/*
  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![25, 5, -11, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {11, 11, w + 1}>;

primesArray := [
[5, 5, 1/5*w^3 + 4/5*w^2 - 11/5*w - 7],
[5, 5, 1/5*w^3 - 6/5*w^2 - 1/5*w + 4],
[11, 11, w + 1],
[11, 11, 1/5*w^3 - 1/5*w^2 - 11/5*w + 2],
[16, 2, 2],
[19, 19, -1/5*w^3 + 1/5*w^2 + 1/5*w + 1],
[19, 19, 1/5*w^3 - 1/5*w^2 - 11/5*w],
[19, 19, -2/5*w^3 + 2/5*w^2 + 17/5*w],
[19, 19, w - 1],
[29, 29, w^2 - w - 8],
[29, 29, 2/5*w^3 - 2/5*w^2 - 7/5*w + 2],
[31, 31, 1/5*w^3 - 1/5*w^2 - 1/5*w + 2],
[31, 31, 2/5*w^3 - 2/5*w^2 - 17/5*w + 3],
[61, 61, -2/5*w^3 - 3/5*w^2 + 12/5*w + 5],
[61, 61, -4/5*w^3 - 6/5*w^2 + 39/5*w + 15],
[61, 61, 3/5*w^3 + 2/5*w^2 - 18/5*w - 3],
[61, 61, 7/5*w^3 + 3/5*w^2 - 62/5*w - 13],
[71, 71, 1/5*w^3 - 6/5*w^2 - 6/5*w + 4],
[71, 71, w^2 - 8],
[79, 79, 3/5*w^3 + 12/5*w^2 - 33/5*w - 22],
[79, 79, 1/5*w^3 + 4/5*w^2 - 11/5*w - 9],
[81, 3, -3],
[101, 101, 3/5*w^3 + 7/5*w^2 - 28/5*w - 15],
[101, 101, 1/5*w^3 + 9/5*w^2 - 16/5*w - 16],
[121, 11, 3/5*w^3 - 3/5*w^2 - 18/5*w + 1],
[131, 131, 1/5*w^3 + 4/5*w^2 - 16/5*w - 8],
[131, 131, w^2 - 2*w - 2],
[149, 149, 4/5*w^3 + 1/5*w^2 - 29/5*w - 5],
[149, 149, 4/5*w^3 - 9/5*w^2 - 19/5*w + 6],
[151, 151, 2/5*w^3 + 3/5*w^2 - 17/5*w - 3],
[151, 151, -2/5*w^3 + 7/5*w^2 + 7/5*w - 8],
[151, 151, 4/5*w^3 + 1/5*w^2 - 29/5*w - 2],
[151, 151, -4/5*w^3 + 9/5*w^2 + 19/5*w - 9],
[179, 179, -3/5*w^3 + 8/5*w^2 + 18/5*w - 6],
[179, 179, 2/5*w^3 + 3/5*w^2 - 12/5*w - 6],
[211, 211, 1/5*w^3 + 4/5*w^2 - 6/5*w - 10],
[211, 211, 2/5*w^3 - 7/5*w^2 - 12/5*w + 2],
[229, 229, 6/5*w^3 - 11/5*w^2 - 41/5*w + 11],
[229, 229, 2*w^2 - 2*w - 9],
[229, 229, -4/5*w^3 - 1/5*w^2 + 19/5*w + 2],
[229, 229, -3/5*w^3 + 3/5*w^2 + 28/5*w - 3],
[239, 239, 11/5*w^3 + 4/5*w^2 - 96/5*w - 19],
[239, 239, -6/5*w^3 + 1/5*w^2 + 51/5*w + 3],
[241, 241, 1/5*w^3 + 4/5*w^2 - 1/5*w - 6],
[241, 241, 4/5*w^3 - 4/5*w^2 - 29/5*w - 1],
[251, 251, -1/5*w^3 + 6/5*w^2 + 11/5*w - 12],
[251, 251, -1/5*w^3 + 11/5*w^2 + 1/5*w - 15],
[269, 269, 2/5*w^3 + 3/5*w^2 - 17/5*w - 2],
[269, 269, -2/5*w^3 + 7/5*w^2 + 7/5*w - 9],
[271, 271, 1/5*w^3 - 1/5*w^2 + 4/5*w],
[271, 271, 3/5*w^3 - 3/5*w^2 - 28/5*w + 2],
[281, 281, w^3 - 3*w^2 - 4*w + 13],
[281, 281, -3/5*w^3 + 3/5*w^2 + 28/5*w - 6],
[281, 281, -6/5*w^3 + 26/5*w^2 + 11/5*w - 16],
[281, 281, w^3 + w^2 - 8*w - 9],
[289, 17, 2/5*w^3 + 3/5*w^2 - 7/5*w - 5],
[289, 17, 4/5*w^3 - 9/5*w^2 - 29/5*w + 8],
[311, 311, -w^2 + 2*w + 11],
[311, 311, -1/5*w^3 + 6/5*w^2 + 1/5*w - 1],
[311, 311, 1/5*w^3 + 4/5*w^2 - 16/5*w + 1],
[311, 311, -1/5*w^3 - 4/5*w^2 + 11/5*w + 10],
[349, 349, -1/5*w^3 - 9/5*w^2 + 11/5*w + 16],
[349, 349, 2/5*w^3 - 12/5*w^2 - 7/5*w + 7],
[359, 359, -4/5*w^3 + 4/5*w^2 + 19/5*w - 1],
[359, 359, -w^3 + 3*w^2 + 5*w - 12],
[359, 359, w^3 - w^2 - 7*w + 2],
[359, 359, 4/5*w^3 + 6/5*w^2 - 29/5*w - 11],
[379, 379, 4/5*w^3 + 6/5*w^2 - 34/5*w - 9],
[379, 379, 3/5*w^3 - 3/5*w^2 - 28/5*w + 7],
[389, 389, w^2 - 3*w - 8],
[389, 389, -4/5*w^3 + 14/5*w^2 + 19/5*w - 16],
[401, 401, -3*w^2 + 2*w + 23],
[401, 401, 1/5*w^3 - 16/5*w^2 + 4/5*w + 11],
[409, 409, -w^3 + 2*w^2 + 3*w - 7],
[409, 409, 7/5*w^3 - 2/5*w^2 - 57/5*w - 2],
[419, 419, -1/5*w^3 + 1/5*w^2 + 16/5*w - 2],
[419, 419, 1/5*w^3 - 1/5*w^2 - 16/5*w],
[421, 421, 2/5*w^3 + 3/5*w^2 - 27/5*w - 3],
[421, 421, 1/5*w^3 + 4/5*w^2 + 4/5*w - 3],
[439, 439, -w^3 + 3*w^2 + 5*w - 16],
[439, 439, -4/5*w^3 - 6/5*w^2 + 29/5*w + 7],
[439, 439, -3/5*w^3 + 13/5*w^2 + 3/5*w - 11],
[439, 439, 4/5*w^3 + 6/5*w^2 - 39/5*w - 10],
[449, 449, 2/5*w^3 - 12/5*w^2 - 2/5*w + 5],
[449, 449, 4/5*w^3 - 24/5*w^2 - 4/5*w + 15],
[461, 461, -1/5*w^3 - 4/5*w^2 - 4/5*w + 4],
[461, 461, 2*w^2 - 2*w - 11],
[461, 461, 3/5*w^3 + 2/5*w^2 - 33/5*w - 4],
[479, 479, -w^3 + 3*w^2 + 6*w - 14],
[479, 479, -3/5*w^3 + 3/5*w^2 + 28/5*w + 3],
[499, 499, 2/5*w^3 + 8/5*w^2 - 17/5*w - 6],
[499, 499, -3/5*w^3 + 13/5*w^2 + 13/5*w - 17],
[509, 509, -w^3 + 2*w^2 + 4*w - 6],
[509, 509, 6/5*w^3 - 1/5*w^2 - 46/5*w - 4],
[521, 521, -2/5*w^3 + 2/5*w^2 + 7/5*w - 7],
[521, 521, -3/5*w^3 + 3/5*w^2 + 23/5*w - 8],
[529, 23, -7/5*w^3 + 17/5*w^2 + 42/5*w - 15],
[529, 23, 2/5*w^3 + 13/5*w^2 - 27/5*w - 21],
[541, 541, 1/5*w^3 - 11/5*w^2 - 1/5*w + 10],
[541, 541, 3/5*w^3 - 8/5*w^2 - 13/5*w + 13],
[541, 541, -2*w^2 + w + 13],
[541, 541, 3/5*w^3 + 2/5*w^2 - 23/5*w + 2],
[569, 569, w^3 - 2*w^2 - 5*w + 7],
[569, 569, -1/5*w^3 + 1/5*w^2 + 1/5*w - 6],
[571, 571, w^2 + w - 9],
[571, 571, -2/5*w^3 + 12/5*w^2 + 7/5*w - 12],
[571, 571, 1/5*w^3 + 9/5*w^2 - 11/5*w - 11],
[571, 571, -2/5*w^3 + 7/5*w^2 + 17/5*w - 4],
[601, 601, -3/5*w^3 + 13/5*w^2 + 18/5*w - 11],
[601, 601, -4/5*w^3 + 4/5*w^2 + 34/5*w + 1],
[631, 631, 2/5*w^3 + 18/5*w^2 - 27/5*w - 30],
[631, 631, 3/5*w^3 - 23/5*w^2 - 3/5*w + 15],
[661, 661, -2*w^3 + 17*w + 12],
[661, 661, -9/5*w^3 + 4/5*w^2 + 74/5*w + 1],
[691, 691, -7/5*w^3 - 18/5*w^2 + 72/5*w + 35],
[691, 691, 6/5*w^3 - 31/5*w^2 - 6/5*w + 19],
[701, 701, -1/5*w^3 + 6/5*w^2 + 16/5*w - 6],
[701, 701, 2/5*w^3 - 7/5*w^2 - 22/5*w + 8],
[709, 709, -3/5*w^3 + 13/5*w^2 + 13/5*w - 18],
[709, 709, 1/5*w^3 + 14/5*w^2 - 11/5*w - 20],
[709, 709, -2/5*w^3 - 8/5*w^2 + 17/5*w + 5],
[709, 709, 1/5*w^3 + 4/5*w^2 - 21/5*w - 10],
[719, 719, -3/5*w^3 + 8/5*w^2 + 13/5*w - 12],
[719, 719, 3/5*w^3 + 2/5*w^2 - 23/5*w + 1],
[719, 719, -7/5*w^3 + 17/5*w^2 + 32/5*w - 14],
[719, 719, 7/5*w^3 + 3/5*w^2 - 52/5*w - 8],
[739, 739, 1/5*w^3 - 1/5*w^2 - 11/5*w - 5],
[739, 739, w - 6],
[761, 761, -4/5*w^3 + 9/5*w^2 + 19/5*w - 11],
[761, 761, 4/5*w^3 + 1/5*w^2 - 29/5*w],
[769, 769, 3/5*w^3 - 13/5*w^2 - 13/5*w + 8],
[769, 769, -2/5*w^3 - 8/5*w^2 + 17/5*w + 15],
[769, 769, 7/5*w^3 - 17/5*w^2 - 37/5*w + 18],
[769, 769, -6/5*w^3 - 4/5*w^2 + 41/5*w + 5],
[809, 809, -1/5*w^3 + 6/5*w^2 - 4/5*w - 14],
[809, 809, -2/5*w^3 - 3/5*w^2 + 22/5*w - 4],
[809, 809, -6/5*w^3 + 6/5*w^2 + 41/5*w - 5],
[809, 809, w^3 - w^2 - 5*w + 4],
[821, 821, -3/5*w^3 - 17/5*w^2 + 33/5*w + 30],
[821, 821, -4/5*w^3 + 24/5*w^2 + 9/5*w - 15],
[829, 829, 7/5*w^3 - 12/5*w^2 - 52/5*w + 10],
[829, 829, -1/5*w^3 + 11/5*w^2 - 4/5*w - 19],
[829, 829, -13/5*w^3 + 3/5*w^2 + 108/5*w + 10],
[829, 829, -w^3 + 4*w^2 + 5*w - 23],
[839, 839, 4/5*w^3 - 9/5*w^2 - 24/5*w + 4],
[839, 839, 3/5*w^3 + 2/5*w^2 - 18/5*w - 8],
[841, 29, -w^3 + w^2 + 6*w - 4],
[859, 859, 6/5*w^3 - 11/5*w^2 - 36/5*w + 9],
[859, 859, w^3 - 6*w - 3],
[881, 881, 4/5*w^3 - 4/5*w^2 - 9/5*w - 1],
[881, 881, 9/5*w^3 - 24/5*w^2 - 19/5*w + 14],
[881, 881, -w^3 + 6*w + 2],
[881, 881, 1/5*w^3 - 1/5*w^2 - 1/5*w - 6],
[911, 911, w^3 - w^2 - 5*w + 1],
[911, 911, 6/5*w^3 - 6/5*w^2 - 41/5*w + 2],
[919, 919, -7/5*w^3 + 12/5*w^2 + 42/5*w - 7],
[919, 919, -6/5*w^3 + 1/5*w^2 + 36/5*w + 5],
[929, 929, -2/5*w^3 - 8/5*w^2 + 17/5*w + 16],
[929, 929, -3/5*w^3 + 13/5*w^2 + 13/5*w - 7],
[961, 31, w^3 - w^2 - 6*w + 2],
[971, 971, 8/5*w^3 - 13/5*w^2 - 48/5*w + 6],
[971, 971, -2/5*w^3 - 3/5*w^2 + 22/5*w],
[971, 971, 7/5*w^3 - 7/5*w^2 - 57/5*w + 7],
[971, 971, -1/5*w^3 + 6/5*w^2 - 4/5*w - 10],
[991, 991, 2/5*w^3 + 8/5*w^2 - 22/5*w - 7],
[991, 991, -2/5*w^3 + 12/5*w^2 + 2/5*w - 15]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^4 - 20*x^2 - 4*x + 52;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [-3/34*e^3 - 2/17*e^2 + 16/17*e + 50/17, e, -1, 3/17*e^3 - 9/34*e^2 - 32/17*e + 70/17, -3/34*e^3 + 13/34*e^2 + 16/17*e - 86/17, 1/34*e^3 - 5/17*e^2 + 6/17*e + 40/17, -1/17*e^3 - 7/17*e^2 + 5/17*e + 90/17, 1/34*e^3 - 5/17*e^2 + 6/17*e + 40/17, 9/34*e^3 - 5/34*e^2 - 82/17*e + 20/17, 5/34*e^3 + 1/34*e^2 - 38/17*e + 64/17, -3/34*e^3 - 2/17*e^2 + 50/17*e + 84/17, -4/17*e^3 + 6/17*e^2 + 37/17*e - 48/17, -2/17*e^3 + 3/17*e^2 + 10/17*e + 44/17, -2/17*e^3 + 3/17*e^2 + 10/17*e - 109/17, -2/17*e^3 + 3/17*e^2 + 61/17*e + 78/17, 1/17*e^3 + 31/34*e^2 - 22/17*e - 124/17, 9/17*e^3 - 5/17*e^2 - 164/17*e - 11/17, -3/34*e^3 + 13/34*e^2 + 16/17*e - 188/17, 3/34*e^3 + 2/17*e^2 - 16/17*e + 86/17, 5/17*e^3 - 16/17*e^2 - 76/17*e + 60/17, 1/17*e^3 + 7/17*e^2 - 22/17*e - 73/17, 5/17*e^3 + 1/17*e^2 - 59/17*e - 8/17, -27/34*e^3 + 15/34*e^2 + 178/17*e + 76/17, 8/17*e^3 + 5/17*e^2 - 142/17*e - 108/17, -4/17*e^3 + 23/17*e^2 + 54/17*e - 354/17, 15/34*e^3 + 3/34*e^2 - 46/17*e - 80/17, -e^2 + e + 17, -3/17*e^3 + 13/17*e^2 + 32/17*e - 104/17, 8/17*e^3 - 12/17*e^2 - 74/17*e + 113/17, 3/17*e^3 + 4/17*e^2 - 117/17*e - 117/17, -16/17*e^3 - 10/17*e^2 + 233/17*e + 63/17, -10/17*e^3 - 19/17*e^2 + 169/17*e + 101/17, -3/34*e^3 - 21/34*e^2 - 18/17*e + 186/17, 9/17*e^3 + 12/17*e^2 - 62/17*e - 249/17, -9/34*e^3 - 23/17*e^2 + 14/17*e + 252/17, 15/17*e^3 - 14/17*e^2 - 262/17*e + 146/17, 3/17*e^3 + 38/17*e^2 - 32/17*e - 474/17, -11/34*e^3 + 4/17*e^2 + 206/17*e - 66/17, -4/17*e^3 + 6/17*e^2 + 3/17*e + 105/17, 7/34*e^3 + 16/17*e^2 - 94/17*e - 230/17, -2/17*e^3 - 48/17*e^2 + 44/17*e + 503/17, -4/17*e^3 + 40/17*e^2 + 54/17*e - 252/17, 1/2*e^3 + e^2 - 10*e - 6, -3/17*e^3 - 38/17*e^2 + 83/17*e + 423/17, 9/17*e^3 + 29/17*e^2 - 164/17*e - 130/17, -1/17*e^3 - 24/17*e^2 - 12/17*e + 413/17, 5/17*e^3 + 1/17*e^2 - 59/17*e + 77/17, -1/34*e^3 + 27/34*e^2 + 62/17*e - 278/17, 2/17*e^3 - 20/17*e^2 - 78/17*e + 92/17, -1/34*e^3 + 5/17*e^2 - 40/17*e + 28/17, 21/34*e^3 - 37/17*e^2 - 146/17*e + 194/17, -11/17*e^3 + 25/17*e^2 + 140/17*e + 4/17, -29/34*e^3 + 9/17*e^2 + 234/17*e + 166/17, -1/17*e^3 + 27/17*e^2 + 5/17*e - 318/17, -19/17*e^3 + 57/34*e^2 + 282/17*e - 160/17, 21/34*e^3 - 20/17*e^2 - 146/17*e + 58/17, 7/17*e^3 - 2/17*e^2 - 120/17*e + 67/17, -3/17*e^3 - 21/17*e^2 + 32/17*e + 474/17, 4/17*e^3 - 6/17*e^2 - 54/17*e + 507/17, 11/17*e^3 - 8/17*e^2 - 140/17*e + 234/17, -6/17*e^3 - 25/17*e^2 + 64/17*e + 302/17, 8/17*e^3 - 75/34*e^2 - 74/17*e + 640/17, -47/34*e^3 + 14/17*e^2 + 262/17*e - 180/17, 9/17*e^3 + 12/17*e^2 - 198/17*e - 96/17, 10/17*e^3 + 55/34*e^2 - 84/17*e - 322/17, 37/34*e^3 - 47/34*e^2 - 322/17*e + 188/17, 23/34*e^3 - 13/17*e^2 - 100/17*e + 172/17, -1/17*e^3 + 78/17*e^2 - 12/17*e - 726/17, -14/17*e^3 - 9/34*e^2 + 274/17*e + 138/17, 2/17*e^3 - 54/17*e^2 - 27/17*e + 313/17, 2/17*e^3 - 54/17*e^2 - 44/17*e + 670/17, -21/34*e^3 + 37/17*e^2 + 78/17*e - 534/17, -e^3 - e^2 + 12*e + 22, 7/17*e^3 + 32/17*e^2 - 86/17*e - 222/17, -5/17*e^3 - 1/17*e^2 + 8/17*e - 128/17, -5/34*e^3 + 25/17*e^2 - 30/17*e - 268/17, 3/17*e^3 + 4/17*e^2 - 100/17*e + 172/17, -21/17*e^3 + 63/34*e^2 + 292/17*e - 48/17, -15/17*e^3 - 3/17*e^2 + 194/17*e + 194/17, -11/34*e^3 + 21/17*e^2 + 70/17*e - 100/17, -23/17*e^3 - 33/34*e^2 + 404/17*e - 4/17, 1/17*e^3 - 10/17*e^2 - 73/17*e + 148/17, -7/34*e^3 - 50/17*e^2 + 26/17*e + 570/17, -4/17*e^3 - 11/17*e^2 + 71/17*e - 31/17, -33/34*e^3 + 12/17*e^2 + 176/17*e - 198/17, -5/17*e^3 + 16/17*e^2 + 110/17*e + 8/17, -41/34*e^3 + 53/34*e^2 + 332/17*e - 76/17, 4*e^2 + 2*e - 39, -21/17*e^3 - 28/17*e^2 + 326/17*e + 71/17, 6/17*e^3 + 8/17*e^2 - 200/17*e - 98/17, -4/17*e^3 - 11/17*e^2 + 71/17*e - 14/17, 24/17*e^3 + 15/17*e^2 - 341/17*e - 324/17, -29/34*e^3 + 69/34*e^2 + 268/17*e - 446/17, 22/17*e^3 - 15/34*e^2 - 280/17*e + 230/17, 1/17*e^3 - 27/17*e^2 - 73/17*e + 352/17, -19/34*e^3 + 44/17*e^2 + 192/17*e - 352/17, 5/17*e^3 + 52/17*e^2 - 93/17*e - 569/17, 29/17*e^3 - 1/17*e^2 - 451/17*e - 60/17, -1/17*e^3 + 10/17*e^2 + 22/17*e - 318/17, -5/17*e^3 + 84/17*e^2 + 25/17*e - 910/17, -2/17*e^3 - 14/17*e^2 - 24/17*e + 180/17, 10/17*e^3 + 89/34*e^2 - 220/17*e - 322/17, -12/17*e^3 + 18/17*e^2 + 145/17*e - 178/17, -29/34*e^3 + 26/17*e^2 + 200/17*e - 412/17, -16/17*e^3 + 7/17*e^2 + 216/17*e + 182/17, 4/17*e^3 + 28/17*e^2 - 156/17*e - 377/17, 10/17*e^3 - 15/17*e^2 - 203/17*e - 288/17, 9/34*e^3 + 29/34*e^2 - 184/17*e + 54/17, -39/34*e^3 + 42/17*e^2 + 310/17*e - 166/17, 11/34*e^3 + 77/34*e^2 - 138/17*e - 682/17, -3/2*e^2 - 6*e + 14, 5/17*e^3 + 35/17*e^2 - 76/17*e - 348/17, 18/17*e^3 + 41/17*e^2 - 192/17*e - 481/17, 25/17*e^3 - 46/17*e^2 - 329/17*e + 504/17, -36/17*e^3 + 37/17*e^2 + 435/17*e - 381/17, 2/17*e^3 + 31/17*e^2 - 44/17*e - 452/17, -5/34*e^3 - 137/34*e^2 + 72/17*e + 820/17, 7/17*e^3 - 36/17*e^2 - 171/17*e + 33/17, -27/34*e^3 + 50/17*e^2 + 178/17*e - 706/17, 21/17*e^3 - 40/17*e^2 - 360/17*e + 354/17, 27/17*e^3 - 15/17*e^2 - 475/17*e - 84/17, 20/17*e^3 - 64/17*e^2 - 236/17*e + 427/17, 10/17*e^3 - 15/17*e^2 - 220/17*e + 528/17, 6/17*e^3 - 43/17*e^2 + 89/17*e + 548/17, 2/17*e^3 + 48/17*e^2 - 10/17*e - 724/17, 23/34*e^3 + 4/17*e^2 - 100/17*e - 236/17, -6/17*e^3 + 26/17*e^2 + 132/17*e + 200/17, -41/34*e^3 + 53/34*e^2 + 366/17*e - 76/17, -38/17*e^3 + 57/17*e^2 + 530/17*e - 388/17, -4/17*e^3 - 28/17*e^2 + 224/17*e + 377/17, e^3 + 2*e^2 - 13*e - 24, 24/17*e^3 - 21/34*e^2 - 290/17*e + 220/17, 11/17*e^3 - 42/17*e^2 - 191/17*e - 4/17, 20/17*e^3 + 4/17*e^2 - 355/17*e - 406/17, -4/17*e^3 - 11/17*e^2 + 37/17*e - 150/17, -27/34*e^3 - 52/17*e^2 + 178/17*e + 552/17, 37/34*e^3 + 2/17*e^2 - 322/17*e + 358/17, 31/34*e^3 + 49/17*e^2 - 358/17*e - 562/17, -30/17*e^3 - 6/17*e^2 + 490/17*e + 65/17, -21/17*e^3 + 6/17*e^2 + 394/17*e + 20/17, -13/17*e^3 - 57/17*e^2 + 184/17*e + 660/17, -15/17*e^3 + 11/34*e^2 + 160/17*e + 262/17, e^3 - 3*e^2 - 20*e + 18, -3/17*e^3 - 38/17*e^2 + 15/17*e + 678/17, -16/17*e^3 + 24/17*e^2 + 233/17*e - 311/17, 21/17*e^3 + 11/17*e^2 - 224/17*e - 632/17, 37/17*e^3 - 64/17*e^2 - 474/17*e + 546/17, -9/17*e^3 - 63/17*e^2 - 6/17*e + 1014/17, -27/34*e^3 + 67/17*e^2 + 178/17*e - 468/17, -9/17*e^3 + 22/17*e^2 + 45/17*e + 79/17, -5/17*e^3 - 1/17*e^2 + 229/17*e + 246/17, 11/17*e^3 + 43/17*e^2 - 174/17*e - 718/17, -35/17*e^3 + 44/17*e^2 + 464/17*e - 250/17, -26/17*e^3 + 56/17*e^2 + 300/17*e - 414/17, -e^3 + 7/2*e^2 + 16*e - 20, -16/17*e^3 + 58/17*e^2 + 216/17*e - 685/17, 25/17*e^3 + 5/17*e^2 - 414/17*e - 312/17, -37/34*e^3 + 13/34*e^2 + 356/17*e + 220/17, 4/17*e^3 - 57/17*e^2 - 54/17*e + 252/17, 49/34*e^3 - 58/17*e^2 - 352/17*e + 396/17, -11/34*e^3 - 13/17*e^2 + 308/17*e + 138/17, -9/17*e^3 + 5/17*e^2 - 6/17*e + 28/17, -23/17*e^3 + 77/17*e^2 + 336/17*e - 820/17, 24/17*e^3 - 2/17*e^2 - 256/17*e + 84/17, 20/17*e^3 + 21/17*e^2 - 202/17*e - 287/17, 33/34*e^3 + 5/17*e^2 - 278/17*e + 504/17];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {11, 11, 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;