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

NN := ideal<ZF | {12, 6, w^3 - 3*w^2 - 3*w + 6}>;

primesArray := [
[3, 3, w],
[4, 2, -w^3 + 3*w^2 + w - 2],
[11, 11, -w^3 + 3*w^2 + 2*w - 5],
[13, 13, w^3 - 2*w^2 - 4*w + 2],
[13, 13, -w + 2],
[17, 17, w^3 - 3*w^2 - 2*w + 2],
[19, 19, -w^2 + w + 4],
[19, 19, w^3 - 2*w^2 - 3*w + 2],
[23, 23, w^2 - 2*w - 1],
[27, 3, w^3 - 2*w^2 - 5*w - 1],
[29, 29, -w^3 + 2*w^2 + 5*w - 1],
[37, 37, -w^3 + 2*w^2 + 5*w - 4],
[41, 41, 2*w^3 - 5*w^2 - 6*w + 4],
[53, 53, w^3 - 2*w^2 - 3*w - 2],
[59, 59, w - 4],
[67, 67, 2*w^2 - 3*w - 8],
[73, 73, 2*w^3 - 5*w^2 - 6*w + 5],
[89, 89, -2*w^3 + 6*w^2 + 5*w - 7],
[97, 97, -2*w^3 + 6*w^2 + 3*w - 7],
[97, 97, -w^3 + 3*w^2 + 3*w - 1],
[103, 103, w^2 - 4*w - 4],
[113, 113, 2*w^3 - 5*w^2 - 4*w + 1],
[113, 113, -w^3 + 3*w^2 + w - 5],
[139, 139, 2*w^2 - 3*w - 4],
[139, 139, w^3 - w^2 - 6*w - 1],
[149, 149, 2*w^3 - 5*w^2 - 5*w + 2],
[149, 149, w^3 - 4*w^2 + 10],
[151, 151, -3*w - 1],
[157, 157, w^2 - w - 8],
[157, 157, w^3 - 7*w - 8],
[163, 163, -3*w^3 + 8*w^2 + 8*w - 10],
[163, 163, w^3 - 3*w^2 - 4*w + 7],
[163, 163, -w^3 + 4*w^2 + w - 8],
[163, 163, -2*w^3 + 5*w^2 + 7*w - 4],
[167, 167, -2*w^3 + 5*w^2 + 8*w - 4],
[169, 13, -2*w^3 + 4*w^2 + 7*w - 5],
[173, 173, 2*w^3 - 4*w^2 - 9*w + 4],
[173, 173, -w^3 + 5*w^2 - 2*w - 5],
[181, 181, 3*w^3 - 7*w^2 - 12*w + 7],
[191, 191, w^3 - 3*w^2 - 3*w - 1],
[199, 199, 2*w^3 - 3*w^2 - 10*w - 1],
[199, 199, 2*w^3 - 6*w^2 - 6*w + 13],
[223, 223, -2*w^3 + 6*w^2 + 4*w - 11],
[223, 223, 3*w^3 - 7*w^2 - 12*w + 8],
[223, 223, -2*w^3 + 5*w^2 + 6*w - 1],
[229, 229, -w^3 + 3*w^2 + 2*w + 1],
[229, 229, 3*w^3 - 4*w^2 - 15*w - 5],
[233, 233, -3*w^3 + 7*w^2 + 11*w - 5],
[233, 233, 2*w^3 - 5*w^2 - 4*w + 5],
[241, 241, w^3 - 3*w^2 - 5*w + 7],
[257, 257, w^3 - 3*w^2 - 4*w + 8],
[263, 263, w^2 - 4*w - 5],
[271, 271, -2*w^3 + 6*w^2 + 5*w - 13],
[277, 277, -2*w^3 + 6*w^2 + 3*w - 8],
[277, 277, w^3 - 2*w^2 - 2*w - 2],
[281, 281, 2*w^3 - 5*w^2 - 4*w + 2],
[283, 283, -3*w^2 + 7*w + 10],
[283, 283, -w^3 + 4*w^2 - w - 7],
[293, 293, 2*w^3 - 5*w^2 - 3*w + 4],
[293, 293, w^3 - 8*w - 4],
[307, 307, -w^3 + 2*w^2 + 6*w - 2],
[311, 311, w^2 - 5],
[317, 317, -w^3 + 5*w^2 - 2*w - 11],
[331, 331, -w^3 + 3*w^2 - 5],
[349, 349, -2*w^2 + 5*w + 2],
[353, 353, -2*w^3 + 5*w^2 + 4*w - 4],
[353, 353, 3*w^3 - 7*w^2 - 10*w + 2],
[353, 353, -2*w^3 + 7*w^2 + 2*w - 10],
[353, 353, w^3 + w^2 - 9*w - 13],
[361, 19, -w^3 + w^2 + 5*w - 1],
[373, 373, -w^3 + 3*w^2 + 4*w - 11],
[373, 373, 2*w^3 - 5*w^2 - 5*w - 2],
[383, 383, w^3 - 5*w^2 + w + 13],
[389, 389, 3*w^3 - 10*w^2 - 4*w + 16],
[397, 397, -w^3 + 2*w^2 + 3*w + 5],
[397, 397, -3*w^3 + 7*w^2 + 10*w - 7],
[419, 419, -w^3 + 2*w^2 + 4*w + 4],
[419, 419, 3*w^3 - 7*w^2 - 12*w + 5],
[421, 421, -w^2 + 5*w - 2],
[421, 421, 3*w^3 - 7*w^2 - 10*w + 5],
[431, 431, w^2 - 10],
[439, 439, w^3 - w^2 - 7*w - 7],
[461, 461, -2*w^3 + 7*w^2 + 5*w - 10],
[467, 467, -3*w^3 + 6*w^2 + 14*w - 4],
[479, 479, 3*w^2 - 6*w - 5],
[479, 479, -4*w^3 + 11*w^2 + 10*w - 10],
[491, 491, -w^3 + w^2 + 6*w - 1],
[499, 499, -4*w^3 + 9*w^2 + 15*w - 10],
[499, 499, 2*w^3 - 5*w^2 - 7*w + 2],
[503, 503, -2*w^3 + 7*w^2 - 8],
[521, 521, 2*w^2 - w - 7],
[521, 521, -3*w^3 + 7*w^2 + 11*w - 11],
[569, 569, 2*w^2 - 5*w - 1],
[569, 569, -4*w - 1],
[571, 571, -w^3 + 3*w^2 - 7],
[577, 577, -w^3 + w^2 + 8*w - 4],
[587, 587, -3*w^3 + 8*w^2 + 7*w - 8],
[593, 593, 3*w^2 - 4*w - 11],
[593, 593, -3*w^3 + 9*w^2 + 6*w - 14],
[601, 601, -3*w^3 + 9*w^2 + 4*w - 10],
[601, 601, 5*w^3 - 9*w^2 - 24*w - 2],
[607, 607, 2*w^2 - 7*w - 5],
[613, 613, w^3 - 6*w - 2],
[613, 613, -w^2 + 3*w + 8],
[617, 617, -w^2 + 4*w - 5],
[619, 619, 3*w^3 - 6*w^2 - 13*w + 5],
[625, 5, -5],
[631, 631, -2*w^3 + 4*w^2 + 8*w - 7],
[647, 647, -2*w^3 + 4*w^2 + 5*w - 1],
[647, 647, w^3 - 3*w^2 - 2*w - 2],
[653, 653, w^3 - 4*w^2 - w + 13],
[653, 653, w^3 - 2*w^2 - 7*w + 5],
[659, 659, -4*w^3 + 10*w^2 + 15*w - 14],
[659, 659, 5*w^3 - 12*w^2 - 17*w + 14],
[661, 661, w^2 - 5*w - 2],
[661, 661, -w^3 + 5*w^2 - 2*w - 7],
[673, 673, -w^3 + w^2 + 6*w - 2],
[683, 683, 2*w^3 - 8*w^2 + w + 8],
[683, 683, 5*w^3 - 12*w^2 - 19*w + 14],
[701, 701, -2*w^3 + 6*w^2 + 5*w - 4],
[709, 709, -w^2 + 3*w - 4],
[709, 709, -2*w^3 + 5*w^2 + 9*w - 8],
[719, 719, w^2 - 7],
[719, 719, -2*w^3 + 3*w^2 + 8*w + 5],
[733, 733, -2*w^3 + 7*w^2 + 3*w - 10],
[739, 739, 4*w^3 - 11*w^2 - 13*w + 14],
[739, 739, -5*w^3 + 10*w^2 + 21*w - 4],
[739, 739, 3*w^3 - 8*w^2 - 6*w + 8],
[739, 739, -w^3 + w^2 + 8*w - 5],
[743, 743, -w - 5],
[743, 743, w^3 - w^2 - 8*w - 8],
[751, 751, -2*w^3 + 5*w^2 + 8*w - 2],
[751, 751, -2*w^3 + 6*w^2 - 1],
[757, 757, -2*w^3 + 6*w^2 + 3*w - 10],
[757, 757, 5*w^2 - 9*w - 22],
[761, 761, -2*w^3 + 4*w^2 + 9*w - 7],
[761, 761, 2*w^3 - 3*w^2 - 12*w - 1],
[769, 769, 5*w^3 - 13*w^2 - 14*w + 10],
[773, 773, -3*w^3 + 5*w^2 + 16*w + 1],
[773, 773, w^3 - 11*w + 2],
[787, 787, -w^3 + 4*w^2 - w - 11],
[797, 797, -5*w^3 + 12*w^2 + 17*w - 11],
[809, 809, -2*w^3 + 7*w^2 + 4*w - 11],
[809, 809, 4*w^3 - 9*w^2 - 14*w + 4],
[821, 821, -w^3 + 5*w^2 - 2*w - 17],
[827, 827, -3*w^3 + 8*w^2 + 9*w - 7],
[829, 829, 2*w^3 - 6*w^2 - 5*w + 2],
[829, 829, -w^3 + 4*w^2 - w - 10],
[839, 839, w^2 - 8],
[853, 853, -5*w^3 + 12*w^2 + 18*w - 16],
[853, 853, w^3 - 6*w - 8],
[863, 863, 5*w^3 - 11*w^2 - 19*w + 7],
[881, 881, -3*w^3 + 9*w^2 + 9*w - 11],
[907, 907, -w^3 + 2*w^2 + 7*w - 4],
[907, 907, -2*w^3 + 3*w^2 + 7*w + 4],
[929, 929, -w^3 + 5*w^2 - 3*w - 11],
[937, 937, 2*w^3 - 5*w^2 - 6*w + 11],
[937, 937, -2*w^3 + 6*w^2 + 3*w - 11],
[941, 941, -3*w^3 + 8*w^2 + 11*w - 8],
[953, 953, -4*w^3 + 10*w^2 + 10*w - 13],
[953, 953, w^3 - 5*w^2 - w + 13],
[967, 967, 3*w^3 - 7*w^2 - 9*w + 7],
[967, 967, -2*w^3 + 5*w^2 + 8*w + 1],
[967, 967, 4*w^3 - 13*w^2 - 6*w + 20],
[967, 967, -2*w^2 + 3*w + 13],
[971, 971, -6*w^3 + 13*w^2 + 25*w - 8],
[977, 977, w^3 - w^2 - 4*w - 10],
[977, 977, 2*w^3 - 6*w^2 - 7*w + 13],
[983, 983, -4*w^3 + 10*w^2 + 11*w - 13],
[983, 983, -w^3 + 5*w^2 - w - 7],
[983, 983, 3*w^3 - 9*w^2 - 7*w + 19],
[983, 983, -2*w^3 + 3*w^2 + 10*w - 1],
[997, 997, 5*w^3 - 12*w^2 - 17*w + 13],
[997, 997, -2*w^3 + 3*w^2 + 13*w - 4],
[997, 997, w^3 - 7*w - 2],
[997, 997, w^3 - 9*w - 2]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^3 - 5*x^2 - 21*x + 98;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [-1, -1, e, -2/7*e^2 + 3/7*e + 4, -1/7*e^2 + 5/7*e + 3, 2/7*e^2 + 4/7*e - 6, 1/7*e^2 + 2/7*e - 2, 4/7*e^2 + 1/7*e - 12, 6/7*e^2 - 2/7*e - 20, -5/7*e^2 - 3/7*e + 21, -1/7*e^2 - 2/7*e + 6, -1/7*e^2 - 2/7*e - 2, -3/7*e^2 + 8/7*e + 7, -5/7*e^2 - 3/7*e + 19, -9/7*e^2 + 10/7*e + 28, -3/7*e^2 + 1/7*e + 17, -10/7*e^2 + 8/7*e + 28, -10/7*e^2 + 15/7*e + 28, 2/7*e^2 + 4/7*e - 2, -4/7*e^2 + 20/7*e + 12, -1/7*e^2 - 9/7*e + 5, 2/7*e^2 - 17/7*e - 3, -2/7*e^2 - 18/7*e + 14, -13/7*e^2 + 9/7*e + 37, e^2 - 2*e - 22, -10, 9/7*e^2 + 18/7*e - 34, -2/7*e^2 + 3/7*e - 1, 15/7*e^2 - 12/7*e - 44, -1/7*e^2 - 16/7*e - 6, -13/7*e^2 + 23/7*e + 31, 10/7*e^2 + 6/7*e - 32, 6/7*e^2 - 9/7*e - 16, 1/7*e^2 - 5/7*e + 1, -2*e^2 + 2*e + 40, 2*e^2 - 38, 1/7*e^2 + 2/7*e, 10/7*e^2 - 8/7*e - 26, 5/7*e^2 + 10/7*e - 12, -13/7*e^2 + 2/7*e + 45, 2/7*e^2 + 4/7*e - 6, -1/7*e^2 + 12/7*e - 12, 8/7*e^2 - 5/7*e - 21, -20/7*e^2 + 2/7*e + 60, e^2 + 2*e - 35, 8/7*e^2 - 33/7*e - 16, -6/7*e^2 + 2/7*e + 18, 1/7*e^2 - 12/7*e + 6, -4/7*e^2 - 1/7*e + 31, -2*e^2 + 40, 15/7*e^2 + 2/7*e - 45, 20/7*e^2 - 30/7*e - 56, 2/7*e^2 - 10/7*e, -8/7*e^2 + 5/7*e + 26, 5/7*e^2 - 11/7*e - 27, 1/7*e^2 + 37/7*e - 7, -e^2 - e + 13, 6/7*e^2 - 2/7*e - 6, -e^2 + e + 27, 18/7*e^2 - 13/7*e - 60, -8/7*e^2 + 19/7*e + 32, -18/7*e^2 + 20/7*e + 38, 9/7*e^2 - 10/7*e - 28, -1/7*e^2 - 2/7*e + 18, 5/7*e^2 - 11/7*e + 3, -6/7*e^2 + 16/7*e + 10, -2/7*e^2 - 4/7*e - 2, 18/7*e^2 - 34/7*e - 48, -4/7*e^2 - 1/7*e + 3, 2*e^2 - 48, -4/7*e^2 - 1/7*e + 30, 3/7*e^2 - 29/7*e - 17, 2*e^2 + 3*e - 59, 16/7*e^2 - 3/7*e - 42, 10/7*e^2 - 22/7*e - 30, 11/7*e^2 - 41/7*e - 21, 22/7*e^2 - 19/7*e - 54, 11/7*e^2 - 6/7*e - 40, -5/7*e^2 - 3/7*e + 19, -2*e^2 - 5*e + 52, 13/7*e^2 - 44/7*e - 35, -10/7*e^2 + 29/7*e + 30, 26/7*e^2 - 25/7*e - 62, -11/7*e^2 - 15/7*e + 21, 17/7*e^2 - 15/7*e - 49, -6/7*e^2 - 12/7*e + 16, -23/7*e^2 + 10/7*e + 58, -9/7*e^2 + 45/7*e + 31, 19/7*e^2 - 39/7*e - 51, 18/7*e^2 - 6/7*e - 44, -2*e^2 + 54, 22/7*e^2 - 12/7*e - 56, -3/7*e^2 + 36/7*e - 9, -8/7*e^2 - 2/7*e + 16, -19/7*e^2 - 24/7*e + 84, 3*e^2 - 5*e - 47, 18/7*e^2 - 27/7*e - 36, -24/7*e^2 + 1/7*e + 65, 10/7*e^2 + 13/7*e - 41, 10/7*e^2 - 8/7*e - 32, 16/7*e^2 - 24/7*e - 46, -20/7*e^2 - 12/7*e + 64, -8/7*e^2 - 16/7*e + 10, 30/7*e^2 - 3/7*e - 106, 2/7*e^2 + 18/7*e + 18, 5/7*e^2 - 11/7*e - 15, 8/7*e^2 + 2/7*e - 44, 9/7*e^2 - 24/7*e - 44, 8/7*e^2 + 16/7*e - 26, -5*e^2 + 2*e + 105, 3/7*e^2 - 15/7*e + 1, 12/7*e^2 - 39/7*e - 40, 2/7*e^2 - 3/7*e + 36, 10/7*e^2 + 34/7*e - 36, 3*e^2 + 2*e - 96, -4*e^2 + 4*e + 70, -17/7*e^2 + 15/7*e + 71, 34/7*e^2 - 30/7*e - 106, 10/7*e^2 + 13/7*e - 24, -3/7*e^2 + 15/7*e - 21, 10/7*e^2 + 27/7*e - 26, e^2 + 7*e - 21, -34/7*e^2 + 30/7*e + 82, -15/7*e^2 - 2/7*e + 29, -16/7*e^2 - 18/7*e + 82, -2/7*e^2 - 46/7*e + 20, 13/7*e^2 - 44/7*e - 24, e^2 - 2*e - 20, 31/7*e^2 + 6/7*e - 94, -6/7*e^2 + 30/7*e + 20, -22/7*e^2 + 33/7*e + 74, -6/7*e^2 + 23/7*e + 34, -5/7*e^2 - 24/7*e + 39, -31/7*e^2 + 15/7*e + 97, 3/7*e^2 + 13/7*e - 13, 2/7*e^2 - 10/7*e - 12, -6/7*e^2 - 12/7*e + 14, 34/7*e^2 - 44/7*e - 94, -1/7*e^2 - 16/7*e + 28, 15/7*e^2 - 5/7*e - 71, -31/7*e^2 + 22/7*e + 108, 26/7*e^2 - 25/7*e - 62, 8/7*e^2 - 26/7*e + 6, -16/7*e^2 + 10/7*e + 72, 4*e^2 - 4*e - 98, -5/7*e^2 + 67/7*e + 11, 32/7*e^2 - 20/7*e - 102, 26/7*e^2 - 32/7*e - 46, 30/7*e^2 - 38/7*e - 86, -7*e + 14, -5/7*e^2 + 18/7*e - 22, -2/7*e^2 - 4/7*e + 20, 2/7*e^2 - 24/7*e + 18, -19/7*e^2 - 31/7*e + 65, 23/7*e^2 - 31/7*e - 47, -23/7*e^2 - 4/7*e + 69, 2/7*e^2 - 38/7*e - 14, -2*e^2 - 6*e + 52, 10/7*e^2 + 41/7*e - 36, 2*e^2 + 5*e - 49, -8*e - 14, 2*e^2 - e - 82, -22/7*e^2 + 26/7*e + 68, -30/7*e^2 + 24/7*e + 70, -10/7*e^2 + 36/7*e + 28, 19/7*e^2 - 18/7*e - 34, -8/7*e^2 - 30/7*e + 30, -15/7*e^2 + 5/7*e + 37, 18/7*e^2 + 8/7*e - 58, 24/7*e^2 - 1/7*e - 55, 8/7*e^2 + 2/7*e - 36, -15/7*e^2 + 26/7*e + 76, -10/7*e^2 + 22/7*e + 32, -2/7*e^2 + 3/7*e - 14, -12/7*e^2 - 3/7*e, 31/7*e^2 - 36/7*e - 84];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {3, 3, w}>] := 1;
ALEigenvalues[ideal<ZF | {4, 2, -w^3 + 3*w^2 + w - 2}>] := 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;