, -3*w^3 + w^2 + 13*w - 8],
[839, 839, 3*w^3 + w^2 - 13*w - 8],
[839, 839, 4*w^2 - w - 12],
[841, 29, 2*w^3 + 4*w^2 - 7*w - 16],
[841, 29, -2*w^3 + 4*w^2 + 7*w - 16],
[857, 857, 3*w^2 - 3*w - 11],
[857, 857, 3*w^2 + 3*w - 11],
[863, 863, w^3 + 3*w^2 - 5*w - 6],
[863, 863, -w^3 + 3*w^2 + 5*w - 6],
[881, 881, w^3 - w^2 - 5*w - 4],
[881, 881, 2*w^2 + 4*w + 3],
[887, 887, w^2 - w - 9],
[887, 887, 3*w^3 + 4*w^2 - 11*w - 11],
[887, 887, -3*w^3 + 4*w^2 + 11*w - 11],
[887, 887, w^2 + w - 9],
[911, 911, w^2 - 4*w - 8],
[911, 911, -3*w^3 - 4*w^2 + 9*w + 13],
[919, 919, 2*w^3 - w^2 - 8*w - 2],
[919, 919, -2*w^3 - w^2 + 8*w - 2],
[929, 929, 4*w^3 + w^2 - 16*w - 2],
[929, 929, 2*w^3 - 11*w - 6],
[937, 937, 3*w^3 - 3*w^2 - 10*w + 5],
[937, 937, -3*w^3 - 3*w^2 + 10*w + 5],
[953, 953, w^3 - 9*w - 11],
[953, 953, 5*w^2 + 3*w - 17],
[953, 953, 5*w^2 - 3*w - 17],
[953, 953, -2*w^3 - w^2 + 8*w + 10],
[961, 31, 4*w^2 - 11],
[961, 31, 4*w^2 - 13],
[967, 967, w^3 - w^2 - 6*w - 5],
[967, 967, 2*w^2 + 3*w + 2],
[983, 983, -w^2 + w - 3],
[983, 983, 2*w^3 - 2*w^2 - 11*w + 10],
[983, 983, 2*w^3 + 2*w^2 - 11*w - 10],
[983, 983, -w^2 - w - 3],
[991, 991, w^3 + 3*w^2 - 10*w - 15],
[991, 991, -2*w^3 - 4*w^2 + w + 6]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^4 - 6*x^2 + 2*x + 5;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, -e^3 - 2*e^2 + 4*e + 3, e^3 + 2*e^2 - 3*e - 8, -2*e^3 - 3*e^2 + 7*e + 6, 2*e^3 + 3*e^2 - 9*e - 6, 1, 3*e^3 + 3*e^2 - 10*e - 6, e^3 + 3*e^2 - 5*e - 9, -e^3 - 3*e^2 + 5*e + 7, 2*e^2 - 10, 2*e^3 - 2*e^2 - 9*e + 7, e^3 + e^2 - 9*e - 1, -e^3 - 2*e^2 + e, -4*e^3 - 2*e^2 + 18*e - 2, 3*e^3 + 6*e^2 - 7*e - 14, 3*e^3 + 8*e^2 - 11*e - 18, -3*e^3 - 6*e^2 + 5*e + 16, -2*e^3 - 4*e^2 + 2*e + 8, -e^3 - 2*e^2 + 2*e - 5, 5*e^3 + 8*e^2 - 15*e - 20, -2*e^3 - 4*e^2 + 6*e - 2, -e^3 + 7*e - 4, 3*e^3 + 3*e^2 - 15*e - 1, -6*e^3 - 4*e^2 + 24*e + 6, 5*e^3 + 2*e^2 - 22*e + 3, -3*e^2 + 4*e + 19, -3*e^3 - 5*e^2 + 10*e + 6, -4*e^2 - 4*e + 8, 2*e^3 + 4*e^2 - 10*e - 12, 3*e^3 - e^2 - 10*e + 8, 2*e^3 - e^2 - 14*e + 7, -6*e^3 - 12*e^2 + 20*e + 28, 4*e^3 + 4*e^2 - 20*e - 2, -5*e^3 - 3*e^2 + 20*e - 2, e^3 + 4*e^2 + 3*e - 10, -6*e^3 - 8*e^2 + 24*e + 8, -8*e^3 - 8*e^2 + 30*e + 12, 2*e^3 + 11*e^2 - 5*e - 36, e^2 + 5*e - 2, 4*e^3 - 20*e - 2, e^3 + 7*e^2 - 3*e - 21, -5*e^3 - 10*e^2 + 18*e + 17, -e^3 + 8*e + 1, -6*e^3 - 10*e^2 + 20*e + 18, 8*e^3 + 8*e^2 - 34*e - 14, -2*e^3 + 14*e + 4, -6*e^3 - 2*e^2 + 28*e + 6, 5*e^3 + 9*e^2 - 15*e - 27, -3*e^3 + 2*e^2 + 20*e - 13, e^3 - 8*e - 3, -3*e^3 + 7*e^2 + 18*e - 14, -4*e + 6, 4*e^3 + 3*e^2 - 18*e - 15, 5*e^3 + 6*e^2 - 27*e - 10, -3*e^3 - 2*e^2 + 17*e - 12, -4*e^3 - 4*e^2 + 14*e + 20, 4*e^3 + e^2 - 24*e + 3, 9*e^3 + 8*e^2 - 33*e - 22, 5*e^3 + 8*e^2 - 24*e - 5, e^3 - 2*e^2 - 5*e + 16, -e^3 - e^2 + 10*e - 12, -e^3 - 13*e^2 - 3*e + 29, -4*e^3 - 2*e^2 + 24*e - 4, -9*e^3 - 7*e^2 + 40*e + 2, 10*e^3 + 10*e^2 - 39*e - 17, -e^3 - 3*e^2 + 14*e + 18, -2*e^3 + 10*e^2 + 17*e - 23, 8*e^3 + 4*e^2 - 30*e + 2, -6*e^3 - 2*e^2 + 20*e + 2, 6*e^3 + 18*e^2 - 20*e - 54, 16*e^3 + 12*e^2 - 58*e - 10, -11*e^3 - 18*e^2 + 42*e + 25, -8*e^3 - 14*e^2 + 30*e + 28, -3*e^3 - 6*e^2 + 12*e + 19, 5*e^3 + 14*e^2 - 18*e - 43, -4*e^3 - 8*e^2 + 10*e + 26, -2*e^3 - 10*e^2 - 2*e + 20, -e^3 + 6*e^2 - 3*e - 42, 8*e^3 + 8*e^2 - 24*e - 20, -e^3 - 8*e^2 - 2*e + 7, -e^3 - 5*e^2 + 6*e - 4, 2*e^3 + 14*e^2 - 6*e - 38, 2*e^3 - 8*e^2 - 12*e + 16, -11*e^3 - 7*e^2 + 45*e + 5, 9*e^3 + 8*e^2 - 23*e - 8, -6*e^3 - 4*e^2 + 18*e - 2, 5*e^3 - 2*e^2 - 19*e + 22, 5*e^3 + 2*e^2 - 14*e - 7, -8*e^3 - 15*e^2 + 22*e + 33, 12*e^3 + 20*e^2 - 30*e - 56, 10*e^2 + 4*e - 22, 18*e^3 + 18*e^2 - 64*e - 28, -10*e^3 - 12*e^2 + 42*e + 2, -3*e^3 - 17*e^2 + 11*e + 39, -18*e^3 - 13*e^2 + 68*e + 21, -10*e^3 - 14*e^2 + 41*e + 5, e^3 + 7*e^2 - 4*e - 42, -e^3 + 2*e^2 - 2*e - 15, -5*e^3 + 21*e + 4, 3*e^3 - 5*e^2 - 17*e - 7, -7*e^3 - 14*e^2 + 21*e + 36, 2*e^3 - 5*e^2 - 21*e + 8, 6*e^3 + 10*e^2 - 26*e - 16, -2*e^3 - e^2 + 22*e + 9, -3*e^3 + 10*e^2 + 15*e - 34, -2*e^3 - 24, 3*e^3 + e^2 - 15*e + 9, -7*e^3 - 4*e^2 + 29*e, -9*e^3 - 10*e^2 + 37*e + 36, -14*e^3 - 12*e^2 + 60*e + 16, 7*e^3 + 10*e^2 - 27*e + 14, 3*e^3 - 4*e^2 - 15*e + 28, 3*e^3 + 11*e^2 - 9*e - 41, e^3 + 3*e^2 + 13*e + 3, -3*e^3 + 6*e^2 + 19*e - 30, -9*e^2 - 7*e + 18, 2*e^3 - 10*e^2 - 18*e + 28, 9*e^3 + 14*e^2 - 25*e - 24, 7*e^3 - 4*e^2 - 23*e + 22, -8*e^3 - 3*e^2 + 38*e + 1, 9*e^3 + 6*e^2 - 30*e + 11, 15*e^3 + 15*e^2 - 55*e - 41, 4*e^3 + 14*e^2 + 11*e - 47, -4*e^3 - 9*e^2 + 11*e + 24, 10*e^2 + 3*e - 7, -14*e^3 - 6*e^2 + 54*e - 6, 8*e^3 + 13*e^2 - 22*e - 33, -14*e^3 - 20*e^2 + 53*e + 49, 2*e^3 + 12*e^2 - 16*e - 30, -13*e^3 - 19*e^2 + 44*e + 48, -9*e^3 - 5*e^2 + 35*e - 5, 11*e^3 + 6*e^2 - 43*e - 12, -6*e^3 + 2*e^2 + 36*e - 6, 7*e^3 + 14*e^2 - 31*e - 36, -2*e^3 + 6*e^2 + 4*e - 44, 3*e^3 + 9*e^2 + 5*e - 21, 8*e^3 + 6*e^2 - 50*e - 18, -6*e^3 - 18*e^2 + 20*e + 30, 12*e^3 + 18*e^2 - 46*e - 30, -8*e^3 - 8*e^2 + 25*e - 1, -6*e^3 - 8*e^2 + 20*e + 18, -8*e^3 - 4*e^2 + 28*e + 32, 10*e^3 + 10*e^2 - 42*e + 12, 3*e^3 + 3*e^2 - 7*e - 1, 4*e^3 + 8*e^2 + 4*e - 24, 4*e^3 + 2*e^2 - 20*e + 32, -14*e^3 - 22*e^2 + 50*e + 48, -4*e^3 - 11*e^2 + 11*e + 58, -3*e^3 - 4*e^2 + 25*e + 14, 9*e^3 + 17*e^2 - 22*e - 30, -2*e^3 - 16*e^2 - 16*e + 42, 6*e^3 - e^2 - 28*e + 9, 23*e^3 + 26*e^2 - 78*e - 59, 13*e^3 + 18*e^2 - 30*e - 41, -6*e^3 - 8*e^2 + 18*e + 30, 18*e^3 + 11*e^2 - 61*e - 6, -2*e^3 + 16*e - 30, 11*e^3 + 12*e^2 - 54*e + 9, -7*e^3 + 11*e^2 + 37*e - 37, -2*e^3 - 6*e^2 + 5*e + 19, 2*e^3 + 14*e^2 - 15*e - 49, -5*e^3 - 15*e^2 + 21*e + 27, 4*e^3 - 4*e^2 - 26*e, -8*e^3 - 10*e^2 + 46*e + 14, 7*e^3 + 31*e^2 - 22*e - 94, 6*e^3 + 10*e^2 - 2*e - 32, -4*e^2 + 16*e + 26, 4*e^3 - 14*e^2 - 18*e + 42, -7*e^3 - 3*e^2 + 29*e - 3, -22*e^3 - 39*e^2 + 74*e + 89, -7*e^3 - 2*e^2 + 25*e - 8, e^3 - 6*e^2 - e + 56];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {17,17,w^3 - 3*w - 1}>] := -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;