/*
  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 | {11, 11, -w^3 + 3*w^2 + 2*w - 5}>;

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^6 - 19*x^4 + 116*x^2 - 225;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, 1/15*e^5 - 4/15*e^3 - 19/15*e, -1, 4/15*e^5 - 46/15*e^3 + 134/15*e, e^4 - 13*e^2 + 41, -2/3*e^5 + 26/3*e^3 - 82/3*e, e^4 - 12*e^2 + 34, -13/15*e^5 + 157/15*e^3 - 443/15*e, -11/15*e^5 + 149/15*e^3 - 481/15*e, -4/15*e^5 + 46/15*e^3 - 104/15*e, 2/3*e^5 - 26/3*e^3 + 79/3*e, -2/15*e^5 + 38/15*e^3 - 172/15*e, -16/15*e^5 + 169/15*e^3 - 371/15*e, -e^4 + 8*e^2 - 6, -2*e^4 + 29*e^2 - 96, 8, 14/15*e^5 - 206/15*e^3 + 694/15*e, -28/15*e^5 + 352/15*e^3 - 1028/15*e, e^4 - 13*e^2 + 37, 22/15*e^5 - 238/15*e^3 + 542/15*e, 3*e^4 - 42*e^2 + 132, -2/5*e^5 + 23/5*e^3 - 37/5*e, -3*e^4 + 37*e^2 - 99, -8*e^4 + 94*e^2 - 248, -2*e^3 + 16*e, -e^3 + 7*e, 2*e^4 - 27*e^2 + 96, 4/3*e^5 - 55/3*e^3 + 179/3*e, 8*e^4 - 97*e^2 + 270, 4*e^4 - 48*e^2 + 138, -2*e^5 + 22*e^3 - 52*e, 5*e^4 - 62*e^2 + 176, 6*e^4 - 78*e^2 + 234, -2/5*e^5 + 8/5*e^3 + 38/5*e, 6/5*e^5 - 64/5*e^3 + 156/5*e, -12/5*e^5 + 158/5*e^3 - 482/5*e, 4/3*e^5 - 58/3*e^3 + 191/3*e, 2*e^4 - 26*e^2 + 86, -28/15*e^5 + 322/15*e^3 - 818/15*e, -3*e^4 + 38*e^2 - 114, -6/5*e^5 + 64/5*e^3 - 141/5*e, -10*e^4 + 123*e^2 - 356, e^4 - 10*e^2 + 40, 6/5*e^5 - 59/5*e^3 + 81/5*e, -4/15*e^5 + 46/15*e^3 - 194/15*e, -4*e^4 + 48*e^2 - 128, 4/5*e^5 - 61/5*e^3 + 249/5*e, 32/15*e^5 - 353/15*e^3 + 787/15*e, -9/5*e^5 + 111/5*e^3 - 319/5*e, 2*e^4 - 21*e^2 + 54, 5*e^2 - 40, 3*e^2 - 12, -6*e^4 + 66*e^2 - 156, -4*e^2 + 26, -12*e^4 + 144*e^2 - 396, -56/15*e^5 + 644/15*e^3 - 1666/15*e, -e^4 + 15*e^2 - 39, 2*e^4 - 30*e^2 + 96, -34/15*e^5 + 436/15*e^3 - 1334/15*e, 26/15*e^5 - 374/15*e^3 + 1276/15*e, -10/3*e^5 + 124/3*e^3 - 350/3*e, -7*e^4 + 83*e^2 - 213, 8, 2*e^4 - 34*e^2 + 136, 3*e^5 - 37*e^3 + 109*e, 12/5*e^5 - 158/5*e^3 + 462/5*e, 22/15*e^5 - 208/15*e^3 + 392/15*e, 4*e^4 - 44*e^2 + 120, -2*e^4 + 38*e^2 - 150, -11/5*e^5 + 139/5*e^3 - 381/5*e, 12*e^4 - 132*e^2 + 310, 8*e^4 - 89*e^2 + 224, 10*e^4 - 126*e^2 + 376, -11*e^4 + 133*e^2 - 379, -4*e^4 + 45*e^2 - 118, 56/15*e^5 - 734/15*e^3 + 2266/15*e, 10*e^4 - 112*e^2 + 280, -26/15*e^5 + 344/15*e^3 - 1096/15*e, -2*e^5 + 24*e^3 - 67*e, 14/5*e^5 - 156/5*e^3 + 364/5*e, 11*e^4 - 143*e^2 + 435, -12*e^4 + 151*e^2 - 446, -4*e^4 + 30*e^2 - 14, 6/5*e^5 - 94/5*e^3 + 306/5*e, 58/15*e^5 - 742/15*e^3 + 2258/15*e, 7/3*e^5 - 97/3*e^3 + 323/3*e, 16/5*e^5 - 194/5*e^3 + 556/5*e, -34/15*e^5 + 361/15*e^3 - 809/15*e, -5*e^5 + 63*e^3 - 189*e, -7*e^4 + 85*e^2 - 243, -6*e^4 + 64*e^2 - 136, 14/15*e^5 - 146/15*e^3 + 229/15*e, -16/5*e^5 + 174/5*e^3 - 361/5*e, 16/3*e^5 - 196/3*e^3 + 563/3*e, -2*e^4 + 32*e^2 - 86, -3/5*e^5 + 27/5*e^3 - 33/5*e, 14/15*e^5 - 191/15*e^3 + 439/15*e, 16*e^4 - 190*e^2 + 518, e^4 - 17*e^2 + 29, -5*e^4 + 52*e^2 - 104, -6/5*e^5 + 94/5*e^3 - 341/5*e, -14/15*e^5 + 206/15*e^3 - 964/15*e, -4/5*e^5 + 56/5*e^3 - 244/5*e, -10*e^4 + 118*e^2 - 322, 6*e^4 - 84*e^2 + 268, 26/15*e^5 - 224/15*e^3 + 286/15*e, -11*e^4 + 134*e^2 - 364, 86/15*e^5 - 1064/15*e^3 + 3076/15*e, -26/15*e^5 + 269/15*e^3 - 541/15*e, -6*e^4 + 74*e^2 - 176, 20*e^2 - 126, 23/15*e^5 - 317/15*e^3 + 943/15*e, -14/15*e^5 + 146/15*e^3 - 124/15*e, 71/15*e^5 - 839/15*e^3 + 2251/15*e, 88/15*e^5 - 1027/15*e^3 + 2663/15*e, -e^4 + 18*e^2 - 74, 22/15*e^5 - 358/15*e^3 + 1412/15*e, -6*e^4 + 78*e^2 - 222, 92/15*e^5 - 1088/15*e^3 + 2902/15*e, 14/3*e^5 - 176/3*e^3 + 508/3*e, 5*e^4 - 48*e^2 + 100, 7/5*e^5 - 73/5*e^3 + 187/5*e, -4*e^4 + 48*e^2 - 104, 2*e^4 - 38*e^2 + 132, 2*e^2 + 2, 4/15*e^5 - 46/15*e^3 - 31/15*e, 28/15*e^5 - 352/15*e^3 + 878/15*e, 4/5*e^5 - 56/5*e^3 + 154/5*e, 37/5*e^5 - 463/5*e^3 + 1377/5*e, -e^4 + 25*e^2 - 135, 8*e^4 - 100*e^2 + 314, 82/15*e^5 - 1078/15*e^3 + 3392/15*e, 10*e^4 - 132*e^2 + 400, -4*e^4 + 62*e^2 - 236, 13*e^4 - 165*e^2 + 471, -14/3*e^5 + 176/3*e^3 - 508/3*e, -6/5*e^5 + 84/5*e^3 - 266/5*e, 22/5*e^5 - 283/5*e^3 + 907/5*e, 46/15*e^5 - 589/15*e^3 + 1751/15*e, 32/15*e^5 - 518/15*e^3 + 1942/15*e, 5*e^4 - 65*e^2 + 205, -14/3*e^5 + 182/3*e^3 - 532/3*e, -4*e^4 + 44*e^2 - 134, -8/3*e^5 + 110/3*e^3 - 358/3*e, 6*e^4 - 60*e^2 + 120, -16/15*e^5 + 109/15*e^3 + 139/15*e, -83/15*e^5 + 1037/15*e^3 - 3073/15*e, -8*e^4 + 95*e^2 - 280, 5*e^4 - 50*e^2 + 98, -64/15*e^5 + 886/15*e^3 - 2834/15*e, -14*e^4 + 182*e^2 - 544, 64/15*e^5 - 796/15*e^3 + 2324/15*e, -38/15*e^5 + 362/15*e^3 - 658/15*e, -4/5*e^5 + 86/5*e^3 - 344/5*e, 9*e^4 - 91*e^2 + 187, -2*e^2, 16*e^4 - 202*e^2 + 590, -17*e^4 + 213*e^2 - 631, -92/15*e^5 + 1238/15*e^3 - 3832/15*e, 22/5*e^5 - 288/5*e^3 + 947/5*e, 4*e^2 - 18, 56/15*e^5 - 764/15*e^3 + 2596/15*e, 86/15*e^5 - 1004/15*e^3 + 2701/15*e, 17*e^4 - 215*e^2 + 631, 2*e^4 - 2*e^2 - 82, 7/3*e^5 - 73/3*e^3 + 149/3*e, -2*e^4 + 44*e^2 - 178, 6*e^4 - 76*e^2 + 254, -1/3*e^5 + 25/3*e^3 - 89/3*e, -8*e^4 + 84*e^2 - 168, -4*e^4 + 66*e^2 - 244, 2*e^5 - 24*e^3 + 68*e, 8/5*e^5 - 82/5*e^3 + 103/5*e, 36/5*e^5 - 449/5*e^3 + 1321/5*e, 6*e^5 - 68*e^3 + 170*e, -4/5*e^5 + 66/5*e^3 - 174/5*e];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {11, 11, -w^3 + 3*w^2 + 2*w - 5}>] := 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;