/*
  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![11, 7, -6, -2, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {16, 2, 2}>;

primesArray := [
[5, 5, -w^2 + 2*w + 3],
[9, 3, w^3 - 3*w^2 - 2*w + 9],
[9, 3, -w^3 + 5*w + 5],
[11, 11, w],
[11, 11, w - 1],
[16, 2, 2],
[19, 19, -w^3 + 2*w^2 + 3*w - 2],
[19, 19, w^3 - w^2 - 4*w + 2],
[29, 29, w^3 - 4*w^2 - w + 10],
[29, 29, -w^3 + 3*w^2 + w - 7],
[41, 41, 3*w^2 - 2*w - 10],
[49, 7, -2*w^2 + 3*w + 8],
[49, 7, w^3 - 2*w^2 - 2*w + 5],
[71, 71, -w - 3],
[71, 71, w - 4],
[79, 79, -w^3 + w^2 + 3*w + 3],
[79, 79, -w^3 + 2*w^2 + 2*w - 6],
[89, 89, w^3 - 3*w^2 - 3*w + 7],
[89, 89, w^3 - 6*w - 2],
[101, 101, 2*w^3 - 5*w^2 - 3*w + 9],
[101, 101, -w^3 + 3*w^2 + 3*w - 12],
[109, 109, -w^3 + 4*w^2 + w - 9],
[109, 109, -w^3 + 2*w^2 + 4*w - 2],
[121, 11, -3*w^2 + 3*w + 10],
[131, 131, -2*w^3 + 5*w^2 + 4*w - 9],
[131, 131, -w^3 + w^2 + 3*w + 5],
[131, 131, -w^3 + 2*w^2 + 2*w - 8],
[131, 131, -w^3 + 6*w^2 - 18],
[149, 149, -w^3 - 2*w^2 + 7*w + 8],
[149, 149, -2*w^3 + 6*w^2 + 5*w - 16],
[151, 151, w^2 + w - 5],
[151, 151, w^2 - 3*w - 3],
[169, 13, w^3 - 4*w^2 - w + 6],
[169, 13, 2*w^3 - 5*w^2 - 4*w + 6],
[179, 179, w^3 - 4*w - 6],
[179, 179, -w^3 + 3*w^2 + w - 9],
[181, 181, w^3 + 3*w^2 - 7*w - 15],
[181, 181, w^3 - 6*w^2 + 2*w + 18],
[191, 191, 5*w^2 - 7*w - 15],
[191, 191, -4*w^2 + 5*w + 12],
[199, 199, -w^3 + 3*w^2 + 3*w - 6],
[199, 199, w^3 - 6*w - 1],
[211, 211, w^3 - 7*w - 4],
[211, 211, -w^3 + 3*w^2 + 4*w - 10],
[229, 229, -w^3 - w^2 + 5*w + 9],
[229, 229, -w^3 + 4*w^2 - 12],
[239, 239, 4*w^2 - 5*w - 17],
[239, 239, 4*w^2 - 3*w - 18],
[241, 241, -w^3 + 3*w^2 + w - 2],
[241, 241, -w^3 + 3*w^2 + 4*w - 9],
[241, 241, w^3 - 7*w - 3],
[241, 241, -w^3 + 4*w - 1],
[271, 271, -w - 4],
[271, 271, w^3 - 3*w^2 - 2*w + 2],
[271, 271, -2*w^3 + 6*w^2 + 2*w - 7],
[271, 271, w - 5],
[281, 281, w^3 - 2*w^2 - 3*w - 1],
[281, 281, -w^3 + w^2 + 4*w - 5],
[289, 17, -2*w^3 + w^2 + 9*w + 1],
[289, 17, -2*w^3 + 5*w^2 + 5*w - 9],
[311, 311, w^3 - w^2 - 6*w + 1],
[311, 311, -w^3 + 2*w^2 + 5*w - 5],
[331, 331, -3*w^3 + 5*w^2 + 10*w - 5],
[331, 331, 3*w^3 - 2*w^2 - 12*w - 6],
[349, 349, -2*w^3 + 2*w^2 + 10*w + 3],
[349, 349, -2*w^3 - w^2 + 12*w + 10],
[349, 349, 2*w^3 + w^2 - 10*w - 12],
[349, 349, -2*w^3 + 4*w^2 + 8*w - 13],
[361, 19, -4*w^2 + 4*w + 13],
[379, 379, w^3 + w^2 - 5*w - 10],
[379, 379, -3*w^3 + 4*w^2 + 12*w - 7],
[379, 379, w^3 + 4*w^2 - 8*w - 21],
[379, 379, w^3 - 3*w - 6],
[389, 389, 2*w^3 - 6*w^2 - 4*w + 13],
[389, 389, -5*w^2 + 7*w + 13],
[389, 389, 2*w^3 - 7*w^2 - 5*w + 20],
[389, 389, 2*w^3 - 10*w - 5],
[419, 419, -w^3 + 5*w^2 + w - 13],
[419, 419, 4*w^2 - 7*w - 14],
[419, 419, 2*w^3 - 3*w^2 - 5*w - 1],
[419, 419, -w^3 - 2*w^2 + 8*w + 8],
[421, 421, -2*w^2 + 5*w + 7],
[421, 421, w^3 - 6*w^2 + 5*w + 13],
[439, 439, 2*w^3 - 2*w^2 - 9*w + 3],
[439, 439, w^3 + w^2 - 7*w - 4],
[449, 449, 3*w^3 - 6*w^2 - 9*w + 8],
[449, 449, 5*w^2 - 8*w - 17],
[449, 449, 2*w^3 - w^2 - 8*w - 7],
[449, 449, -3*w^3 + 3*w^2 + 12*w - 4],
[479, 479, -2*w^3 + 3*w^2 + 5*w + 3],
[479, 479, 3*w^3 - 7*w^2 - 9*w + 15],
[499, 499, -w^3 + 4*w^2 + 4*w - 9],
[499, 499, w^3 + w^2 - 9*w - 2],
[509, 509, 2*w^3 + 2*w^2 - 11*w - 16],
[509, 509, -2*w^3 + 8*w^2 + w - 23],
[521, 521, -3*w^3 + w^2 + 13*w + 6],
[521, 521, w^3 + 4*w^2 - 10*w - 16],
[541, 541, 3*w^3 - 2*w^2 - 13*w - 3],
[541, 541, -w^3 + 7*w^2 - w - 25],
[541, 541, w^3 + 4*w^2 - 10*w - 20],
[541, 541, -3*w^3 + 7*w^2 + 8*w - 15],
[571, 571, w^3 + 4*w^2 - 9*w - 15],
[571, 571, w^3 - 7*w^2 + 2*w + 19],
[601, 601, -3*w^3 + 2*w^2 + 13*w + 5],
[601, 601, 3*w^3 - 7*w^2 - 8*w + 17],
[619, 619, w^3 - 3*w - 7],
[619, 619, w^3 - 5*w^2 + 3*w + 13],
[619, 619, 3*w^3 - 4*w^2 - 9*w - 2],
[619, 619, -w^3 + 3*w^2 - 9],
[641, 641, 6*w^2 - 4*w - 23],
[641, 641, -3*w^3 + w^2 + 12*w + 5],
[659, 659, -2*w^3 + 6*w^2 + 3*w - 17],
[659, 659, -2*w^3 + 9*w + 10],
[661, 661, 3*w^3 - 6*w^2 - 9*w + 14],
[661, 661, 5*w^2 - 6*w - 18],
[661, 661, -5*w^2 + 4*w + 19],
[661, 661, 3*w^3 - 3*w^2 - 12*w - 2],
[691, 691, -3*w^3 + 5*w^2 + 10*w - 6],
[691, 691, -3*w^3 + 4*w^2 + 11*w - 6],
[701, 701, -w^3 - 4*w^2 + 9*w + 20],
[701, 701, -w^3 + 4*w^2 + 3*w - 19],
[701, 701, -w^3 + 4*w^2 + 4*w - 15],
[701, 701, w^3 - 7*w^2 + 2*w + 24],
[709, 709, 3*w^2 - 5*w - 13],
[709, 709, 3*w^2 - w - 15],
[719, 719, -w^3 + 3*w^2 + 4*w - 4],
[719, 719, -w^3 + 7*w - 2],
[739, 739, -w^3 + 4*w^2 + 3*w - 9],
[739, 739, -2*w^3 - w^2 + 11*w + 8],
[739, 739, 2*w^3 - 7*w^2 - 3*w + 16],
[739, 739, -w^3 - w^2 + 8*w + 3],
[751, 751, 3*w^3 - 5*w^2 - 10*w + 10],
[751, 751, 2*w^2 - 5*w - 5],
[761, 761, 3*w^3 - 3*w^2 - 11*w + 3],
[761, 761, w^3 - 2*w^2 - 3*w - 2],
[761, 761, -w^3 + w^2 + 4*w - 6],
[761, 761, -3*w^3 + 6*w^2 + 8*w - 8],
[769, 769, -w^3 - w^2 + 7*w + 1],
[769, 769, -2*w^3 + 6*w^2 + 6*w - 15],
[769, 769, -2*w^3 + 5*w^2 + 5*w - 7],
[769, 769, 2*w^3 + w^2 - 12*w - 9],
[809, 809, -2*w^3 + 8*w^2 + 4*w - 23],
[809, 809, -3*w^3 + 11*w^2 + 2*w - 28],
[829, 829, -3*w^3 + 7*w^2 + 7*w - 9],
[829, 829, w^3 + 2*w^2 - 6*w - 15],
[829, 829, -w^3 + 5*w^2 - w - 18],
[829, 829, -3*w^3 + 2*w^2 + 12*w - 2],
[839, 839, 2*w^3 - 9*w^2 + 20],
[839, 839, -3*w^3 + 10*w^2 + 5*w - 30],
[841, 29, 5*w^2 - 5*w - 16],
[881, 881, -w^3 - 4*w^2 + 8*w + 18],
[881, 881, 2*w^3 - 7*w^2 - w + 9],
[881, 881, 2*w^3 - 6*w^2 - 3*w + 6],
[881, 881, -w^3 + 7*w^2 - 3*w - 21],
[911, 911, -w^3 - 4*w^2 + 8*w + 16],
[911, 911, -w^3 - w^2 + 9*w + 5],
[911, 911, -w^3 + 4*w^2 + 4*w - 12],
[911, 911, w^3 - 7*w^2 + 3*w + 19],
[919, 919, -4*w^3 + 7*w^2 + 14*w - 9],
[919, 919, 4*w^3 - 5*w^2 - 16*w + 8],
[929, 929, 3*w^2 - w - 16],
[929, 929, 3*w^2 - 5*w - 14],
[941, 941, -w^3 - w^2 + 9*w + 7],
[941, 941, -3*w^3 + 7*w^2 + 7*w - 15],
[941, 941, 3*w^3 - 2*w^2 - 12*w - 4],
[941, 941, w^3 + 4*w^2 - 13*w - 16],
[961, 31, -2*w^2 + 2*w + 13],
[961, 31, 5*w^2 - 5*w - 18],
[971, 971, w^3 + 4*w^2 - 9*w - 19],
[971, 971, w^3 - 7*w^2 + 2*w + 23],
[991, 991, 3*w^3 - 7*w^2 - 7*w + 14],
[991, 991, -3*w^3 + 2*w^2 + 12*w + 3]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^6 - 34*x^4 + 296*x^2 - 148;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [-5/29*e^4 + 84/29*e^2 - 12/29, e, e, -1/29*e^4 + 11/29*e^2 + 102/29, -1/29*e^4 + 11/29*e^2 + 102/29, -1, 1/58*e^5 + 9/29*e^3 - 283/29*e, 1/58*e^5 + 9/29*e^3 - 283/29*e, 7/58*e^5 - 82/29*e^3 + 368/29*e, 7/58*e^5 - 82/29*e^3 + 368/29*e, 14/29*e^4 - 270/29*e^2 + 486/29, -2/29*e^5 + 51/29*e^3 - 289/29*e, -2/29*e^5 + 51/29*e^3 - 289/29*e, -4/29*e^4 + 44/29*e^2 + 118/29, -4/29*e^4 + 44/29*e^2 + 118/29, 1/29*e^5 - 40/29*e^3 + 420/29*e, 1/29*e^5 - 40/29*e^3 + 420/29*e, -1/29*e^5 + 11/29*e^3 + 15/29*e, -1/29*e^5 + 11/29*e^3 + 15/29*e, -1/29*e^4 + 40/29*e^2 - 188/29, -1/29*e^4 + 40/29*e^2 - 188/29, 9/58*e^5 - 122/29*e^3 + 788/29*e, 9/58*e^5 - 122/29*e^3 + 788/29*e, 1/29*e^4 + 18/29*e^2 - 102/29, -11/29*e^4 + 237/29*e^2 - 618/29, 1/29*e^4 - 40/29*e^2 + 72/29, 1/29*e^4 - 40/29*e^2 + 72/29, -11/29*e^4 + 237/29*e^2 - 618/29, -1/58*e^5 - 38/29*e^3 + 718/29*e, -1/58*e^5 - 38/29*e^3 + 718/29*e, -10/29*e^4 + 226/29*e^2 - 778/29, -10/29*e^4 + 226/29*e^2 - 778/29, -2/29*e^5 + 80/29*e^3 - 869/29*e, -2/29*e^5 + 80/29*e^3 - 869/29*e, 1/58*e^5 - 20/29*e^3 + 123/29*e, 1/58*e^5 - 20/29*e^3 + 123/29*e, -3/29*e^4 + 62/29*e^2 - 332/29, -3/29*e^4 + 62/29*e^2 - 332/29, -16/29*e^4 + 292/29*e^2 - 108/29, -16/29*e^4 + 292/29*e^2 - 108/29, 4/29*e^5 - 44/29*e^3 - 408/29*e, 4/29*e^5 - 44/29*e^3 - 408/29*e, 25/29*e^4 - 420/29*e^2 + 176/29, 25/29*e^4 - 420/29*e^2 + 176/29, -21/58*e^5 + 246/29*e^3 - 1162/29*e, -21/58*e^5 + 246/29*e^3 - 1162/29*e, -2/29*e^5 + 22/29*e^3 + 204/29*e, -2/29*e^5 + 22/29*e^3 + 204/29*e, 12/29*e^4 - 277/29*e^2 + 806/29, -13/29*e^4 + 259/29*e^2 - 878/29, -13/29*e^4 + 259/29*e^2 - 878/29, 12/29*e^4 - 277/29*e^2 + 806/29, -2*e^2 + 30, -18/29*e^4 + 314/29*e^2 - 252/29, -18/29*e^4 + 314/29*e^2 - 252/29, -2*e^2 + 30, -17/29*e^4 + 303/29*e^2 - 702/29, -17/29*e^4 + 303/29*e^2 - 702/29, -1/29*e^5 + 40/29*e^3 - 333/29*e, -1/29*e^5 + 40/29*e^3 - 333/29*e, 26/29*e^4 - 460/29*e^2 + 306/29, 26/29*e^4 - 460/29*e^2 + 306/29, -15/29*e^4 + 252/29*e^2 + 196/29, -15/29*e^4 + 252/29*e^2 + 196/29, 5/58*e^5 - 42/29*e^3 + 122/29*e, -19/58*e^5 + 148/29*e^3 + 244/29*e, -19/58*e^5 + 148/29*e^3 + 244/29*e, 5/58*e^5 - 42/29*e^3 + 122/29*e, -24/29*e^4 + 380/29*e^2 + 882/29, 7/58*e^5 - 24/29*e^3 - 473/29*e, -1/58*e^5 - 9/29*e^3 + 457/29*e, -1/58*e^5 - 9/29*e^3 + 457/29*e, 7/58*e^5 - 24/29*e^3 - 473/29*e, -17/58*e^5 + 224/29*e^3 - 1192/29*e, -17/58*e^5 + 166/29*e^3 - 322/29*e, -17/58*e^5 + 166/29*e^3 - 322/29*e, -17/58*e^5 + 224/29*e^3 - 1192/29*e, -13/58*e^5 + 202/29*e^3 - 1599/29*e, 11/58*e^5 - 104/29*e^3 + 19/29*e, 11/58*e^5 - 104/29*e^3 + 19/29*e, -13/58*e^5 + 202/29*e^3 - 1599/29*e, -3/29*e^4 + 4/29*e^2 + 596/29, -3/29*e^4 + 4/29*e^2 + 596/29, 7/29*e^5 - 164/29*e^3 + 852/29*e, 7/29*e^5 - 164/29*e^3 + 852/29*e, 5/29*e^5 - 113/29*e^3 + 447/29*e, -2/29*e^5 - 65/29*e^3 + 1625/29*e, -2/29*e^5 - 65/29*e^3 + 1625/29*e, 5/29*e^5 - 113/29*e^3 + 447/29*e, 2/29*e^5 - 80/29*e^3 + 782/29*e, 2/29*e^5 - 80/29*e^3 + 782/29*e, 23/58*e^5 - 286/29*e^3 + 1495/29*e, 23/58*e^5 - 286/29*e^3 + 1495/29*e, -3/58*e^5 + 2/29*e^3 + 588/29*e, -3/58*e^5 + 2/29*e^3 + 588/29*e, 13/29*e^4 - 201/29*e^2 - 514/29, 13/29*e^4 - 201/29*e^2 - 514/29, 23/29*e^4 - 398/29*e^2 + 380/29, -33/29*e^4 + 624/29*e^2 - 752/29, -33/29*e^4 + 624/29*e^2 - 752/29, 23/29*e^4 - 398/29*e^2 + 380/29, -7/29*e^4 + 193/29*e^2 - 678/29, -7/29*e^4 + 193/29*e^2 - 678/29, 47/29*e^4 - 836/29*e^2 + 774/29, 47/29*e^4 - 836/29*e^2 + 774/29, -15/58*e^5 + 97/29*e^3 + 475/29*e, -21/58*e^5 + 188/29*e^3 - 147/29*e, -21/58*e^5 + 188/29*e^3 - 147/29*e, -15/58*e^5 + 97/29*e^3 + 475/29*e, 19/29*e^4 - 412/29*e^2 + 1194/29, 19/29*e^4 - 412/29*e^2 + 1194/29, 11/58*e^5 - 75/29*e^3 - 213/29*e, 11/58*e^5 - 75/29*e^3 - 213/29*e, -20/29*e^4 + 278/29*e^2 + 590/29, 13/29*e^4 - 288/29*e^2 + 1400/29, 13/29*e^4 - 288/29*e^2 + 1400/29, -20/29*e^4 + 278/29*e^2 + 590/29, -33/29*e^4 + 537/29*e^2 + 466/29, -33/29*e^4 + 537/29*e^2 + 466/29, 46/29*e^4 - 854/29*e^2 + 1398/29, 36/29*e^4 - 570/29*e^2 - 714/29, 36/29*e^4 - 570/29*e^2 - 714/29, 46/29*e^4 - 854/29*e^2 + 1398/29, 7/58*e^5 - 82/29*e^3 + 368/29*e, 7/58*e^5 - 82/29*e^3 + 368/29*e, 1/29*e^5 + 76/29*e^3 - 1436/29*e, 1/29*e^5 + 76/29*e^3 - 1436/29*e, -17/58*e^5 + 224/29*e^3 - 1163/29*e, -1/58*e^5 + 78/29*e^3 - 1283/29*e, -1/58*e^5 + 78/29*e^3 - 1283/29*e, -17/58*e^5 + 224/29*e^3 - 1163/29*e, -32/29*e^4 + 526/29*e^2 + 480/29, -32/29*e^4 + 526/29*e^2 + 480/29, e^4 - 17*e^2 + 10, -30/29*e^4 + 562/29*e^2 - 594/29, -30/29*e^4 + 562/29*e^2 - 594/29, e^4 - 17*e^2 + 10, -4/29*e^5 + 160/29*e^3 - 1477/29*e, 5/29*e^5 - 55/29*e^3 - 481/29*e, 5/29*e^5 - 55/29*e^3 - 481/29*e, -4/29*e^5 + 160/29*e^3 - 1477/29*e, 7/29*e^5 - 222/29*e^3 + 1809/29*e, 7/29*e^5 - 222/29*e^3 + 1809/29*e, -3/58*e^5 + 2/29*e^3 + 530/29*e, -23/58*e^5 + 286/29*e^3 - 1582/29*e, -23/58*e^5 + 286/29*e^3 - 1582/29*e, -3/58*e^5 + 2/29*e^3 + 530/29*e, -1/29*e^5 + 40/29*e^3 - 710/29*e, -1/29*e^5 + 40/29*e^3 - 710/29*e, -5/29*e^4 + 84/29*e^2 + 1554/29, 22/29*e^4 - 445/29*e^2 + 18/29, -e^4 + 14*e^2 + 30, -e^4 + 14*e^2 + 30, 22/29*e^4 - 445/29*e^2 + 18/29, -16/29*e^4 + 350/29*e^2 - 804/29, 72/29*e^4 - 1256/29*e^2 + 834/29, 72/29*e^4 - 1256/29*e^2 + 834/29, -16/29*e^4 + 350/29*e^2 - 804/29, -13/29*e^5 + 288/29*e^3 - 1226/29*e, -13/29*e^5 + 288/29*e^3 - 1226/29*e, 8/29*e^5 - 117/29*e^3 - 323/29*e, 8/29*e^5 - 117/29*e^3 - 323/29*e, 27/29*e^4 - 442/29*e^2 - 840/29, -14/29*e^4 + 328/29*e^2 - 1414/29, -14/29*e^4 + 328/29*e^2 - 1414/29, 27/29*e^4 - 442/29*e^2 - 840/29, -4/29*e^4 + 131/29*e^2 + 466/29, -32/29*e^4 + 555/29*e^2 - 274/29, -42/29*e^4 + 752/29*e^2 - 1284/29, -42/29*e^4 + 752/29*e^2 - 1284/29, 8/29*e^4 - 204/29*e^2 - 62/29, 8/29*e^4 - 204/29*e^2 - 62/29];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {16, 2, 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;