/*
  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![3, -4, -8, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {9,9,w - 1}>;

primesArray := [
[3, 3, w],
[3, 3, w^3 - 2*w^2 - 6*w + 1],
[9, 3, w^3 - 2*w^2 - 5*w + 1],
[16, 2, 2],
[17, 17, w^3 - w^2 - 8*w - 5],
[17, 17, -2*w^3 + 4*w^2 + 11*w - 2],
[17, 17, -w^3 + 3*w^2 + 2*w - 2],
[17, 17, w^3 - 2*w^2 - 4*w + 1],
[25, 5, w^3 - 3*w^2 - 3*w + 1],
[25, 5, w^2 - 2*w - 7],
[29, 29, w^3 - 2*w^2 - 6*w - 1],
[29, 29, w - 2],
[53, 53, w^3 - 3*w^2 - 4*w + 4],
[53, 53, -w^3 + 3*w^2 + 4*w - 5],
[61, 61, -w^3 + 2*w^2 + 7*w - 4],
[61, 61, -4*w^3 + 11*w^2 + 14*w - 8],
[101, 101, -w^3 + 2*w^2 + 7*w - 1],
[103, 103, -w^3 + 3*w^2 + 2*w - 5],
[103, 103, -w^3 + w^2 + 8*w + 2],
[113, 113, w^3 - 3*w^2 - 3*w + 7],
[113, 113, w^2 - 2*w - 1],
[127, 127, w^3 - 2*w^2 - 4*w - 2],
[127, 127, -2*w^3 + 4*w^2 + 11*w + 1],
[131, 131, -3*w^3 + 7*w^2 + 14*w - 8],
[131, 131, -w^2 + w + 8],
[131, 131, 2*w^3 - 5*w^2 - 9*w + 1],
[131, 131, -3*w^3 + 6*w^2 + 16*w + 1],
[139, 139, -2*w^3 + 5*w^2 + 9*w - 2],
[139, 139, -w^2 + w + 7],
[157, 157, -2*w^3 + 5*w^2 + 7*w - 5],
[157, 157, -2*w^3 + 3*w^2 + 13*w + 2],
[157, 157, -2*w^3 + 5*w^2 + 6*w - 2],
[157, 157, -3*w^3 + 5*w^2 + 19*w + 4],
[169, 13, -2*w^3 + 4*w^2 + 10*w - 1],
[173, 173, 3*w^3 - 8*w^2 - 10*w + 7],
[173, 173, 2*w^3 - 2*w^2 - 15*w - 8],
[179, 179, -3*w^2 + 6*w + 17],
[179, 179, -2*w^3 + 3*w^2 + 12*w + 1],
[179, 179, -3*w^3 + 7*w^2 + 13*w - 7],
[179, 179, -2*w^3 + 3*w^2 + 14*w + 4],
[191, 191, w^3 - 3*w^2 - w + 4],
[191, 191, -2*w^3 + 3*w^2 + 14*w + 2],
[199, 199, -w^3 + 8*w + 10],
[199, 199, 5*w^3 - 15*w^2 - 14*w + 14],
[211, 211, -w - 4],
[211, 211, -w^3 + 2*w^2 + 6*w - 5],
[211, 211, 2*w^3 - 3*w^2 - 12*w - 4],
[211, 211, -3*w^3 + 7*w^2 + 13*w - 4],
[257, 257, -3*w^3 + 8*w^2 + 10*w - 2],
[257, 257, 2*w^3 - 2*w^2 - 16*w - 7],
[263, 263, 3*w^3 - 6*w^2 - 17*w + 1],
[263, 263, w^3 - 2*w^2 - 3*w - 1],
[269, 269, 5*w^3 - 13*w^2 - 19*w + 7],
[269, 269, -3*w^3 + 6*w^2 + 18*w - 1],
[277, 277, w^3 - 4*w^2 - w + 5],
[277, 277, -w^3 + 4*w^2 + w - 11],
[283, 283, -w^3 + 5*w^2 - w - 14],
[283, 283, 2*w^3 - 7*w^2 - 4*w + 10],
[283, 283, -w^3 + w^2 + 9*w + 1],
[283, 283, w^2 - 4*w - 5],
[311, 311, -w^3 + w^2 + 9*w + 5],
[311, 311, w^2 - 4*w - 1],
[313, 313, -4*w^3 + 9*w^2 + 19*w - 10],
[313, 313, -2*w^2 + 5*w + 11],
[337, 337, -3*w^3 + 6*w^2 + 14*w + 2],
[337, 337, -4*w^3 + 8*w^2 + 21*w + 1],
[347, 347, -2*w^3 + 6*w^2 + 7*w - 10],
[347, 347, -w^3 + 4*w^2 + 2*w - 7],
[361, 19, -4*w^3 + 11*w^2 + 13*w - 10],
[361, 19, -2*w^3 + 5*w^2 + 8*w - 10],
[373, 373, 2*w^3 - 5*w^2 - 10*w + 2],
[373, 373, -w^3 + 2*w^2 + 8*w - 2],
[373, 373, -2*w^3 + 4*w^2 + 13*w - 1],
[373, 373, -w^3 + 3*w^2 + 5*w - 8],
[389, 389, -w^3 + 4*w^2 + 2*w - 10],
[389, 389, -2*w^3 + 6*w^2 + 7*w - 7],
[419, 419, -2*w^3 + 5*w^2 + 6*w - 1],
[419, 419, -3*w^3 + 5*w^2 + 19*w + 5],
[433, 433, w^2 - 4*w - 2],
[433, 433, w^3 - w^2 - 9*w - 4],
[439, 439, -w^3 + 4*w^2 + w - 2],
[439, 439, w^3 - 4*w^2 - w + 14],
[443, 443, -2*w^3 + 5*w^2 + 9*w + 1],
[443, 443, -w^2 + w + 10],
[467, 467, w^3 - w^2 - 8*w + 4],
[467, 467, w^3 - 4*w^2 + 2],
[491, 491, -4*w^3 + 12*w^2 + 10*w - 11],
[491, 491, 2*w^3 - w^2 - 18*w - 11],
[491, 491, 3*w^3 - 9*w^2 - 7*w + 11],
[491, 491, 2*w^3 - 20*w - 19],
[503, 503, w^3 - 3*w^2 - 4*w + 10],
[503, 503, -w^3 + w^2 + 10*w - 2],
[523, 523, -4*w^3 + 9*w^2 + 22*w - 7],
[523, 523, w^3 - 3*w^2 - 7*w + 5],
[529, 23, -2*w^3 + 4*w^2 + 10*w + 5],
[529, 23, -2*w^3 + 4*w^2 + 10*w - 7],
[547, 547, 2*w^3 - w^2 - 18*w - 14],
[547, 547, 3*w^3 - 8*w^2 - 14*w + 8],
[571, 571, 5*w^3 - 11*w^2 - 25*w + 5],
[571, 571, -4*w^3 + 9*w^2 + 19*w - 11],
[599, 599, -5*w^3 + 12*w^2 + 20*w - 4],
[599, 599, 4*w^3 - 6*w^2 - 25*w - 11],
[601, 601, -4*w^3 + 9*w^2 + 20*w - 5],
[601, 601, w^3 - 9*w - 14],
[601, 601, 3*w^3 - 8*w^2 - 11*w + 2],
[601, 601, -2*w^3 + 3*w^2 + 14*w - 1],
[641, 641, -2*w^3 + 3*w^2 + 15*w - 5],
[641, 641, -4*w^3 + 8*w^2 + 23*w - 2],
[647, 647, -3*w^3 + 8*w^2 + 13*w - 11],
[647, 647, -w^3 + 4*w^2 + 3*w - 7],
[659, 659, 3*w^3 - 7*w^2 - 14*w - 2],
[659, 659, w^3 - w^2 - 6*w - 11],
[673, 673, -3*w^3 + 7*w^2 + 11*w - 1],
[673, 673, -4*w^3 + 7*w^2 + 24*w + 5],
[677, 677, 5*w^3 - 12*w^2 - 21*w + 11],
[677, 677, -2*w^3 + 6*w^2 + 5*w - 1],
[677, 677, -2*w^3 + 6*w^2 + 8*w - 7],
[677, 677, 2*w^3 - 6*w^2 - 8*w + 11],
[719, 719, -2*w^3 + 6*w^2 + 10*w - 7],
[719, 719, -4*w^3 + 10*w^2 + 20*w - 13],
[727, 727, 3*w^3 - 10*w^2 - 8*w + 10],
[727, 727, 3*w^3 - 8*w^2 - 13*w + 10],
[727, 727, 2*w^3 - 8*w^2 - 3*w + 23],
[727, 727, -w^3 + 4*w^2 + 3*w - 8],
[751, 751, -7*w^3 + 18*w^2 + 25*w - 7],
[751, 751, w^3 - 7*w^2 + 5*w + 26],
[757, 757, -4*w^3 + 9*w^2 + 18*w - 7],
[757, 757, -3*w^3 + 5*w^2 + 17*w + 1],
[797, 797, 2*w^2 - 4*w - 5],
[797, 797, 2*w^3 - 6*w^2 - 6*w + 11],
[823, 823, -2*w^3 + 2*w^2 + 15*w + 14],
[823, 823, -3*w^3 + 8*w^2 + 10*w - 1],
[829, 829, -6*w^3 + 16*w^2 + 22*w - 7],
[829, 829, 5*w^3 - 13*w^2 - 20*w + 8],
[829, 829, -w^3 + 7*w + 4],
[829, 829, -5*w^3 + 12*w^2 + 23*w - 14],
[841, 29, -3*w^3 + 6*w^2 + 15*w - 2],
[859, 859, -4*w^3 + 7*w^2 + 24*w - 1],
[859, 859, -5*w^3 + 15*w^2 + 14*w - 13],
[881, 881, 3*w^3 - 7*w^2 - 16*w + 4],
[881, 881, -w^3 + 3*w^2 + 6*w - 7],
[883, 883, -2*w^3 + 3*w^2 + 12*w + 10],
[883, 883, -3*w^3 + 7*w^2 + 13*w + 2],
[887, 887, 4*w^3 - 11*w^2 - 11*w + 11],
[887, 887, 2*w^3 - 7*w^2 - 3*w + 7],
[887, 887, 4*w^3 - 5*w^2 - 29*w - 10],
[887, 887, 3*w^2 - 7*w - 16],
[907, 907, -w^3 + 3*w^2 + 8*w - 5],
[907, 907, -5*w^3 + 11*w^2 + 28*w - 8],
[911, 911, -4*w^3 + 10*w^2 + 18*w - 17],
[911, 911, 2*w^2 - 2*w - 1],
[919, 919, 4*w^3 - 7*w^2 - 25*w - 4],
[919, 919, 2*w^3 - 5*w^2 - 5*w + 1],
[937, 937, -w^3 + 3*w^2 + w - 8],
[937, 937, -2*w^3 + 3*w^2 + 14*w - 2],
[953, 953, -2*w^3 + w^2 + 17*w + 19],
[953, 953, 4*w^3 - 11*w^2 - 13*w + 4],
[961, 31, 2*w^3 - 3*w^2 - 11*w - 8],
[961, 31, w^3 - 4*w^2 + 2*w + 11],
[971, 971, 2*w^2 - 5*w - 4],
[971, 971, -w^3 + 4*w^2 - 11],
[991, 991, -2*w^3 + 6*w^2 + 9*w - 8],
[991, 991, -2*w^3 + 3*w^2 + 13*w - 1],
[991, 991, 3*w^3 - 8*w^2 - 14*w + 11],
[991, 991, -2*w^3 + 5*w^2 + 7*w - 8],
[997, 997, -2*w^3 + 5*w^2 + 6*w + 1],
[997, 997, -3*w^3 + 5*w^2 + 19*w + 7],
[997, 997, -6*w^3 + 12*w^2 + 33*w - 4],
[997, 997, 4*w^3 - 11*w^2 - 15*w + 11]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^6 - 17*x^4 + 68*x^2 - 16;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, 0, 1/2*e^3 - 9/2*e, -1/8*e^5 + 13/8*e^3 - 5*e, -1/8*e^5 + 17/8*e^3 - 19/2*e, -1/4*e^5 + 15/4*e^3 - 25/2*e, -1/8*e^5 + 17/8*e^3 - 13/2*e, -1/4*e^4 + 9/4*e^2 + 1, -1/4*e^4 + 17/4*e^2 - 9, -1/4*e^5 + 19/4*e^3 - 41/2*e, -1/8*e^5 + 9/8*e^3 + 1/2*e, -1/4*e^4 + 9/4*e^2 + 4, 1/8*e^5 - 9/8*e^3 - 1/2*e, 1/4*e^4 - 9/4*e^2 + 2, 3/8*e^5 - 43/8*e^3 + 35/2*e, 1/4*e^4 - 13/4*e^2 + 4, 1/2*e^5 - 17/2*e^3 + 32*e, -1/2*e^5 + 13/2*e^3 - 14*e, 1/4*e^5 - 13/4*e^3 + 10*e, 3/4*e^4 - 39/4*e^2 + 15, -3/4*e^4 + 27/4*e^2 + 3, -e^2 + 8, e^3 - 5*e, e^4 - 12*e^2 + 14, -1/4*e^5 + 17/4*e^3 - 19*e, 1/2*e^5 - 17/2*e^3 + 38*e, 3*e, -1/2*e^4 + 7/2*e^2 + 14, -1/2*e^4 + 13/2*e^2 - 4, 1/4*e^5 - 19/4*e^3 + 35/2*e, e^5 - 29/2*e^3 + 83/2*e, -5/4*e^4 + 53/4*e^2 - 16, 1/4*e^4 - 13/4*e^2 + 2, -1/2*e^5 + 13/2*e^3 - 14*e, 5/8*e^5 - 69/8*e^3 + 49/2*e, -3/4*e^4 + 39/4*e^2 - 12, e^4 - 12*e^2 + 20, 3*e^2 - 12, 3*e^2 - 24, 1/2*e^5 - 15/2*e^3 + 22*e, -1/2*e^5 + 15/2*e^3 - 28*e, 12, -1/4*e^5 + 13/4*e^3 - 10*e, e^4 - 13*e^2 + 28, -3/4*e^5 + 43/4*e^3 - 35*e, 1/2*e^4 - 11/2*e^2 + 18, 1/4*e^5 - 17/4*e^3 + 9*e, e^4 - 13*e^2 + 22, -3/4*e^4 + 39/4*e^2 - 30, 3/4*e^4 - 27/4*e^2 + 6, 6, -1/4*e^5 + 21/4*e^3 - 26*e, e^5 - 31/2*e^3 + 95/2*e, -5/4*e^5 + 67/4*e^3 - 85/2*e, 1/4*e^5 - 19/4*e^3 + 29/2*e, 1/4*e^5 - 19/4*e^3 + 59/2*e, 2*e^2 - 10, 1/4*e^5 - 17/4*e^3 + 15*e, 3/4*e^5 - 47/4*e^3 + 34*e, 2*e^2 + 2, 24, -1/4*e^5 + 9/4*e^3 - 2*e, 5/4*e^5 - 83/4*e^3 + 145/2*e, -5/4*e^4 + 53/4*e^2 - 11, 3/4*e^4 - 43/4*e^2 + 23, -e^5 + 35/2*e^3 - 131/2*e, -6, 3/2*e^4 - 27/2*e^2 + 6, -3/4*e^4 + 27/4*e^2 + 11, -3/4*e^4 + 39/4*e^2 - 19, e^5 - 17*e^3 + 60*e, 4*e^3 - 38*e, -e^4 + 11*e^2 - 18, e^4 - 7*e^2 - 26, -3/4*e^4 + 27/4*e^2 - 9, -3/4*e^4 + 27/4*e^2 - 15, -1/4*e^5 + 29/4*e^3 - 43*e, -1/4*e^5 + 29/4*e^3 - 43*e, -7/4*e^4 + 71/4*e^2 - 6, 3/8*e^5 - 91/8*e^3 + 137/2*e, e^5 - 13*e^3 + 28*e, -5/4*e^5 + 77/4*e^3 - 59*e, -24, -3*e^2 + 30, 1/2*e^5 - 21/2*e^3 + 52*e, 12, 1/2*e^4 - 9/2*e^2 - 8, 5/4*e^5 - 85/4*e^3 + 71*e, -1/4*e^5 + 5/4*e^3 + 5*e, -3/4*e^5 + 39/4*e^3 - 24*e, -3/2*e^4 + 33/2*e^2 - 24, -3*e^2 + 18, 1/4*e^5 - 37/4*e^3 + 61*e, -5/4*e^5 + 77/4*e^3 - 56*e, 5/8*e^5 - 77/8*e^3 + 47/2*e, -5/4*e^4 + 65/4*e^2 - 40, e^5 - 16*e^3 + 55*e, 1/4*e^5 - 13/4*e^3 - 2*e, -3/4*e^5 + 43/4*e^3 - 23*e, -1/2*e^4 + 13/2*e^2 - 44, -1/4*e^5 + 29/4*e^3 - 43*e, -1/4*e^5 + 5/4*e^3 + 14*e, 5/8*e^5 - 85/8*e^3 + 57/2*e, -7/4*e^4 + 59/4*e^2 + 6, -1/8*e^5 + 17/8*e^3 - 9/2*e, 1/4*e^4 + 11/4*e^2 - 38, 6*e^2 - 18, 3*e^2 - 42, 1/2*e^4 - 3/2*e^2 - 14, 3/4*e^5 - 51/4*e^3 + 51*e, -5/4*e^5 + 73/4*e^3 - 50*e, 1/4*e^5 - 5/4*e^3 - 14*e, 5/8*e^5 - 53/8*e^3 + 5/2*e, -1/8*e^5 + 49/8*e^3 - 91/2*e, 5/8*e^5 - 61/8*e^3 + 29/2*e, -7/8*e^5 + 143/8*e^3 - 151/2*e, 1/2*e^5 - 9*e^3 + 89/2*e, -9/4*e^4 + 93/4*e^2 - 15, 3/2*e^4 - 27/2*e^2 + 18, 3*e^2 - 18, -7/4*e^5 + 115/4*e^3 - 97*e, -1/4*e^5 + 13/4*e^3 - 4*e, 3/2*e^4 - 41/2*e^2 + 26, 1/2*e^4 + 1/2*e^2 - 36, -1/2*e^5 + 13/2*e^3 - 23*e, -5/4*e^5 + 65/4*e^3 - 32*e, 3/4*e^4 - 19/4*e^2 - 34, 1/8*e^5 + 23/8*e^3 - 65/2*e, 1/8*e^5 - 41/8*e^3 + 55/2*e, 1/8*e^5 - 17/8*e^3 + 25/2*e, -2*e^5 + 29*e^3 - 83*e, -1/2*e^5 + 13/2*e^3 - 11*e, 1/4*e^4 - 1/4*e^2 + 7, -1/2*e^3 + 5/2*e, -e^5 + 23/2*e^3 - 29/2*e, -3/4*e^4 + 23/4*e^2 + 23, -2*e^4 + 26*e^2 - 46, 3/2*e^4 - 29/2*e^2 + 2, e^3 - 17*e, -7/4*e^4 + 63/4*e^2 - 11, 5/4*e^5 - 87/4*e^3 + 155/2*e, -1/4*e^5 + 25/4*e^3 - 49*e, 3*e^4 - 34*e^2 + 56, 6*e^2 - 30, -3/2*e^4 + 39/2*e^2 - 54, 1/2*e^5 - 9/2*e^3 - 14*e, -3*e^2 - 12, -1/2*e^4 + 1/2*e^2 + 34, 2*e^5 - 32*e^3 + 110*e, 1/2*e^4 - 15/2*e^2 + 34, -3/4*e^5 + 27/4*e^3 + 9*e, e^3 - 23*e, 2*e^4 - 22*e^2 + 36, -1/4*e^4 + 17/4*e^2 - 3, -1/2*e^3 + 41/2*e, 7/8*e^5 - 87/8*e^3 + 47/2*e, 5/4*e^4 - 57/4*e^2 + 40, -1/8*e^5 + 49/8*e^3 - 73/2*e, -1/8*e^5 + 1/8*e^3 - 7/2*e, -5/4*e^5 + 81/4*e^3 - 73*e, 3/2*e^4 - 27/2*e^2 + 18, -2*e^5 + 32*e^3 - 104*e, 1/4*e^5 - 25/4*e^3 + 31*e, 7/4*e^5 - 115/4*e^3 + 94*e, 1/4*e^5 - 25/4*e^3 + 25*e, 7/4*e^4 - 67/4*e^2 - 4, 1/4*e^4 + 11/4*e^2 - 4, 7/4*e^5 - 121/4*e^3 + 233/2*e, -5/4*e^5 + 71/4*e^3 - 121/2*e];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {3,3,w^3 - 2*w^2 - 6*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;