/*
  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![9, 6, -7, -2, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {7,7,w - 1}>;

primesArray := [
[4, 2, -1/3*w^3 + 2/3*w^2 + 4/3*w - 1],
[7, 7, 1/3*w^3 - 2/3*w^2 - 7/3*w + 1],
[7, 7, w - 2],
[7, 7, w - 1],
[9, 3, w],
[9, 3, -1/3*w^3 + 2/3*w^2 + 7/3*w - 2],
[23, 23, -1/3*w^3 + 2/3*w^2 + 1/3*w - 2],
[23, 23, -2/3*w^3 + 4/3*w^2 + 11/3*w - 4],
[31, 31, -2/3*w^3 + 1/3*w^2 + 14/3*w + 2],
[41, 41, -1/3*w^3 + 5/3*w^2 + 4/3*w - 7],
[41, 41, -1/3*w^3 + 5/3*w^2 + 4/3*w - 4],
[47, 47, 1/3*w^3 - 5/3*w^2 - 1/3*w + 5],
[47, 47, w^2 - w - 4],
[73, 73, -w^3 + 3*w^2 + 4*w - 8],
[73, 73, -2/3*w^3 + 7/3*w^2 - 1/3*w - 4],
[73, 73, -w^3 + w^2 + 7*w + 1],
[73, 73, 2/3*w^3 - 4/3*w^2 - 14/3*w + 5],
[79, 79, w^2 - 5],
[79, 79, -2/3*w^3 + 7/3*w^2 + 8/3*w - 6],
[97, 97, -2/3*w^3 + 7/3*w^2 + 5/3*w - 4],
[97, 97, 1/3*w^3 + 1/3*w^2 - 7/3*w - 5],
[103, 103, w^3 - 3*w^2 - 2*w + 5],
[103, 103, -4/3*w^3 + 11/3*w^2 + 13/3*w - 10],
[113, 113, -1/3*w^3 + 5/3*w^2 + 4/3*w - 3],
[113, 113, w^3 - 2*w^2 - 5*w + 1],
[113, 113, -1/3*w^3 + 5/3*w^2 + 4/3*w - 1],
[113, 113, -1/3*w^3 + 5/3*w^2 + 4/3*w - 10],
[127, 127, -2/3*w^3 + 1/3*w^2 + 14/3*w + 1],
[127, 127, -2/3*w^3 + 7/3*w^2 + 2/3*w - 6],
[137, 137, -1/3*w^3 + 5/3*w^2 - 2/3*w - 6],
[137, 137, 1/3*w^3 + 1/3*w^2 - 10/3*w - 1],
[151, 151, -w^3 + 2*w^2 + 3*w - 5],
[151, 151, 1/3*w^3 + 1/3*w^2 - 10/3*w - 7],
[169, 13, -2/3*w^3 + 7/3*w^2 + 5/3*w - 2],
[169, 13, 1/3*w^3 + 1/3*w^2 - 7/3*w - 7],
[191, 191, 1/3*w^3 - 2/3*w^2 + 2/3*w - 2],
[191, 191, 2*w - 5],
[191, 191, -w^3 + 2*w^2 + 6*w - 2],
[191, 191, 2/3*w^3 - 4/3*w^2 - 14/3*w - 1],
[239, 239, 1/3*w^3 - 2/3*w^2 - 7/3*w - 3],
[239, 239, w - 5],
[241, 241, 1/3*w^3 + 1/3*w^2 - 10/3*w],
[241, 241, 4/3*w^3 - 17/3*w^2 + 2/3*w + 9],
[241, 241, -1/3*w^3 + 5/3*w^2 - 2/3*w - 7],
[241, 241, -4/3*w^3 + 14/3*w^2 + 13/3*w - 14],
[257, 257, -w - 4],
[257, 257, -1/3*w^3 + 2/3*w^2 + 7/3*w - 6],
[263, 263, -4/3*w^3 + 11/3*w^2 + 16/3*w - 8],
[263, 263, 2/3*w^3 - 1/3*w^2 - 8/3*w - 3],
[281, 281, -1/3*w^3 - 1/3*w^2 + 10/3*w - 2],
[281, 281, -1/3*w^3 + 5/3*w^2 - 5/3*w - 5],
[281, 281, -1/3*w^3 + 5/3*w^2 - 2/3*w - 9],
[281, 281, -2/3*w^3 + 1/3*w^2 + 17/3*w],
[289, 17, -w^3 + 2*w^2 + 4*w - 4],
[289, 17, w^3 - 2*w^2 - 4*w + 2],
[311, 311, -2/3*w^3 + 10/3*w^2 - 4/3*w - 9],
[311, 311, -2/3*w^3 - 2/3*w^2 + 20/3*w + 5],
[313, 313, -1/3*w^3 + 5/3*w^2 - 2/3*w - 8],
[313, 313, -1/3*w^3 - 1/3*w^2 + 10/3*w - 1],
[313, 313, w^3 - 4*w^2 - 2*w + 10],
[313, 313, 1/3*w^3 + 4/3*w^2 - 10/3*w - 8],
[337, 337, -4/3*w^3 + 5/3*w^2 + 19/3*w + 4],
[337, 337, 5/3*w^3 - 13/3*w^2 - 17/3*w + 5],
[353, 353, -4/3*w^3 + 8/3*w^2 + 25/3*w - 3],
[353, 353, -1/3*w^3 - 1/3*w^2 + 16/3*w + 6],
[353, 353, -2/3*w^3 + 10/3*w^2 + 2/3*w - 13],
[353, 353, -5/3*w^3 + 7/3*w^2 + 29/3*w - 3],
[367, 367, -5/3*w^3 + 4/3*w^2 + 35/3*w + 4],
[367, 367, 2/3*w^3 - 7/3*w^2 - 8/3*w + 4],
[367, 367, -4/3*w^3 + 14/3*w^2 + 1/3*w - 8],
[367, 367, w^2 - 7],
[383, 383, 4/3*w^3 - 14/3*w^2 - 10/3*w + 15],
[383, 383, -5/3*w^3 + 13/3*w^2 + 20/3*w - 12],
[383, 383, 2*w^2 - 2*w - 13],
[383, 383, w^3 - w^2 - 8*w - 1],
[401, 401, -4/3*w^3 + 11/3*w^2 + 22/3*w - 11],
[401, 401, 5/3*w^3 - 10/3*w^2 - 29/3*w + 5],
[409, 409, -w^3 + 2*w^2 + 3*w - 2],
[409, 409, -1/3*w^3 + 8/3*w^2 - 2/3*w - 6],
[409, 409, 1/3*w^3 - 8/3*w^2 + 2/3*w + 12],
[409, 409, 4/3*w^3 - 8/3*w^2 - 19/3*w + 4],
[433, 433, -w^3 + 3*w^2 - 1],
[433, 433, -1/3*w^3 + 5/3*w^2 - 8/3*w - 5],
[433, 433, -w^3 + w^2 + 8*w - 2],
[433, 433, 5/3*w^3 - 7/3*w^2 - 32/3*w - 2],
[439, 439, w^2 - 10],
[439, 439, -2/3*w^3 + 4/3*w^2 + 17/3*w - 7],
[449, 449, -2/3*w^3 + 4/3*w^2 + 11/3*w - 8],
[449, 449, -4/3*w^3 + 5/3*w^2 + 22/3*w + 2],
[457, 457, 2/3*w^3 - 7/3*w^2 - 14/3*w + 6],
[457, 457, 2/3*w^3 - 7/3*w^2 - 14/3*w + 9],
[463, 463, w^3 - 3*w^2 - 6*w + 11],
[463, 463, -1/3*w^3 + 5/3*w^2 + 10/3*w - 4],
[479, 479, -1/3*w^3 + 5/3*w^2 - 2/3*w + 1],
[479, 479, 5/3*w^3 - 10/3*w^2 - 23/3*w + 9],
[487, 487, -2/3*w^3 + 7/3*w^2 + 2/3*w - 8],
[487, 487, -2/3*w^3 + 1/3*w^2 + 14/3*w - 1],
[503, 503, w^3 - 4*w^2 - 2*w + 16],
[503, 503, -1/3*w^3 + 5/3*w^2 - 2/3*w - 11],
[521, 521, -4/3*w^3 + 11/3*w^2 + 10/3*w - 7],
[521, 521, 4/3*w^3 - 5/3*w^2 - 22/3*w],
[529, 23, -1/3*w^3 + 2/3*w^2 + 4/3*w - 6],
[569, 569, -1/3*w^3 + 8/3*w^2 + 1/3*w - 10],
[569, 569, -2/3*w^3 + 10/3*w^2 + 5/3*w - 10],
[577, 577, -2/3*w^3 + 10/3*w^2 + 5/3*w - 12],
[577, 577, -1/3*w^3 + 8/3*w^2 + 1/3*w - 8],
[593, 593, w^3 - w^2 - 5*w - 5],
[593, 593, 4/3*w^3 - 11/3*w^2 - 13/3*w + 4],
[599, 599, 2*w^2 - 3*w - 7],
[599, 599, 2/3*w^3 - 1/3*w^2 - 8/3*w - 4],
[599, 599, -1/3*w^3 + 8/3*w^2 - 5/3*w - 9],
[599, 599, -4/3*w^3 + 11/3*w^2 + 16/3*w - 7],
[607, 607, 4/3*w^3 - 8/3*w^2 - 22/3*w + 9],
[607, 607, -2/3*w^3 + 4/3*w^2 + 2/3*w - 5],
[625, 5, -5],
[631, 631, -1/3*w^3 + 2/3*w^2 + 13/3*w - 5],
[631, 631, -2/3*w^3 + 4/3*w^2 + 17/3*w - 1],
[641, 641, -2/3*w^3 + 7/3*w^2 + 14/3*w - 7],
[641, 641, 2/3*w^3 - 7/3*w^2 - 14/3*w + 8],
[647, 647, -2/3*w^3 - 5/3*w^2 + 26/3*w + 8],
[647, 647, -1/3*w^3 + 8/3*w^2 + 1/3*w - 9],
[647, 647, -2/3*w^3 + 10/3*w^2 + 5/3*w - 11],
[647, 647, -2/3*w^3 + 13/3*w^2 - 10/3*w - 13],
[673, 673, -4/3*w^3 + 5/3*w^2 + 25/3*w - 2],
[673, 673, -w^3 + 3*w^2 + w - 7],
[727, 727, -2/3*w^3 + 7/3*w^2 + 2/3*w - 11],
[727, 727, -2/3*w^3 + 1/3*w^2 + 14/3*w - 4],
[743, 743, w^2 - w - 10],
[743, 743, 1/3*w^3 - 5/3*w^2 - 1/3*w - 1],
[769, 769, -w^3 + 2*w^2 + 7*w - 4],
[769, 769, 3*w - 2],
[809, 809, -1/3*w^3 + 8/3*w^2 - 2/3*w - 8],
[809, 809, -1/3*w^3 + 8/3*w^2 - 2/3*w - 10],
[823, 823, -w^3 + 2*w^2 + 7*w - 5],
[823, 823, 3*w - 1],
[839, 839, 4/3*w^3 - 2/3*w^2 - 25/3*w - 8],
[839, 839, 2/3*w^3 - 7/3*w^2 - 14/3*w + 12],
[857, 857, -1/3*w^3 + 5/3*w^2 - 5/3*w - 7],
[857, 857, -2/3*w^3 + 1/3*w^2 + 17/3*w - 2],
[863, 863, -2/3*w^3 + 7/3*w^2 - 7/3*w - 2],
[863, 863, 5/3*w^3 - 7/3*w^2 - 35/3*w + 1],
[881, 881, -2/3*w^3 + 1/3*w^2 + 14/3*w + 8],
[881, 881, -2/3*w^3 + 13/3*w^2 - 4/3*w - 18],
[881, 881, -5/3*w^3 + 7/3*w^2 + 35/3*w + 3],
[881, 881, -2/3*w^3 + 7/3*w^2 + 2/3*w + 1],
[887, 887, -1/3*w^3 + 11/3*w^2 - 8/3*w - 7],
[887, 887, -1/3*w^3 + 11/3*w^2 - 8/3*w - 18],
[911, 911, 5/3*w^3 - 4/3*w^2 - 29/3*w - 3],
[911, 911, 2*w^3 - 6*w^2 - 5*w + 13],
[929, 929, 2*w^3 - 4*w^2 - 9*w + 2],
[929, 929, 5/3*w^3 - 10/3*w^2 - 17/3*w],
[929, 929, -4/3*w^3 + 14/3*w^2 + 13/3*w - 11],
[929, 929, 1/3*w^3 + 4/3*w^2 - 7/3*w - 9],
[937, 937, -w^3 + 4*w^2 + w - 13],
[937, 937, 2/3*w^3 + 2/3*w^2 - 17/3*w - 3],
[953, 953, -2*w^3 + w^2 + 15*w + 8],
[953, 953, -5/3*w^3 + 19/3*w^2 - 1/3*w - 11],
[961, 31, 4/3*w^3 - 8/3*w^2 - 16/3*w + 5],
[967, 967, -4/3*w^3 + 2/3*w^2 + 31/3*w + 11],
[967, 967, 7/3*w^3 - 5/3*w^2 - 46/3*w - 9],
[977, 977, 2*w^3 - 5*w^2 - 8*w + 10],
[977, 977, -w^3 + 3*w^2 + 2*w - 14],
[977, 977, -4/3*w^3 + 8/3*w^2 + 25/3*w - 11],
[977, 977, -w^3 + w^2 + 6*w - 7],
[983, 983, -5/3*w^3 + 16/3*w^2 + 8/3*w - 12],
[983, 983, -1/3*w^3 + 11/3*w^2 - 11/3*w - 15],
[983, 983, 3*w^2 - 2*w - 16],
[983, 983, -5/3*w^3 + 16/3*w^2 + 11/3*w - 14],
[991, 991, 4/3*w^3 - 8/3*w^2 - 22/3*w + 1],
[991, 991, 2/3*w^3 - 4/3*w^2 - 2/3*w - 3]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^3 + 2*x^2 - 6*x - 11;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, e^2 - e - 8, e^2 - 7, 1, -2*e^2 - e + 9, e^2 + e - 6, 2*e^2 - e - 11, -e^2 + e + 6, -2*e^2 - e + 5, -3*e^2 - e + 12, 2*e^2 - 4, 2*e^2 - 12, -4*e^2 + 20, 3*e^2 + 5*e - 14, 6, -2*e^2 - 3*e + 5, 3*e + 1, e^2 + 2*e + 3, -e^2 + 3*e + 4, -4*e^2 + 2*e + 26, -e^2 - 4*e + 7, 3*e^2 - 15, -3*e^2 - 4*e + 15, -2*e^2 - e + 9, 2*e^2 - 22, 3*e^2 - 4*e - 27, -e^2 - 5*e + 4, e^2 - 2*e + 5, 3*e^2 + 4*e - 23, -7*e^2 + e + 38, -5*e^2 + 4*e + 39, 3*e^2 - 3*e - 6, 5*e^2 - 5*e - 38, 3*e^2 + e - 28, -3*e^2 - 3*e + 2, e^2 + e - 10, -3*e^2 - 3, 7*e^2 + e - 42, -e^2 + 3*e - 2, e^2 - 5*e - 6, e^2 - 5*e - 6, -2*e^2 - 8, -e^2 + 2*e - 3, -4*e^2 - 6*e + 20, -5*e^2 - 3*e + 26, 2*e^2 - 36, 5*e^2 + 3*e - 20, 7*e^2 - 6*e - 57, 2*e^2 - 2*e - 8, 4*e + 10, 13*e^2 - 63, 7*e^2 - 5*e - 42, -13*e^2 - 3*e + 64, -3*e^2 + 7*e + 16, -5*e^2 - e + 22, e^2 + 5*e - 8, 8*e^2 - 4*e - 60, -4*e^2 - 6*e + 16, 2*e^2 - 10*e - 18, -4*e^2 - 7*e + 31, -4*e^2 + 4*e + 6, -e^2 - 2*e + 9, -7*e^2 + 10*e + 47, -7*e^2 + 2*e + 33, -e^2 + 2*e - 3, -6*e^2 - 3*e + 15, 4*e^2 - 5*e - 23, 6*e^2 + 11*e - 31, 5*e^2 - 4*e - 19, e^2 - 6*e - 23, 7*e^2 + 2*e - 47, -10*e^2 + 11*e + 65, 7*e^2 - e - 52, 3*e^2 - 5*e - 2, -e^2 - 7*e - 2, -6*e^2 + 6*e + 34, -4*e^2, -9*e^2 + 47, e^2 + 10*e + 9, 11*e^2 + 7*e - 56, 7*e^2 + e - 48, -e^2 - 6*e + 23, -2*e^2 + e + 25, 11*e^2 + 4*e - 53, -7*e^2 + 5*e + 30, e^2 + 3, -14*e^2 + 6*e + 86, 7*e^2 + 8*e - 41, -15*e^2 + 3*e + 84, 5*e^2 + 10*e - 27, -6*e^2 + 5*e + 19, 2*e - 20, 2*e^2 - 6*e + 6, 3*e^2 - 6*e - 29, 7*e^2 - 27, e^2 - 7*e - 12, -12*e^2 + 6*e + 74, 3*e^2 + e + 18, 9*e^2 - 8*e - 49, 12*e^2 + 7*e - 43, -3*e^2 + 4*e + 5, 3*e^2 - 7*e - 32, -8*e^2 + 12*e + 60, -5*e^2 - 14*e + 31, -3*e^2 + 17*e + 22, 4*e^2 - 34, 10*e^2 + 8*e - 30, -10*e^2 - 7*e + 59, -9*e^2 + 2*e + 45, -e^2 + 9*e + 6, -2*e^2 + 10*e + 32, -e^2 + 9*e + 6, -6*e^2 + 6*e + 68, -4*e^2 + 4*e + 36, -10*e^2 - 4*e + 52, 3*e^2 - e - 22, 11*e^2 - 10*e - 59, -7*e^2 - 2*e + 5, -e^2 + 11*e + 10, -3*e^2 + 5*e + 20, -5*e^2 - 4*e + 21, e^2 - 5*e - 42, 3*e^2 - 7*e - 18, 3*e^2 + 10*e - 19, -2*e^2 + 6*e + 26, 7*e^2 - 12*e - 51, e^2 + 16*e - 5, 7*e + 15, 8*e^2 - 15*e - 59, 20*e^2 + 6*e - 88, 12*e^2 - 10*e - 94, 20*e^2 + 4*e - 92, -6*e^2 + 6*e + 68, 14*e^2 + 3*e - 61, -16*e^2 - 6*e + 64, 7*e^2 - 5*e - 54, 4*e^2 - 11*e - 41, e^2 + 3*e - 24, 12*e^2 + 8*e - 70, 9*e^2 - 3*e - 46, 7*e^2 - 2*e - 45, -5*e^2 + 4*e + 35, 11*e^2 + 3*e - 50, 17*e^2 + 7*e - 80, -6*e^2 - 4*e + 16, 12*e^2 - e - 87, 3*e^2 - 11*e - 44, 6*e^2 - 5*e - 63, 8*e^2 - 5*e - 33, -18*e^2 - 9*e + 79, -9*e^2 - 4*e + 3, -9*e^2 - 11*e + 50, 5*e^2 + 15*e - 32, -9*e^2 + 11*e + 42, 5*e - 9, 15*e^2 - 2*e - 71, -4*e^2 + 10*e + 16, 13*e^2 - 11*e - 74, 13*e^2 + 9*e - 40, 4*e^2 - 6*e - 30, -11*e^2 - e + 72, 12*e^2 + 11*e - 47, 2*e^2 + 13*e - 31, 3*e^2 + e - 4, 8*e^2 - 5*e - 25, -11*e + 7, -12*e^2 + 13*e + 83, -6*e^2 - 8*e + 38, 21*e^2 - 7*e - 124, e^2 + 7*e + 10];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {7,7,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;