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

NN := ideal<ZF | {23, 23, -w^2 + 8}>;

primesArray := [
[3, 3, w - 2],
[5, 5, w],
[5, 5, -w + 3],
[8, 2, 2],
[9, 3, w^2 + w - 4],
[13, 13, w + 3],
[17, 17, -w^2 + w + 3],
[23, 23, w^2 - 2],
[23, 23, w^2 - 3],
[23, 23, -w^2 + 8],
[29, 29, w - 4],
[37, 37, w^2 + w - 8],
[41, 41, w^2 + 2*w - 4],
[47, 47, 2*w^2 + w - 8],
[59, 59, -2*w^2 - 3*w + 6],
[61, 61, -2*w - 1],
[67, 67, -2*w - 3],
[79, 79, 2*w^2 - 9],
[109, 109, w^2 + 2*w - 6],
[109, 109, 2*w^2 + w - 14],
[109, 109, -w^2 - w - 1],
[113, 113, 2*w^2 + w - 12],
[127, 127, w^2 - 3*w - 2],
[131, 131, -4*w^2 - 5*w + 9],
[137, 137, w^2 + 2*w - 7],
[137, 137, 3*w + 8],
[137, 137, 2*w^2 + w - 17],
[139, 139, 3*w^2 + 2*w - 11],
[149, 149, w - 6],
[151, 151, w^2 + 2*w - 9],
[157, 157, -w^2 - 2*w + 13],
[157, 157, 2*w^2 - 7],
[163, 163, w^2 + 3*w + 3],
[163, 163, 2*w^2 - 3*w - 3],
[163, 163, 2*w^2 - w - 7],
[167, 167, 3*w^2 + w - 13],
[169, 13, 2*w^2 + 5*w - 1],
[173, 173, w^2 + 2*w + 2],
[179, 179, 4*w^2 + 2*w - 19],
[181, 181, w^2 + 3*w - 6],
[191, 191, 3*w^2 + 2*w - 14],
[193, 193, w^2 - 3*w - 3],
[197, 197, 3*w^2 - 2*w - 13],
[211, 211, -w - 6],
[223, 223, -w^2 + w - 3],
[223, 223, -3*w - 2],
[223, 223, 2*w^2 - 2*w - 3],
[229, 229, 2*w^2 - w - 4],
[233, 233, 2*w + 7],
[239, 239, -3*w - 4],
[239, 239, w^2 - 11],
[239, 239, 4*w^2 - w - 21],
[241, 241, 3*w^2 + w - 19],
[251, 251, -w^2 - 4*w - 1],
[257, 257, w - 7],
[263, 263, 3*w^2 + w - 12],
[269, 269, 3*w^2 - 4*w - 11],
[271, 271, 4*w^2 + 6*w - 11],
[277, 277, -4*w^2 - 3*w + 17],
[283, 283, 5*w^2 + 4*w - 18],
[289, 17, 3*w^2 + 2*w - 9],
[307, 307, -2*w^2 - w + 18],
[307, 307, w^2 + 2*w + 3],
[307, 307, -w^2 - 3],
[313, 313, 3*w^2 - 13],
[317, 317, 3*w^2 - 2*w - 12],
[331, 331, w^2 + 3*w - 9],
[331, 331, w^2 + 3*w - 11],
[331, 331, w^2 + 3*w + 4],
[343, 7, -7],
[347, 347, 2*w^2 + 2*w - 13],
[353, 353, 4*w^2 + 5*w - 13],
[359, 359, 3*w^2 + 6*w - 1],
[379, 379, w^2 - 3*w - 6],
[383, 383, 3*w^2 - 23],
[383, 383, w^2 - 12],
[383, 383, 3*w^2 + 3*w - 13],
[401, 401, -6*w^2 - 7*w + 16],
[409, 409, -4*w^2 + 3*w + 21],
[421, 421, w^2 - 3*w - 9],
[433, 433, w^2 - 3*w - 8],
[433, 433, w^2 + 4*w - 8],
[433, 433, 3*w^2 - w - 12],
[439, 439, -4*w - 11],
[449, 449, 2*w^2 + 3*w - 19],
[457, 457, w^2 - 4*w - 3],
[461, 461, 5*w^2 + 6*w - 16],
[461, 461, -2*w^2 + 3*w + 16],
[461, 461, 2*w^2 + 4*w - 9],
[467, 467, 3*w^2 - 4*w - 12],
[479, 479, 3*w^2 + w - 9],
[487, 487, 5*w^2 + 2*w - 22],
[491, 491, 3*w^2 - 11],
[499, 499, 5*w^2 + w - 24],
[503, 503, 3*w^2 + 2*w - 17],
[503, 503, 3*w^2 + w - 7],
[503, 503, 2*w^2 - 4*w - 7],
[509, 509, 3*w - 11],
[521, 521, 3*w^2 - 3*w - 16],
[523, 523, -w - 8],
[541, 541, 6*w^2 + 7*w - 19],
[547, 547, 3*w^2 + 4*w - 12],
[557, 557, 2*w^2 - 3*w - 12],
[563, 563, 4*w^2 + 3*w - 12],
[563, 563, 4*w^2 + w - 17],
[563, 563, -w^2 - 5*w - 3],
[569, 569, w^2 - 4*w - 4],
[577, 577, w^2 - w - 13],
[587, 587, 4*w^2 + 3*w - 8],
[587, 587, 2*w^2 - 3*w - 13],
[587, 587, 2*w^2 + 3*w - 18],
[593, 593, 4*w - 13],
[599, 599, w - 9],
[613, 613, -w^2 + 3*w - 7],
[613, 613, w^2 + 5*w + 2],
[613, 613, 3*w^2 - 3*w - 17],
[617, 617, w^2 + 5*w + 8],
[641, 641, 4*w^2 + 3*w - 11],
[641, 641, w^2 + w - 14],
[641, 641, 3*w^2 - 2*w - 9],
[647, 647, 2*w^2 + 3*w - 13],
[653, 653, 3*w^2 - 8],
[659, 659, 5*w^2 + 4*w - 21],
[659, 659, 2*w^2 - 4*w - 19],
[659, 659, -2*w^2 - 6*w + 9],
[677, 677, -6*w - 13],
[691, 691, 6*w - 11],
[701, 701, 3*w^2 - 3*w - 19],
[719, 719, 2*w^2 + 3*w - 16],
[727, 727, 3*w^2 + 4*w - 13],
[733, 733, 3*w^2 - w - 8],
[739, 739, 2*w^2 + 4*w - 11],
[739, 739, 4*w^2 - 3*w - 24],
[739, 739, 3*w^2 - w - 6],
[743, 743, 4*w^2 - 2*w - 17],
[751, 751, -w - 9],
[761, 761, 5*w^2 + 5*w - 19],
[769, 769, w^2 - 4*w - 6],
[769, 769, -5*w - 1],
[769, 769, 2*w^2 - 19],
[773, 773, 2*w^2 - 6*w - 3],
[797, 797, 4*w^2 - 17],
[809, 809, -7*w^2 - 4*w + 29],
[811, 811, 6*w^2 + 3*w - 29],
[823, 823, 4*w^2 + 2*w - 13],
[823, 823, -w^2 - 4*w - 7],
[823, 823, 4*w^2 + 4*w - 17],
[827, 827, -2*w^2 - 3*w - 2],
[829, 829, -w^2 + w - 6],
[829, 829, -w^2 + 4*w - 9],
[829, 829, -2*w^2 + 3*w + 21],
[839, 839, 6*w^2 - 2*w - 31],
[841, 29, w^2 + 3*w + 6],
[853, 853, -7*w^2 - 6*w + 27],
[859, 859, 6*w - 1],
[881, 881, w^2 + 5*w - 16],
[907, 907, 2*w^2 + 7*w + 7],
[919, 919, w^2 - 4*w - 9],
[953, 953, -4*w^2 - w + 32],
[967, 967, 5*w^2 - 5*w - 18],
[971, 971, 3*w^2 - 4*w - 24],
[977, 977, 2*w^2 + 4*w + 3],
[997, 997, w^2 + 5*w - 12],
[997, 997, 5*w^2 - 3*w - 22],
[997, 997, 3*w^2 + 8*w - 2]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^5 + 3*x^4 - 6*x^3 - 23*x^2 - 11*x + 7;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, -e^4 - e^3 + 8*e^2 + 8*e - 3, -e - 1, -e - 1, 2*e^4 + 3*e^3 - 16*e^2 - 21*e + 7, -e^2 + 2, e^4 + e^3 - 8*e^2 - 9*e + 2, -e^3 - e^2 + 8*e + 5, -3*e^4 - 5*e^3 + 24*e^2 + 34*e - 10, -1, e^4 + e^3 - 9*e^2 - 6*e + 6, 2*e^4 + 3*e^3 - 16*e^2 - 22*e, e^4 - 8*e^2 + 10, -4*e^4 - 5*e^3 + 33*e^2 + 37*e - 21, 3*e^4 + 3*e^3 - 25*e^2 - 24*e + 16, e^4 + 2*e^3 - 8*e^2 - 14*e - 1, 2*e^3 + 2*e^2 - 12*e - 10, -e^4 - 2*e^3 + 8*e^2 + 16*e + 5, 4*e^4 + 6*e^3 - 30*e^2 - 47*e + 4, e^3 - 2*e^2 - 5*e + 5, -e^2 - e + 11, -e^4 - 2*e^3 + 8*e^2 + 16*e - 11, -6*e^4 - 9*e^3 + 49*e^2 + 62*e - 24, -4*e^4 - 5*e^3 + 35*e^2 + 36*e - 33, 3*e^3 - 23*e - 8, -2*e^4 - 3*e^3 + 18*e^2 + 25*e - 21, e^4 - 8*e^2 - 3*e - 2, 4*e^4 + 4*e^3 - 34*e^2 - 31*e + 25, 4*e^4 + 5*e^3 - 30*e^2 - 39*e + 1, 5*e^4 + 8*e^3 - 41*e^2 - 60*e + 9, -7*e^4 - 9*e^3 + 59*e^2 + 66*e - 42, -8*e^4 - 11*e^3 + 62*e^2 + 76*e - 26, -2*e^4 - 5*e^3 + 15*e^2 + 39*e + 5, 4*e^4 + 7*e^3 - 32*e^2 - 45*e + 18, -e^4 - 3*e^3 + 9*e^2 + 17*e - 14, 7*e^4 + 8*e^3 - 59*e^2 - 53*e + 49, 3*e^2 - 8, 6*e^4 + 10*e^3 - 45*e^2 - 70*e + 15, -8*e^4 - 12*e^3 + 64*e^2 + 88*e - 19, -3*e^4 - e^3 + 25*e^2 + 4*e - 26, e^4 + 2*e^3 - 8*e^2 - 15*e + 3, 2*e^4 + 6*e^3 - 16*e^2 - 44*e - 8, -2*e^4 - 4*e^3 + 17*e^2 + 28*e - 9, -5*e^4 - 9*e^3 + 37*e^2 + 61*e - 6, -4*e^4 - 7*e^3 + 30*e^2 + 44*e - 6, 4*e^4 + 7*e^3 - 29*e^2 - 48*e + 3, 3*e^4 + 4*e^3 - 28*e^2 - 29*e + 32, 8*e^4 + 11*e^3 - 68*e^2 - 76*e + 38, 3*e^3 + 2*e^2 - 24*e - 14, -2*e^4 - 2*e^3 + 15*e^2 + 18*e - 3, -3*e^4 - 5*e^3 + 23*e^2 + 27*e - 7, -2*e^4 + e^3 + 19*e^2 - 3*e - 25, -e^3 + e^2 + 11*e - 1, 3*e^4 + 6*e^3 - 24*e^2 - 39*e - 3, -e^4 - 3*e^3 + 11*e^2 + 20*e - 26, 8*e^4 + 10*e^3 - 64*e^2 - 74*e + 25, -7*e^4 - 10*e^3 + 53*e^2 + 68*e - 21, 4*e^4 + 5*e^3 - 30*e^2 - 32*e + 9, 7*e^4 + 8*e^3 - 59*e^2 - 57*e + 34, -6*e^4 - 7*e^3 + 49*e^2 + 54*e - 18, 12*e^4 + 17*e^3 - 95*e^2 - 122*e + 37, 5*e^4 + 10*e^3 - 43*e^2 - 70*e + 18, -6*e^4 - 12*e^3 + 47*e^2 + 89*e - 11, 5*e^4 + 6*e^3 - 36*e^2 - 43*e + 5, -2*e^3 - 2*e^2 + 13*e + 2, e^4 + 3*e^3 - 8*e^2 - 14*e + 2, 6*e^4 + 5*e^3 - 50*e^2 - 38*e + 43, 3*e^4 + 3*e^3 - 29*e^2 - 19*e + 32, 2*e^4 - 23*e^2 - 4*e + 39, 2*e^4 + 3*e^3 - 14*e^2 - 18*e - 22, 5*e^4 + 8*e^3 - 38*e^2 - 65*e + 6, -2*e^4 - 4*e^3 + 14*e^2 + 32*e - 11, -14*e^4 - 20*e^3 + 119*e^2 + 142*e - 70, -11*e^4 - 17*e^3 + 90*e^2 + 119*e - 42, e^4 + e^3 - 12*e^2 - 9*e + 6, 6*e^4 + 7*e^3 - 51*e^2 - 51*e + 22, -e^4 - e^3 + 9*e^2 + 6*e + 18, -3*e^4 - 3*e^3 + 31*e^2 + 23*e - 50, -8*e^4 - 6*e^3 + 69*e^2 + 49*e - 59, -4*e^3 + e^2 + 28*e, -7*e^4 - 10*e^3 + 61*e^2 + 74*e - 30, -7*e^4 - 10*e^3 + 60*e^2 + 70*e - 39, -3*e^4 - 4*e^3 + 22*e^2 + 29*e + 8, 9*e^4 + 14*e^3 - 73*e^2 - 95*e + 45, -3*e^4 + 2*e^3 + 30*e^2 - 10*e - 35, 4*e^3 - 20*e + 13, -10*e^4 - 14*e^3 + 80*e^2 + 96*e - 30, 3*e^3 + 5*e^2 - 18*e - 26, 16*e^4 + 20*e^3 - 130*e^2 - 140*e + 81, 10*e^4 + 13*e^3 - 76*e^2 - 90*e + 28, -6*e^4 - 7*e^3 + 53*e^2 + 50*e - 58, 10*e^4 + 13*e^3 - 81*e^2 - 90*e + 37, 9*e^4 + 15*e^3 - 72*e^2 - 100*e + 32, -e^4 + 8*e^2 - 2*e - 1, 4*e^3 + e^2 - 33*e - 9, 12*e^4 + 15*e^3 - 101*e^2 - 103*e + 82, -7*e^4 - 6*e^3 + 60*e^2 + 48*e - 35, -13*e^4 - 20*e^3 + 99*e^2 + 141*e - 22, -3*e^4 - 3*e^3 + 20*e^2 + 19*e - 1, 8*e^4 + 11*e^3 - 64*e^2 - 69*e + 33, -6*e^4 - 7*e^3 + 52*e^2 + 45*e - 57, -3*e^4 - 6*e^3 + 28*e^2 + 41*e - 34, 4*e^2 + 2*e - 2, 3*e^4 + 3*e^3 - 25*e^2 - 24*e - 5, -10*e^4 - 16*e^3 + 82*e^2 + 114*e - 31, 9*e^4 + 13*e^3 - 75*e^2 - 91*e + 48, -e^3 - 2*e^2 + e + 17, -3*e^4 + 3*e^3 + 25*e^2 - 15*e - 24, 10*e^4 + 17*e^3 - 82*e^2 - 110*e + 39, -5*e^4 - 8*e^3 + 44*e^2 + 59*e - 27, -12*e^4 - 13*e^3 + 97*e^2 + 89*e - 72, 6*e^4 + 11*e^3 - 48*e^2 - 73*e + 1, -9*e^4 - 8*e^3 + 71*e^2 + 58*e - 45, -4*e^4 - 6*e^3 + 35*e^2 + 46*e - 29, -12*e^4 - 17*e^3 + 93*e^2 + 113*e - 37, -9*e^4 - 14*e^3 + 72*e^2 + 96*e - 37, -e^4 - e^3 + 7*e^2 + 19*e - 6, -3*e^4 - 6*e^3 + 22*e^2 + 42*e - 1, -2*e^4 - 3*e^3 + 20*e^2 + 25*e - 26, 2*e^4 + 4*e^3 - 16*e^2 - 33*e - 9, -e^4 - 2*e^3 + 4*e^2 + 26*e + 39, -7*e^4 - 12*e^3 + 59*e^2 + 76*e - 45, -7*e^4 - 9*e^3 + 48*e^2 + 58*e + 11, -e^4 - 7*e^3 + 6*e^2 + 57*e + 12, -3*e^4 + 27*e^2 + 3*e - 45, 8*e^4 + 12*e^3 - 67*e^2 - 97*e + 40, 17*e^4 + 21*e^3 - 144*e^2 - 154*e + 107, 5*e^4 + 11*e^3 - 36*e^2 - 73*e + 8, 8*e^4 + 10*e^3 - 76*e^2 - 68*e + 82, -14*e^4 - 22*e^3 + 103*e^2 + 157*e - 15, 12*e^4 + 15*e^3 - 100*e^2 - 101*e + 84, -5*e^3 + 3*e^2 + 37*e - 15, -10*e^4 - 17*e^3 + 81*e^2 + 111*e - 39, 10*e^4 + 17*e^3 - 83*e^2 - 124*e + 43, 14*e^4 + 23*e^3 - 115*e^2 - 151*e + 60, -13*e^4 - 17*e^3 + 116*e^2 + 121*e - 91, 14*e^4 + 20*e^3 - 113*e^2 - 135*e + 74, 9*e^4 + 9*e^3 - 76*e^2 - 70*e + 62, -7*e^4 - 15*e^3 + 52*e^2 + 102*e - 5, 6*e^4 + 12*e^3 - 53*e^2 - 91*e + 26, 17*e^4 + 21*e^3 - 135*e^2 - 147*e + 73, 10*e^4 + 16*e^3 - 76*e^2 - 113*e + 41, 8*e^4 + 13*e^3 - 66*e^2 - 90*e + 29, -18*e^4 - 27*e^3 + 146*e^2 + 177*e - 81, 11*e^4 + 17*e^3 - 93*e^2 - 128*e + 54, -e^4 + 2*e^3 + 6*e^2 - e + 13, 4*e^4 + 7*e^3 - 35*e^2 - 61*e + 9, 12*e^4 + 21*e^3 - 100*e^2 - 142*e + 57, e^4 + 3*e^3 - 11*e^2 - 21*e + 26, 3*e^4 + 6*e^3 - 20*e^2 - 48*e - 21, -6*e^4 - 12*e^3 + 48*e^2 + 81*e - 1, -8*e^4 - 8*e^3 + 67*e^2 + 64*e - 60, 7*e^3 + 8*e^2 - 53*e - 31, -13*e^4 - 15*e^3 + 105*e^2 + 100*e - 65, e^4 - 4*e^3 - 15*e^2 + 27*e + 14, -17*e^4 - 24*e^3 + 132*e^2 + 177*e - 49, -6*e^4 - 12*e^3 + 50*e^2 + 87*e - 25, -5*e^4 - 8*e^3 + 32*e^2 + 55*e + 31, 11*e^4 + 10*e^3 - 97*e^2 - 84*e + 90, -7*e^4 - 11*e^3 + 54*e^2 + 63*e - 23, -8*e^4 - 15*e^3 + 63*e^2 + 105*e - 1, -11*e^4 - 14*e^3 + 95*e^2 + 100*e - 70, -7*e^4 - 9*e^3 + 61*e^2 + 64*e - 61, -2*e^4 - e^3 + 7*e^2 + 3*e + 23, -13*e^4 - 23*e^3 + 103*e^2 + 161*e - 32];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {23, 23, -w^2 + 8}>] := 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;