/*
  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, 2, -5, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {29, 29, 2*w^4 - 3*w^3 - 8*w^2 + 7*w + 4}>;

primesArray := [
[3, 3, -w^4 + w^3 + 4*w^2 - 2*w - 2],
[5, 5, w^2 - w - 2],
[19, 19, w^4 - 2*w^3 - 4*w^2 + 5*w + 4],
[23, 23, -w^3 + w^2 + 3*w - 1],
[29, 29, 2*w^4 - 3*w^3 - 8*w^2 + 7*w + 4],
[32, 2, 2],
[37, 37, w^3 - 2*w^2 - 2*w + 2],
[41, 41, -2*w^4 + 3*w^3 + 9*w^2 - 8*w - 6],
[43, 43, -2*w^4 + 3*w^3 + 8*w^2 - 8*w - 6],
[47, 47, w^4 - 2*w^3 - 5*w^2 + 6*w + 5],
[53, 53, -w^4 + w^3 + 4*w^2 - w - 4],
[61, 61, w^2 - 2*w - 3],
[67, 67, w^4 - w^3 - 4*w^2 + 3*w],
[67, 67, -w^4 + w^3 + 5*w^2 - 2*w - 2],
[71, 71, w^4 - w^3 - 4*w^2 + 5],
[71, 71, w^4 - 2*w^3 - 3*w^2 + 5*w + 3],
[71, 71, 2*w^4 - 2*w^3 - 8*w^2 + 5*w + 4],
[73, 73, -2*w^4 + 2*w^3 + 9*w^2 - 5*w - 6],
[81, 3, -2*w^4 + 3*w^3 + 10*w^2 - 9*w - 10],
[97, 97, -2*w^4 + 3*w^3 + 7*w^2 - 5*w - 4],
[101, 101, -w^4 + 2*w^3 + 2*w^2 - 4*w + 2],
[103, 103, -w^4 + w^3 + 3*w^2 - 2],
[103, 103, -2*w^4 + 2*w^3 + 10*w^2 - 5*w - 7],
[107, 107, 3*w^4 - 4*w^3 - 12*w^2 + 10*w + 5],
[107, 107, -w^4 + 6*w^2 + w - 4],
[107, 107, w^4 - 2*w^3 - 5*w^2 + 8*w + 3],
[109, 109, 3*w^4 - 3*w^3 - 13*w^2 + 6*w + 8],
[127, 127, -w^4 + 6*w^2 + 2*w - 5],
[131, 131, -2*w^4 + 3*w^3 + 7*w^2 - 7*w - 5],
[137, 137, w^4 - w^3 - 3*w^2 - 2],
[137, 137, 3*w^4 - 5*w^3 - 12*w^2 + 12*w + 9],
[151, 151, 2*w^4 - 2*w^3 - 8*w^2 + 3*w + 6],
[151, 151, 2*w^4 - 4*w^3 - 8*w^2 + 11*w + 6],
[157, 157, -w^4 + 2*w^3 + 2*w^2 - 5*w],
[167, 167, -w^4 + 6*w^2 - w - 5],
[167, 167, w^4 - w^3 - 6*w^2 + 3*w + 4],
[167, 167, w^4 - 5*w^2 - 3*w + 3],
[169, 13, -w^3 + 4*w - 1],
[173, 173, w^4 - 2*w^3 - 5*w^2 + 7*w + 4],
[181, 181, 2*w^4 - 4*w^3 - 5*w^2 + 6*w + 6],
[181, 181, w^3 - 2*w^2 - 3*w + 3],
[193, 193, -w^4 + w^3 + 4*w^2 - 3*w + 1],
[197, 197, 2*w^4 - 3*w^3 - 9*w^2 + 7*w + 5],
[211, 211, 2*w^4 - 3*w^3 - 8*w^2 + 6*w + 5],
[229, 229, 2*w^2 - 2*w - 5],
[239, 239, w^2 - 2*w - 4],
[257, 257, -w^3 + 4*w - 2],
[257, 257, w^4 - 2*w^3 - 6*w^2 + 7*w + 7],
[263, 263, -2*w^4 + 4*w^3 + 7*w^2 - 10*w - 7],
[271, 271, -w^4 + 2*w^3 + 5*w^2 - 8*w - 6],
[271, 271, 2*w^4 - 2*w^3 - 9*w^2 + 4*w + 3],
[271, 271, -w^3 + 2*w^2 + 3*w - 2],
[277, 277, -w^4 + 3*w^3 + 3*w^2 - 7*w - 3],
[277, 277, 2*w^4 - 2*w^3 - 7*w^2 + 3*w + 2],
[277, 277, -2*w^4 + 4*w^3 + 7*w^2 - 12*w - 5],
[281, 281, -w^4 + 3*w^3 + 4*w^2 - 9*w - 5],
[281, 281, -3*w^4 + 4*w^3 + 12*w^2 - 8*w - 7],
[281, 281, -2*w^4 + 3*w^3 + 8*w^2 - 6*w - 4],
[289, 17, 3*w^4 - 5*w^3 - 13*w^2 + 14*w + 9],
[331, 331, -w^4 + 2*w^3 + 2*w^2 - 3*w + 2],
[337, 337, 2*w^4 - w^3 - 9*w^2 + w + 2],
[337, 337, -w^4 + 2*w^3 + 3*w^2 - 4*w + 2],
[337, 337, -3*w^4 + 3*w^3 + 13*w^2 - 7*w - 8],
[347, 347, -w^4 + 2*w^3 + w^2 - 2*w + 2],
[347, 347, 3*w^4 - 4*w^3 - 11*w^2 + 9*w + 4],
[349, 349, 3*w^4 - 5*w^3 - 12*w^2 + 12*w + 7],
[353, 353, w^4 - w^3 - 5*w^2 + 3*w],
[353, 353, 2*w^2 - w - 5],
[359, 359, 2*w^4 - 2*w^3 - 9*w^2 + 6*w + 5],
[373, 373, w^3 - 2*w - 3],
[373, 373, -3*w^4 + 5*w^3 + 11*w^2 - 12*w - 8],
[379, 379, -3*w^4 + 5*w^3 + 13*w^2 - 14*w - 14],
[383, 383, w^4 - 7*w^2 + 2*w + 2],
[389, 389, w^4 - 3*w^2 - 5*w],
[389, 389, w^4 - w^3 - 3*w^2 + 2*w - 3],
[389, 389, 2*w^3 - 2*w^2 - 6*w + 1],
[397, 397, -w^3 + 4*w^2 + w - 6],
[419, 419, 2*w^4 - 3*w^3 - 10*w^2 + 10*w + 8],
[439, 439, -2*w^4 + 3*w^3 + 9*w^2 - 8*w - 4],
[439, 439, 2*w^4 - 3*w^3 - 7*w^2 + 8*w + 5],
[443, 443, w^4 + w^3 - 7*w^2 - 4*w + 4],
[449, 449, -2*w^4 + 2*w^3 + 7*w^2 - 3*w - 3],
[457, 457, 4*w^4 - 6*w^3 - 15*w^2 + 13*w + 9],
[461, 461, 3*w^4 - 5*w^3 - 11*w^2 + 11*w + 6],
[463, 463, -w^4 + 4*w^2 + 3*w - 2],
[467, 467, 3*w^4 - 5*w^3 - 11*w^2 + 13*w + 7],
[467, 467, 2*w^4 - 2*w^3 - 9*w^2 + 6*w + 4],
[479, 479, -3*w^4 + 3*w^3 + 12*w^2 - 4*w - 7],
[487, 487, w^4 - w^3 - 7*w^2 + 3*w + 11],
[503, 503, 2*w^4 - 4*w^3 - 7*w^2 + 13*w + 1],
[509, 509, w^4 - 3*w^3 - 4*w^2 + 8*w + 5],
[521, 521, -w^4 + 2*w^3 + 2*w^2 - 7*w + 2],
[523, 523, 2*w^4 - 3*w^3 - 6*w^2 + 6*w],
[523, 523, -3*w^4 + 4*w^3 + 15*w^2 - 12*w - 11],
[523, 523, -w^4 + 2*w^3 + 4*w^2 - 8*w - 4],
[547, 547, w^3 - 3*w^2 - w + 7],
[557, 557, -w^4 + 4*w^3 + 4*w^2 - 14*w - 6],
[557, 557, w^4 - 2*w^3 - 6*w^2 + 8*w + 7],
[563, 563, -w^4 + w^3 + 6*w^2 - 4*w - 10],
[563, 563, w^4 - w^3 - 4*w^2 + 4*w - 1],
[569, 569, -3*w^4 + 6*w^3 + 10*w^2 - 14*w - 6],
[569, 569, -w^4 + w^3 + 5*w^2 - 4],
[571, 571, 5*w^4 - 8*w^3 - 19*w^2 + 20*w + 12],
[571, 571, -3*w^4 + 6*w^3 + 12*w^2 - 19*w - 7],
[587, 587, -2*w^4 + 2*w^3 + 7*w^2 - 5*w],
[599, 599, -2*w^4 + 3*w^3 + 9*w^2 - 10*w - 7],
[601, 601, -2*w^4 + 3*w^3 + 8*w^2 - 5*w - 5],
[607, 607, w^4 - 3*w^3 - 3*w^2 + 6*w],
[607, 607, -3*w^4 + 5*w^3 + 14*w^2 - 13*w - 10],
[619, 619, w^4 - w^3 - 2*w^2 - 2],
[619, 619, 3*w^4 - 3*w^3 - 13*w^2 + 4*w + 7],
[619, 619, -w^4 + 4*w^3 + 3*w^2 - 13*w - 6],
[625, 5, -w^4 + 4*w^3 + 3*w^2 - 11*w - 2],
[641, 641, -3*w^4 + 5*w^3 + 14*w^2 - 16*w - 14],
[643, 643, -5*w^4 + 7*w^3 + 19*w^2 - 18*w - 7],
[659, 659, -3*w^4 + 6*w^3 + 12*w^2 - 19*w - 9],
[661, 661, 2*w^3 - 2*w^2 - 5*w + 1],
[673, 673, w^4 - 2*w^3 - 3*w^2 + 3*w - 1],
[677, 677, w^4 - 4*w^2 - 2*w + 3],
[677, 677, -2*w^4 + 3*w^3 + 11*w^2 - 9*w - 14],
[677, 677, -2*w^4 + 5*w^3 + 8*w^2 - 17*w - 7],
[691, 691, 3*w^4 - 4*w^3 - 14*w^2 + 13*w + 6],
[709, 709, w^4 - 2*w^3 - 5*w^2 + 5*w + 8],
[719, 719, w^4 - w^3 - 3*w^2 + 2*w - 4],
[727, 727, -3*w^4 + 4*w^3 + 11*w^2 - 8*w - 5],
[733, 733, w^4 + w^3 - 7*w^2 - 5*w + 6],
[733, 733, -4*w^4 + 6*w^3 + 15*w^2 - 14*w - 8],
[739, 739, 3*w^4 - 3*w^3 - 13*w^2 + 7*w + 7],
[739, 739, -3*w^4 + 4*w^3 + 11*w^2 - 11*w - 2],
[757, 757, w^2 - 4*w + 1],
[757, 757, -2*w^4 + 3*w^3 + 8*w^2 - 5*w - 3],
[773, 773, -5*w^4 + 8*w^3 + 20*w^2 - 19*w - 14],
[811, 811, -2*w^4 + 2*w^3 + 10*w^2 - 8*w - 7],
[811, 811, 2*w^4 - 10*w^2 - 3*w + 6],
[811, 811, 3*w^4 - 3*w^3 - 14*w^2 + 9*w + 10],
[821, 821, w^4 - 3*w^3 + 3*w - 2],
[823, 823, 2*w^4 - 4*w^3 - 9*w^2 + 13*w + 8],
[827, 827, -3*w^4 + 5*w^3 + 11*w^2 - 13*w - 1],
[839, 839, 3*w^4 - 5*w^3 - 13*w^2 + 13*w + 7],
[841, 29, -2*w^3 + 3*w^2 + 3*w - 3],
[841, 29, -4*w^4 + 6*w^3 + 14*w^2 - 13*w - 5],
[853, 853, -w^4 + 4*w^2 + 5*w - 1],
[857, 857, -w^4 + 4*w^3 + 2*w^2 - 11*w - 4],
[863, 863, w^4 - 4*w^3 - w^2 + 11*w],
[877, 877, -4*w^4 + 4*w^3 + 17*w^2 - 8*w - 7],
[881, 881, -4*w^4 + 6*w^3 + 17*w^2 - 17*w - 15],
[881, 881, -2*w^3 + 3*w^2 + 5*w - 4],
[907, 907, -3*w^4 + 6*w^3 + 13*w^2 - 18*w - 12],
[907, 907, -w^4 + 5*w^2 + 4*w - 6],
[919, 919, 3*w^4 - 3*w^3 - 13*w^2 + 7*w + 5],
[919, 919, w^4 - 4*w^3 - 3*w^2 + 13*w + 1],
[919, 919, -4*w^4 + 7*w^3 + 17*w^2 - 20*w - 13],
[929, 929, -w^3 + 5*w - 3],
[941, 941, 4*w^4 - 6*w^3 - 15*w^2 + 16*w + 9],
[947, 947, 3*w^4 - 6*w^3 - 9*w^2 + 16*w + 3],
[953, 953, -w^4 + 2*w^3 + 6*w^2 - 8*w - 6],
[953, 953, 3*w^4 - 3*w^3 - 12*w^2 + 8*w + 6],
[961, 31, -2*w^4 + 4*w^3 + 9*w^2 - 14*w - 5],
[967, 967, -2*w^3 + 5*w + 4],
[983, 983, 2*w^3 - 7*w - 6],
[997, 997, 4*w^4 - 5*w^3 - 15*w^2 + 12*w + 6],
[997, 997, 4*w^4 - 5*w^3 - 16*w^2 + 13*w + 8]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^3 - 6*x - 1;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, -e, -e^2 - 1, -2*e^2 + 8, -1, 2*e^2 - 9, -2*e^2 - e + 7, -e + 1, 2*e - 6, -e^2 - e + 1, 0, 3*e^2 - 4*e - 13, 2*e^2 + 2*e - 12, e^2 - 2*e - 12, 3*e - 3, e^2 + 3*e - 11, e^2 + 2*e - 13, -4, 3*e^2 - e - 15, -2*e^2 + 3*e + 3, e^2 - 4*e - 6, 2*e^2 - 5*e - 9, 2*e^2 - 2*e, -e^2 + 6, -5*e^2 + 3*e + 19, -e^2 - 6, 2*e^2 - 10, e^2 - 3*e + 5, -2*e^2 + 3*e + 2, -e^2 - 2*e - 11, 2*e^2 - 2*e - 8, -4*e^2 + 4*e + 20, -2*e^2 + 7*e + 15, 10, 3*e^2 - 7*e - 19, -e^2 - 4*e + 9, -3*e^2 + 2*e + 16, -2*e^2 + 8, -e^2 + 3*e + 7, 2*e^2 - 5*e - 1, -e^2 + 4*e + 16, -2*e^2 - 2*e + 10, -3*e^2 - e + 19, 4*e^2 - 8*e - 26, -7*e^2 + 6*e + 27, -e^2 + e - 7, e^2 + 3*e - 29, 3*e^2 + e + 3, 4*e^2 - 8*e - 20, 4*e^2 - 4*e - 32, -e^2 - 4*e - 3, 9*e^2 - e - 25, 5*e^2 - 3*e - 23, 2*e^2 + 5*e - 13, 5*e^2 + 2*e - 16, -3*e^2 - 3*e + 3, -3*e^2 + 9, -5*e^2 + 5*e + 25, -e^2 - 5, 4*e^2 - 5*e - 9, 4*e^2 + 9*e - 35, 3*e^2 + 2*e - 4, -5*e^2 + 6*e + 21, -3*e^2 + 9*e + 29, -2*e^2 - e + 22, 9*e^2 - 3*e - 31, e^2 - 4*e - 20, 3*e^2 + 7*e - 31, -5*e^2 + 9*e + 23, -8*e^2 + 4*e + 18, 6*e^2 + 3*e - 32, -2*e^2 + 8*e + 10, 2*e^2 - 3*e - 12, 7*e^2 + 2*e - 22, 6, 4*e^2 - 2*e - 2, -3*e^2 + 4*e + 7, e^2 - 12*e - 7, -e^2 - 11*e - 3, -3*e^2 + e - 1, e^2 + 5*e - 23, -8*e^2 - e + 21, -4*e^2 + 4*e + 30, -8*e^2 + 3*e + 26, -7*e^2 + 4*e + 20, -4*e^2 + 5*e + 19, -5*e^2 + 6*e + 31, -8*e + 10, 5*e^2 + 5*e - 17, 9*e^2 - e - 21, 3*e^2 - 11*e - 25, -14*e + 4, e, 2*e^2 - 6*e - 12, -4*e^2 - 4*e + 20, -6*e^2 + 5*e + 21, -2*e^2 - 12*e + 8, e^2 + 8*e - 11, -3*e^2 - 7*e + 19, 6*e^2 + e - 21, 4*e^2 + 12*e - 22, -5*e^2 + 8*e + 19, 6*e - 4, -4*e^2 - e + 27, 7*e^2 + e - 29, 8*e^2 + 10*e - 40, 7*e^2 - 4*e - 43, 2*e^2 - 2*e - 28, -4*e^2 - 3*e + 16, -8*e^2 + 2*e + 38, 7*e^2 + 5*e - 31, 5*e^2 - 9*e - 43, -e^2 + 6*e + 25, -5*e^2 - 8*e + 47, -5*e^2 - 14*e + 29, -3*e^2 + 19*e + 19, 2*e^2 + 2*e + 2, -6*e^2 + 4*e + 28, -e^2 - 5*e + 21, 7*e^2 + 5*e - 17, 2*e^2 - 6*e + 10, 2*e^2 + 4*e - 28, -e^2 - 13*e + 13, 4*e^2 + 6*e - 22, -12*e^2 + 5*e + 45, 6*e^2 + 8*e - 48, -2*e^2 - 12*e + 8, 7*e^2 - 8*e - 27, 6*e^2 + 8*e - 36, -4*e^2 - e + 26, -5*e^2 + 3, 2*e^2 - e - 9, -2*e^2 + 11*e + 17, -12*e^2 + 10*e + 54, -4*e^2 - 13*e + 26, -2*e^2 - 4*e - 8, 2*e^2 - 16, 6*e^2 + 15*e - 54, 8*e^2 + 3*e - 23, 5*e^2 - 27, -4*e^2 + 6*e + 8, 5*e^2 + 3*e - 5, 7*e^2 + 10*e - 54, 4*e^2 - 6*e - 22, -2*e^2 + 2*e + 12, -4*e^2 - 19*e + 26, -9*e^2 + 11*e + 31, -2*e^2 - 9*e - 3, 10*e^2 - 2*e - 52, -9*e^2 + 20*e + 56, 2*e^2 - 14*e - 6, 3*e^2 - 7*e + 19, 17*e^2 - 6*e - 61, -2*e^2 + 6*e + 8, -3*e^2 - 10*e + 29, -4*e^2 - e + 39, e^2 - 19*e - 3, -8*e^2 - 4*e + 34, -4*e^2 + 6*e, -2*e^2 - 2*e - 14, 4*e^2 + 16*e - 14, 2*e^2 + 7*e - 14];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {29, 29, 2*w^4 - 3*w^3 - 8*w^2 + 7*w + 4}>] := 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;