/*
  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![6, 0, -6, 0, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {22, 22, -w^3 + w^2 + 3*w - 2}>;

primesArray := [
[2, 2, -w^2 + w + 2],
[3, 3, w^2 - w - 3],
[11, 11, -w^2 + w + 1],
[11, 11, -w^2 - w + 1],
[13, 13, w^3 - 4*w + 1],
[13, 13, -w^3 + 4*w + 1],
[25, 5, -w^2 - 2*w + 1],
[25, 5, w^2 - 2*w - 1],
[37, 37, w^3 - 3*w - 1],
[37, 37, w^3 - 3*w + 1],
[59, 59, w^2 - w - 5],
[59, 59, -w^2 - w + 5],
[61, 61, -w^3 + w^2 + 4*w - 7],
[61, 61, w^3 - 3*w^2 - 6*w + 11],
[73, 73, 2*w^2 - w - 5],
[73, 73, 2*w - 1],
[73, 73, -2*w - 1],
[73, 73, 2*w^2 + w - 5],
[83, 83, 2*w^3 + w^2 - 9*w - 7],
[83, 83, -2*w^3 + w^2 + 7*w + 1],
[107, 107, w^3 + w^2 - 4*w - 1],
[107, 107, -w^3 + w^2 + 4*w - 1],
[109, 109, -w^3 + 2*w^2 + 5*w - 11],
[109, 109, w^3 + 2*w^2 - 5*w - 11],
[121, 11, 2*w^2 - 5],
[131, 131, 3*w^3 - 2*w^2 - 14*w + 11],
[131, 131, 3*w^3 - 12*w + 5],
[157, 157, -3*w^3 + 2*w^2 + 13*w - 11],
[157, 157, -w^3 + 2*w^2 + 7*w - 5],
[167, 167, w^2 + 2*w - 5],
[167, 167, -w^3 + w^2 + w + 1],
[167, 167, 3*w^3 - w^2 - 9*w - 5],
[167, 167, w^2 - 2*w - 5],
[169, 13, -w^2 + 7],
[179, 179, -w^3 + w^2 + 6*w - 1],
[179, 179, w^3 + w^2 - 6*w - 1],
[181, 181, -w^3 + 5*w - 5],
[181, 181, w^3 - 5*w - 5],
[191, 191, -2*w^3 + 2*w^2 + 6*w - 1],
[191, 191, -2*w^2 + 2*w + 7],
[191, 191, -4*w^2 + 4*w + 11],
[191, 191, -2*w^3 + 8*w + 5],
[227, 227, -w^3 + 2*w^2 + 5*w - 5],
[227, 227, w^3 + 2*w^2 - 5*w - 5],
[229, 229, -w^3 + 3*w + 5],
[229, 229, -w^3 - 4*w^2 + 7*w + 13],
[239, 239, -w^3 + w^2 + 5*w - 1],
[239, 239, 2*w^3 - 4*w^2 - 12*w + 13],
[239, 239, 2*w^3 - 2*w^2 - 10*w + 7],
[239, 239, w^3 + w^2 - 5*w - 1],
[241, 241, 2*w^3 - w^2 - 8*w + 5],
[241, 241, -3*w^2 + 4*w + 7],
[241, 241, -2*w^3 + w^2 + 6*w + 1],
[241, 241, -2*w^3 - w^2 + 8*w + 5],
[251, 251, -w^3 + 2*w - 5],
[251, 251, 3*w^3 - 2*w^2 - 12*w + 13],
[263, 263, 2*w^3 - 2*w^2 - 8*w + 11],
[263, 263, -w^3 + w^2 + 3*w - 7],
[263, 263, 3*w^3 - w^2 - 11*w + 11],
[263, 263, -2*w^3 + 6*w - 7],
[277, 277, -3*w^2 - w + 11],
[277, 277, 3*w^2 - w - 11],
[311, 311, -w^3 + 5*w^2 + 7*w - 19],
[311, 311, 2*w^3 - 3*w^2 - 8*w + 17],
[311, 311, -2*w^3 + 6*w - 5],
[311, 311, -2*w^3 - 2*w^2 + 8*w + 5],
[337, 337, -2*w^3 + 8*w + 1],
[337, 337, -w^3 + 3*w^2 + 5*w - 11],
[337, 337, w^3 + 3*w^2 - 5*w - 11],
[337, 337, 2*w^3 - 8*w + 1],
[347, 347, w^3 - 3*w^2 - 6*w + 13],
[347, 347, -w^3 - 3*w^2 + 6*w + 13],
[349, 349, -w^3 + 4*w - 5],
[349, 349, w^3 - 4*w - 5],
[361, 19, 4*w^3 - 2*w^2 - 17*w + 13],
[361, 19, 2*w^3 - 2*w^2 - 11*w + 7],
[373, 373, -2*w^3 - 5*w^2 + 13*w + 17],
[373, 373, 2*w^3 - w^2 - 7*w + 1],
[383, 383, 4*w^3 - w^2 - 16*w + 13],
[383, 383, w^2 + 2*w - 7],
[383, 383, w^2 - 2*w - 7],
[383, 383, 2*w^3 + 3*w^2 - 6*w - 5],
[397, 397, -w^3 + 2*w^2 + 7*w - 7],
[397, 397, w^3 + 2*w^2 - 7*w - 7],
[419, 419, 5*w^2 - 5*w - 11],
[419, 419, 3*w^2 - 3*w - 5],
[421, 421, w^3 + 4*w^2 - 5*w - 17],
[421, 421, -w^3 + 4*w^2 + 5*w - 17],
[433, 433, w^3 - w^2 - 7*w + 1],
[433, 433, 3*w - 1],
[433, 433, -3*w - 1],
[433, 433, 5*w^3 - 3*w^2 - 21*w + 19],
[443, 443, w^3 - 4*w^2 - 6*w + 17],
[443, 443, -3*w^3 + 6*w^2 + 16*w - 25],
[467, 467, -w^3 + w^2 + 2*w - 5],
[467, 467, w^3 + w^2 - 2*w - 5],
[491, 491, 3*w^3 - w^2 - 14*w + 7],
[491, 491, -3*w^3 - w^2 + 14*w + 7],
[529, 23, 3*w^2 - 11],
[529, 23, 3*w^2 - 7],
[541, 541, 3*w^3 + 2*w^2 - 10*w - 1],
[541, 541, 3*w^3 - 4*w^2 - 16*w + 17],
[563, 563, w^3 + 2*w^2 - 2*w - 7],
[563, 563, -w^3 + 2*w^2 + 2*w - 7],
[587, 587, -2*w^3 + w^2 + 5*w - 5],
[587, 587, 2*w^3 + w^2 - 5*w - 5],
[613, 613, 3*w^3 - 13*w - 1],
[613, 613, 3*w^3 - 13*w + 1],
[659, 659, w^3 + 3*w^2 - 4*w - 5],
[659, 659, -w^3 + 3*w^2 + 4*w - 5],
[661, 661, w^3 + 4*w^2 - 9*w - 13],
[661, 661, -3*w^3 + 4*w^2 + 7*w - 5],
[673, 673, -2*w^3 + 7*w + 1],
[673, 673, w^3 + w^2 - 7*w - 5],
[673, 673, -w^3 + w^2 + 7*w - 5],
[673, 673, 2*w^3 - 7*w + 1],
[683, 683, -2*w^3 - w^2 + 9*w + 1],
[683, 683, 2*w^3 - w^2 - 9*w + 1],
[709, 709, -3*w^3 + 2*w^2 + 8*w + 1],
[709, 709, -w^3 + 4*w^2 - 2*w - 7],
[733, 733, 2*w^3 - 5*w^2 - 11*w + 19],
[733, 733, -2*w^3 + w^2 + 7*w - 11],
[743, 743, 2*w^3 + 2*w^2 - 11*w - 5],
[743, 743, 2*w^3 - w^2 - 8*w - 1],
[743, 743, -2*w^3 - w^2 + 8*w - 1],
[743, 743, -2*w^3 + 2*w^2 + 11*w - 5],
[757, 757, 2*w^3 + 3*w^2 - 3*w + 1],
[757, 757, -2*w^3 + 5*w^2 + 11*w - 17],
[827, 827, -5*w^3 + 4*w^2 + 21*w - 23],
[827, 827, -w^3 + w - 7],
[829, 829, 2*w^3 - 3*w^2 - 9*w + 17],
[829, 829, -2*w^3 - 3*w^2 + 9*w + 17],
[841, 29, 2*w^2 - w - 13],
[841, 29, 2*w^2 + w - 13],
[853, 853, -w^3 + 6*w - 7],
[853, 853, -2*w^3 + w^2 + 11*w - 5],
[863, 863, w^3 - 5*w^2 - 9*w + 13],
[863, 863, 2*w^3 + 4*w^2 - 5*w - 11],
[863, 863, -2*w^3 + 4*w^2 + 5*w - 11],
[863, 863, w^3 - 3*w^2 - 7*w + 7],
[877, 877, -w^3 - 3*w^2 + 4*w + 17],
[877, 877, w^3 - 3*w^2 - 4*w + 17],
[887, 887, -2*w^3 - w^2 + 10*w + 1],
[887, 887, 2*w^3 + 2*w^2 - 9*w - 5],
[887, 887, -2*w^3 + 2*w^2 + 9*w - 5],
[887, 887, 2*w^3 - w^2 - 10*w + 1],
[911, 911, -2*w^3 + 11*w - 5],
[911, 911, -w^3 + w^2 + w - 5],
[911, 911, w^3 + w^2 - w - 5],
[911, 911, 2*w^3 - 11*w - 5],
[937, 937, -4*w^3 - 2*w^2 + 16*w + 17],
[937, 937, 3*w^3 - w^2 - 11*w - 1],
[937, 937, -3*w^3 - 5*w^2 + 17*w + 19],
[937, 937, -4*w^2 + 2*w + 13],
[947, 947, -3*w^3 + 2*w^2 + 10*w + 1],
[947, 947, 3*w^3 - 12*w - 7],
[971, 971, w^3 + w^2 - 2*w - 7],
[971, 971, -w^3 + w^2 + 2*w - 7],
[983, 983, -w^3 - 3*w^2 + 7*w + 13],
[983, 983, 2*w^3 - 2*w^2 - 7*w + 1],
[983, 983, -2*w^3 - 2*w^2 + 7*w + 1],
[983, 983, w^3 - 7*w^2 + 3*w + 17],
[997, 997, -w^3 + 2*w^2 + 4*w - 13],
[997, 997, 3*w^3 - 8*w^2 - 18*w + 29]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^9 - x^8 - 19*x^7 + 14*x^6 + 119*x^5 - 48*x^4 - 276*x^3 + 13*x^2 + 174*x + 36;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [1, e, 1/8*e^7 - 2*e^5 + 1/4*e^4 + 73/8*e^3 - 17/8*e^2 - 41/4*e + 3/2, 1, -1/24*e^8 - 1/12*e^7 + 2/3*e^6 + 11/12*e^5 - 77/24*e^4 - 11/8*e^3 + 9/2*e^2 - 11/3*e + 1, 1/12*e^8 - 5/24*e^7 - 4/3*e^6 + 19/6*e^5 + 20/3*e^4 - 109/8*e^3 - 101/8*e^2 + 193/12*e + 17/2, 1/12*e^8 - 1/12*e^7 - 4/3*e^6 + 7/6*e^5 + 71/12*e^4 - 9/2*e^3 - 23/4*e^2 + 29/6*e + 1, 1/12*e^8 - 1/12*e^7 - 11/6*e^6 + 7/6*e^5 + 155/12*e^4 - 9/2*e^3 - 117/4*e^2 + 13/3*e + 10, -1/6*e^8 + 1/6*e^7 + 8/3*e^6 - 7/3*e^5 - 77/6*e^4 + 8*e^3 + 39/2*e^2 - 5/3*e - 2, 1/12*e^8 + 1/24*e^7 - 4/3*e^6 - 5/6*e^5 + 37/6*e^4 + 37/8*e^3 - 63/8*e^2 - 65/12*e + 5/2, 1/4*e^7 - 4*e^5 + 1/2*e^4 + 77/4*e^3 - 17/4*e^2 - 55/2*e + 3, -1/4*e^8 + 1/2*e^7 + 9/2*e^6 - 15/2*e^5 - 101/4*e^4 + 123/4*e^3 + 95/2*e^2 - 59/2*e - 21, 5/24*e^8 - 5/24*e^7 - 10/3*e^6 + 41/12*e^5 + 379/24*e^4 - 63/4*e^3 - 183/8*e^2 + 211/12*e + 17/2, -1/24*e^8 + 1/24*e^7 + 7/6*e^6 - 13/12*e^5 - 239/24*e^4 + 31/4*e^3 + 231/8*e^2 - 161/12*e - 37/2, -1/8*e^8 + 1/2*e^7 + 2*e^6 - 29/4*e^5 - 73/8*e^4 + 229/8*e^3 + 43/4*e^2 - 57/2*e + 2, -5/12*e^8 + 13/24*e^7 + 43/6*e^6 - 47/6*e^5 - 115/3*e^4 + 237/8*e^3 + 553/8*e^2 - 287/12*e - 49/2, 1/12*e^8 - 1/3*e^7 - 4/3*e^6 + 31/6*e^5 + 77/12*e^4 - 95/4*e^3 - 23/2*e^2 + 97/3*e + 10, 1/8*e^8 - 1/2*e^7 - 5/2*e^6 + 29/4*e^5 + 129/8*e^4 - 221/8*e^3 - 141/4*e^2 + 20*e + 14, 1/8*e^7 - 2*e^5 + 1/4*e^4 + 65/8*e^3 - 25/8*e^2 - 17/4*e + 9/2, 1/8*e^8 - 3/8*e^7 - 2*e^6 + 25/4*e^5 + 83/8*e^4 - 61/2*e^3 - 183/8*e^2 + 161/4*e + 33/2, 1/2*e^6 - 7*e^4 - e^3 + 51/2*e^2 + 13/2*e - 18, -1/8*e^8 + 1/8*e^7 + 2*e^6 - 5/4*e^5 - 71/8*e^4 + 1/4*e^3 + 57/8*e^2 + 37/4*e + 15/2, 1/8*e^8 - 1/8*e^7 - 2*e^6 + 9/4*e^5 + 87/8*e^4 - 49/4*e^3 - 217/8*e^2 + 79/4*e + 43/2, 1/4*e^8 + 1/8*e^7 - 4*e^6 - 3/2*e^5 + 37/2*e^4 + 39/8*e^3 - 181/8*e^2 - 29/4*e + 7/2, 1/12*e^8 + 1/24*e^7 - 4/3*e^6 - 5/6*e^5 + 37/6*e^4 + 29/8*e^3 - 71/8*e^2 + 43/12*e + 11/2, 1/8*e^8 - 1/8*e^7 - 5/2*e^6 + 5/4*e^5 + 127/8*e^4 - 1/4*e^3 - 261/8*e^2 - 23/4*e + 15/2, -1/8*e^7 + 2*e^5 - 1/4*e^4 - 65/8*e^3 + 9/8*e^2 + 9/4*e + 3/2, -1/6*e^8 + 1/24*e^7 + 8/3*e^6 - 1/3*e^5 - 157/12*e^4 - 1/8*e^3 + 165/8*e^2 - 29/12*e - 1/2, -7/24*e^8 + 7/24*e^7 + 14/3*e^6 - 55/12*e^5 - 545/24*e^4 + 81/4*e^3 + 285/8*e^2 - 233/12*e - 13/2, 3/8*e^8 - 1/2*e^7 - 13/2*e^6 + 31/4*e^5 + 283/8*e^4 - 255/8*e^3 - 259/4*e^2 + 24*e + 21, -1/8*e^8 + 3/8*e^7 + 5/2*e^6 - 21/4*e^5 - 123/8*e^4 + 37/2*e^3 + 235/8*e^2 - 43/4*e - 21/2, 1/4*e^8 - 1/8*e^7 - 4*e^6 + 3/2*e^5 + 18*e^4 - 19/8*e^3 - 155/8*e^2 - 27/4*e - 3/2, 1/8*e^8 - 5/8*e^7 - 5/2*e^6 + 37/4*e^5 + 127/8*e^4 - 151/4*e^3 - 249/8*e^2 + 145/4*e + 3/2, -3/8*e^8 - 1/8*e^7 + 13/2*e^6 + 5/4*e^5 - 277/8*e^4 - 3/4*e^3 + 459/8*e^2 - 39/4*e - 29/2, -1/8*e^8 + 2*e^6 - 1/4*e^5 - 73/8*e^4 + 17/8*e^3 + 33/4*e^2 - 7/2*e + 18, -1/2*e^8 + 1/2*e^7 + 17/2*e^6 - 7*e^5 - 89/2*e^4 + 26*e^3 + 76*e^2 - 53/2*e - 24, 5/24*e^8 - 11/24*e^7 - 10/3*e^6 + 89/12*e^5 + 391/24*e^4 - 33*e^3 - 229/8*e^2 + 373/12*e + 35/2, -7/24*e^8 + 5/12*e^7 + 14/3*e^6 - 67/12*e^5 - 515/24*e^4 + 139/8*e^3 + 55/2*e^2 - 17/3*e - 5, -1/8*e^8 + 3/8*e^7 + 5/2*e^6 - 21/4*e^5 - 139/8*e^4 + 37/2*e^3 + 379/8*e^2 - 27/4*e - 57/2, 1/4*e^8 - 1/2*e^7 - 9/2*e^6 + 15/2*e^5 + 105/4*e^4 - 123/4*e^3 - 111/2*e^2 + 57/2*e + 24, 1/8*e^8 - 3/8*e^7 - 3/2*e^6 + 21/4*e^5 + 19/8*e^4 - 39/2*e^3 + 93/8*e^2 + 67/4*e - 27/2, -1/4*e^8 + 5*e^6 - 1/2*e^5 - 125/4*e^4 + 29/4*e^3 + 127/2*e^2 - 23*e - 30, -1/8*e^8 + 5/8*e^7 + 5/2*e^6 - 37/4*e^5 - 135/8*e^4 + 151/4*e^3 + 345/8*e^2 - 161/4*e - 51/2, 1/4*e^8 - 7/8*e^7 - 9/2*e^6 + 25/2*e^5 + 53/2*e^4 - 377/8*e^3 - 473/8*e^2 + 149/4*e + 57/2, -1/6*e^8 + 5/12*e^7 + 19/6*e^6 - 19/3*e^5 - 61/3*e^4 + 109/4*e^3 + 199/4*e^2 - 101/3*e - 26, 5/24*e^8 - 1/12*e^7 - 10/3*e^6 + 5/12*e^5 + 385/24*e^4 + 27/8*e^3 - 27*e^2 - 29/3*e + 25, 3/8*e^8 - 1/2*e^7 - 6*e^6 + 27/4*e^5 + 227/8*e^4 - 167/8*e^3 - 169/4*e^2 + 3/2*e + 12, 1/8*e^8 + 1/8*e^7 - 3/2*e^6 - 3/4*e^5 + 19/8*e^4 - 4*e^3 + 105/8*e^2 + 51/4*e - 27/2, -1/4*e^8 + 3/8*e^7 + 4*e^6 - 11/2*e^5 - 39/2*e^4 + 197/8*e^3 + 249/8*e^2 - 159/4*e - 15/2, -3/8*e^8 + 1/8*e^7 + 7*e^6 - 7/4*e^5 - 329/8*e^4 + 11/2*e^3 + 637/8*e^2 + 9/4*e - 63/2, -1/24*e^8 + 5/12*e^7 + 5/3*e^6 - 85/12*e^5 - 413/24*e^4 + 281/8*e^3 + 57*e^2 - 137/3*e - 41, -1/24*e^8 + 5/12*e^7 + 2/3*e^6 - 85/12*e^5 - 53/24*e^4 + 273/8*e^3 - 6*e^2 - 119/3*e + 13, -1/24*e^8 + 5/12*e^7 + 5/3*e^6 - 85/12*e^5 - 413/24*e^4 + 281/8*e^3 + 57*e^2 - 137/3*e - 41, -1/6*e^8 + 5/12*e^7 + 19/6*e^6 - 16/3*e^5 - 58/3*e^4 + 61/4*e^3 + 163/4*e^2 - 2/3*e - 14, 1/8*e^8 - 7/8*e^7 - 2*e^6 + 49/4*e^5 + 75/8*e^4 - 45*e^3 - 83/8*e^2 + 141/4*e - 27/2, -1/8*e^8 + 1/4*e^7 + 2*e^6 - 13/4*e^5 - 93/8*e^4 + 75/8*e^3 + 31*e^2 + e - 15, 1/8*e^8 - 3/8*e^7 - 2*e^6 + 25/4*e^5 + 83/8*e^4 - 63/2*e^3 - 199/8*e^2 + 181/4*e + 45/2, 1/8*e^8 - 5/8*e^7 - 5/2*e^6 + 37/4*e^5 + 119/8*e^4 - 155/4*e^3 - 185/8*e^2 + 177/4*e + 15/2, 1/8*e^8 - 2*e^6 + 5/4*e^5 + 81/8*e^4 - 113/8*e^3 - 85/4*e^2 + 59/2*e + 18, -1/8*e^8 + 5/8*e^7 + 5/2*e^6 - 37/4*e^5 - 127/8*e^4 + 143/4*e^3 + 265/8*e^2 - 105/4*e - 27/2, 1/12*e^8 - 1/12*e^7 - 11/6*e^6 + 13/6*e^5 + 167/12*e^4 - 33/2*e^3 - 149/4*e^2 + 97/3*e + 16, -13/24*e^8 + 2/3*e^7 + 55/6*e^6 - 121/12*e^5 - 1181/24*e^4 + 327/8*e^3 + 373/4*e^2 - 101/3*e - 35, -1/2*e^8 + 9/8*e^7 + 9*e^6 - 16*e^5 - 205/4*e^4 + 485/8*e^3 + 807/8*e^2 - 225/4*e - 87/2, -1/8*e^8 - 3/8*e^7 + 3/2*e^6 + 19/4*e^5 - 23/8*e^4 - 49/4*e^3 - 55/8*e^2 - 41/4*e - 3/2, 1/8*e^8 + 1/2*e^7 - 2*e^6 - 27/4*e^5 + 81/8*e^4 + 203/8*e^3 - 79/4*e^2 - 65/2*e + 12, -e^5 - e^4 + 12*e^3 + 13*e^2 - 26*e - 30, 3/8*e^8 - 3/8*e^7 - 7*e^6 + 27/4*e^5 + 341/8*e^4 - 147/4*e^3 - 739/8*e^2 + 229/4*e + 85/2, 7/12*e^8 - 17/24*e^7 - 59/6*e^6 + 67/6*e^5 + 307/6*e^4 - 389/8*e^3 - 685/8*e^2 + 571/12*e + 41/2, 11/24*e^8 - 5/6*e^7 - 25/3*e^6 + 143/12*e^5 + 1123/24*e^4 - 357/8*e^3 - 331/4*e^2 + 203/6*e + 34, -1/8*e^8 - 3/8*e^7 + 2*e^6 + 23/4*e^5 - 79/8*e^4 - 93/4*e^3 + 109/8*e^2 + 73/4*e + 13/2, -1/4*e^8 + 1/4*e^7 + 9/2*e^6 - 7/2*e^5 - 107/4*e^4 + 29/2*e^3 + 235/4*e^2 - 20*e - 27, 1/4*e^8 - 3/4*e^7 - 7/2*e^6 + 21/2*e^5 + 47/4*e^4 - 39*e^3 - 13/4*e^2 + 33*e - 6, -1/24*e^8 - 1/3*e^7 + 1/6*e^6 + 59/12*e^5 + 103/24*e^4 - 181/8*e^3 - 99/4*e^2 + 103/3*e + 16, 11/24*e^8 - 23/24*e^7 - 47/6*e^6 + 179/12*e^5 + 1021/24*e^4 - 263/4*e^3 - 649/8*e^2 + 883/12*e + 83/2, -5/12*e^8 + 1/6*e^7 + 43/6*e^6 - 11/6*e^5 - 457/12*e^4 + 9/4*e^3 + 127/2*e^2 + 29/6*e - 11, 5/24*e^8 - 17/24*e^7 - 23/6*e^6 + 125/12*e^5 + 547/24*e^4 - 165/4*e^3 - 375/8*e^2 + 481/12*e + 29/2, -7/24*e^8 - 5/24*e^7 + 31/6*e^6 + 29/12*e^5 - 665/24*e^4 - 13/4*e^3 + 333/8*e^2 - 227/12*e - 7/2, 1/3*e^8 - 23/24*e^7 - 35/6*e^6 + 41/3*e^5 + 377/12*e^4 - 421/8*e^3 - 455/8*e^2 + 637/12*e + 95/2, -3/8*e^8 + 3/4*e^7 + 11/2*e^6 - 43/4*e^5 - 183/8*e^4 + 321/8*e^3 + 55/2*e^2 - 61/2*e - 3, 1/8*e^7 - e^5 + 9/4*e^4 - 15/8*e^3 - 185/8*e^2 + 55/4*e + 63/2, 3/8*e^8 - 3/4*e^7 - 13/2*e^6 + 51/4*e^5 + 295/8*e^4 - 497/8*e^3 - 161/2*e^2 + 159/2*e + 45, 1/8*e^8 - 1/4*e^7 - e^6 + 17/4*e^5 - 35/8*e^4 - 171/8*e^3 + 35*e^2 + 28*e - 27, 1/3*e^8 - 1/12*e^7 - 16/3*e^6 + 5/3*e^5 + 139/6*e^4 - 43/4*e^3 - 81/4*e^2 + 143/6*e + 1, -1/24*e^8 + 11/12*e^7 + 7/6*e^6 - 157/12*e^5 - 221/24*e^4 + 413/8*e^3 + 22*e^2 - 343/6*e - 8, 3/8*e^8 - 5/8*e^7 - 13/2*e^6 + 39/4*e^5 + 273/8*e^4 - 44*e^3 - 429/8*e^2 + 205/4*e + 39/2, 1/8*e^8 + 1/2*e^7 - 3*e^6 - 31/4*e^5 + 185/8*e^4 + 267/8*e^3 - 227/4*e^2 - 75/2*e + 12, -3/8*e^8 + 1/8*e^7 + 13/2*e^6 - 7/4*e^5 - 281/8*e^4 + 11/2*e^3 + 481/8*e^2 - 17/4*e + 1/2, 1/8*e^8 - 3/8*e^7 - 3*e^6 + 21/4*e^5 + 187/8*e^4 - 39/2*e^3 - 511/8*e^2 + 101/4*e + 67/2, -13/24*e^8 + 19/24*e^7 + 55/6*e^6 - 133/12*e^5 - 1151/24*e^4 + 42*e^3 + 657/8*e^2 - 503/12*e - 43/2, 5/24*e^8 - 17/24*e^7 - 13/3*e^6 + 113/12*e^5 + 643/24*e^4 - 121/4*e^3 - 355/8*e^2 + 163/12*e - 7/2, 1/12*e^8 - 1/3*e^7 - 1/3*e^6 + 37/6*e^5 - 79/12*e^4 - 135/4*e^3 + 61/2*e^2 + 142/3*e - 14, -1/6*e^8 + 5/12*e^7 + 19/6*e^6 - 19/3*e^5 - 64/3*e^4 + 113/4*e^3 + 243/4*e^2 - 110/3*e - 44, 1/4*e^8 - 3/8*e^7 - 11/2*e^6 + 13/2*e^5 + 81/2*e^4 - 269/8*e^3 - 853/8*e^2 + 201/4*e + 105/2, 1/4*e^8 + 1/8*e^7 - 9/2*e^6 - 5/2*e^5 + 53/2*e^4 + 103/8*e^3 - 449/8*e^2 - 35/4*e + 33/2, -1/8*e^8 + 5/8*e^7 + 2*e^6 - 37/4*e^5 - 63/8*e^4 + 155/4*e^3 - 27/8*e^2 - 167/4*e + 39/2, 1/2*e^8 - 3/4*e^7 - 15/2*e^6 + 11*e^5 + 30*e^4 - 173/4*e^3 - 95/4*e^2 + 42*e - 6, 1/8*e^8 - 3*e^6 + 5/4*e^5 + 177/8*e^4 - 121/8*e^3 - 209/4*e^2 + 87/2*e + 24, 1/4*e^8 - e^7 - 5*e^6 + 27/2*e^5 + 125/4*e^4 - 189/4*e^3 - 131/2*e^2 + 46*e + 36, 11/24*e^8 - 17/24*e^7 - 47/6*e^6 + 131/12*e^5 + 985/24*e^4 - 87/2*e^3 - 539/8*e^2 + 349/12*e + 29/2, 5/24*e^8 + 7/24*e^7 - 10/3*e^6 - 31/12*e^5 + 403/24*e^4 - 5/4*e^3 - 275/8*e^2 + 283/12*e + 59/2, -1/4*e^8 + 5/8*e^7 + 9/2*e^6 - 21/2*e^5 - 28*e^4 + 415/8*e^3 + 571/8*e^2 - 303/4*e - 95/2, -5/8*e^8 + 1/2*e^7 + 11*e^6 - 29/4*e^5 - 485/8*e^4 + 233/8*e^3 + 439/4*e^2 - 69/2*e - 34, -7/8*e^8 + 7/8*e^7 + 29/2*e^6 - 55/4*e^5 - 577/8*e^4 + 239/4*e^3 + 843/8*e^2 - 207/4*e - 33/2, 1/4*e^8 - 3/8*e^7 - 5*e^6 + 11/2*e^5 + 67/2*e^4 - 181/8*e^3 - 625/8*e^2 + 87/4*e + 39/2, 1/4*e^8 - 7/2*e^6 - 1/2*e^5 + 53/4*e^4 + 31/4*e^3 - 16*e^2 - 55/2*e + 30, -1/2*e^7 + 8*e^5 - e^4 - 77/2*e^3 + 11/2*e^2 + 55*e, -1/8*e^8 + 1/8*e^7 + 5/2*e^6 - 1/4*e^5 - 127/8*e^4 - 35/4*e^3 + 245/8*e^2 + 95/4*e + 1/2, -1/4*e^7 - 1/2*e^6 + 4*e^5 + 9/2*e^4 - 69/4*e^3 - 13/4*e^2 + 12*e - 7, -1/8*e^8 + 1/4*e^7 + 3*e^6 - 13/4*e^5 - 189/8*e^4 + 99/8*e^3 + 64*e^2 - 19*e - 21, -9/8*e^7 - e^6 + 17*e^5 + 51/4*e^4 - 609/8*e^3 - 311/8*e^2 + 413/4*e + 69/2, 3/8*e^8 - 5/8*e^7 - 8*e^6 + 39/4*e^5 + 441/8*e^4 - 42*e^3 - 1033/8*e^2 + 135/4*e + 109/2, -5/8*e^8 + 1/2*e^7 + 23/2*e^6 - 29/4*e^5 - 525/8*e^4 + 225/8*e^3 + 469/4*e^2 - 23*e - 34, 1/12*e^8 + 37/24*e^7 - 11/6*e^6 - 143/6*e^5 + 85/6*e^4 + 841/8*e^3 - 311/8*e^2 - 1439/12*e + 23/2, 1/12*e^8 - 1/12*e^7 - 17/6*e^6 + 13/6*e^5 + 323/12*e^4 - 35/2*e^3 - 317/4*e^2 + 100/3*e + 37, -5/12*e^8 + 7/24*e^7 + 31/6*e^6 - 35/6*e^5 - 77/6*e^4 + 267/8*e^3 - 69/8*e^2 - 545/12*e + 5/2, 1/3*e^8 - 11/24*e^7 - 29/6*e^6 + 23/3*e^5 + 197/12*e^4 - 289/8*e^3 + 69/8*e^2 + 565/12*e - 73/2, -5/8*e^8 + 23/2*e^6 - 1/4*e^5 - 533/8*e^4 + 21/8*e^3 + 499/4*e^2 - 5*e - 42, 1/4*e^7 + e^6 - 2*e^5 - 27/2*e^4 - 7/4*e^3 + 179/4*e^2 + 19/2*e - 21, 1/8*e^8 - 7/8*e^7 - 7/2*e^6 + 49/4*e^5 + 235/8*e^4 - 44*e^3 - 631/8*e^2 + 111/4*e + 61/2, 1/4*e^7 + e^6 - 3*e^5 - 23/2*e^4 + 33/4*e^3 + 103/4*e^2 - 15/2*e - 1, -5/12*e^8 + 7/6*e^7 + 20/3*e^6 - 107/6*e^5 - 373/12*e^4 + 305/4*e^3 + 39*e^2 - 248/3*e - 2, -1/24*e^8 + 13/24*e^7 - 5/6*e^6 - 97/12*e^5 + 409/24*e^4 + 129/4*e^3 - 485/8*e^2 - 233/12*e + 95/2, 1/2*e^8 - 1/4*e^7 - 15/2*e^6 + 4*e^5 + 30*e^4 - 63/4*e^3 - 85/4*e^2 + e - 15, -3/4*e^8 + 11/8*e^7 + 27/2*e^6 - 41/2*e^5 - 78*e^4 + 681/8*e^3 + 1293/8*e^2 - 369/4*e - 141/2, 1/8*e^8 - 1/2*e^7 - 9/2*e^6 + 29/4*e^5 + 337/8*e^4 - 237/8*e^3 - 445/4*e^2 + 30*e + 24, 3/4*e^8 - 13/8*e^7 - 14*e^6 + 49/2*e^5 + 167/2*e^4 - 803/8*e^3 - 1367/8*e^2 + 365/4*e + 153/2, -1/24*e^8 - 5/24*e^7 - 5/6*e^6 + 35/12*e^5 + 397/24*e^4 - 13/2*e^3 - 455/8*e^2 - 275/12*e + 65/2, -1/6*e^8 - 1/12*e^7 + 8/3*e^6 - 1/3*e^5 - 37/3*e^4 + 43/4*e^3 + 55/4*e^2 - 109/6*e - 5, -5/8*e^8 + 1/2*e^7 + 23/2*e^6 - 29/4*e^5 - 549/8*e^4 + 257/8*e^3 + 577/4*e^2 - 48*e - 63, 3/8*e^7 + e^6 - 4*e^5 - 45/4*e^4 + 27/8*e^3 + 229/8*e^2 + 97/4*e - 39/2, -1/24*e^8 + 5/12*e^7 - 1/3*e^6 - 85/12*e^5 + 235/24*e^4 + 273/8*e^3 - 30*e^2 - 104/3*e - 11, 11/24*e^8 - 23/24*e^7 - 53/6*e^6 + 191/12*e^5 + 1333/24*e^4 - 311/4*e^3 - 985/8*e^2 + 1303/12*e + 131/2, 11/24*e^8 - 5/24*e^7 - 41/6*e^6 + 23/12*e^5 + 721/24*e^4 + 4*e^3 - 399/8*e^2 - 479/12*e + 53/2, -19/24*e^8 + 17/12*e^7 + 41/3*e^6 - 247/12*e^5 - 1799/24*e^4 + 623/8*e^3 + 295/2*e^2 - 197/3*e - 71, -1/6*e^8 - 5/24*e^7 + 25/6*e^6 + 11/3*e^5 - 403/12*e^4 - 163/8*e^3 + 715/8*e^2 + 451/12*e - 79/2, -5/12*e^8 + 1/6*e^7 + 26/3*e^6 - 17/6*e^5 - 709/12*e^4 + 61/4*e^3 + 140*e^2 - 80/3*e - 68, -3/4*e^7 - 1/2*e^6 + 10*e^5 + 13/2*e^4 - 139/4*e^3 - 103/4*e^2 + 35*e + 42, 1/8*e^8 + 3/8*e^7 - 3/2*e^6 - 23/4*e^5 + 7/8*e^4 + 101/4*e^3 + 223/8*e^2 - 123/4*e - 81/2, 3/8*e^8 - 5/8*e^7 - 5*e^6 + 35/4*e^5 + 113/8*e^4 - 33*e^3 + 55/8*e^2 + 135/4*e - 27/2, -3/8*e^8 + 3/4*e^7 + 7*e^6 - 51/4*e^5 - 343/8*e^4 + 489/8*e^3 + 97*e^2 - 69*e - 51, -5/12*e^8 + 17/12*e^7 + 23/3*e^6 - 131/6*e^5 - 535/12*e^4 + 191/2*e^3 + 363/4*e^2 - 691/6*e - 53, -7/24*e^8 + 25/24*e^7 + 14/3*e^6 - 199/12*e^5 - 605/24*e^4 + 74*e^3 + 527/8*e^2 - 995/12*e - 133/2, -1/8*e^7 + 1/2*e^6 + e^5 - 29/4*e^4 + 15/8*e^3 + 221/8*e^2 - 1/4*e - 57/2, 3/2*e^7 - 22*e^5 + 177/2*e^3 + 5/2*e^2 - 88*e - 12, 3/8*e^8 - 7/8*e^7 - 13/2*e^6 + 51/4*e^5 + 285/8*e^4 - 189/4*e^3 - 515/8*e^2 + 83/4*e + 9/2, -3/8*e^8 - 3/8*e^7 + 11/2*e^6 + 25/4*e^5 - 161/8*e^4 - 29*e^3 - 3/8*e^2 + 79/4*e + 105/2, 1/8*e^8 - 1/2*e^7 - 2*e^6 + 25/4*e^5 + 73/8*e^4 - 125/8*e^3 - 23/4*e^2 - 17/2*e - 6, 1/4*e^7 + 1/2*e^6 - 4*e^5 - 15/2*e^4 + 65/4*e^3 + 93/4*e^2 - 6*e + 21, -1/4*e^8 + 1/4*e^7 + 7/2*e^6 - 9/2*e^5 - 55/4*e^4 + 43/2*e^3 + 83/4*e^2 - 16*e - 27, 3/8*e^8 - 3/4*e^7 - 11/2*e^6 + 39/4*e^5 + 175/8*e^4 - 241/8*e^3 - 41/2*e^2 + 31/2*e + 12, -1/24*e^8 - 1/12*e^7 - 1/3*e^6 + 23/12*e^5 + 187/24*e^4 - 131/8*e^3 - 35/2*e^2 + 175/3*e - 5, -7/8*e^8 + 5/4*e^7 + 29/2*e^6 - 75/4*e^5 - 579/8*e^4 + 617/8*e^3 + 112*e^2 - 153/2*e - 34, -3/8*e^8 + 6*e^6 - 3/4*e^5 - 243/8*e^4 + 91/8*e^3 + 251/4*e^2 - 99/2*e - 52, -1/24*e^8 - 1/3*e^7 + 5/3*e^6 + 47/12*e^5 - 353/24*e^4 - 85/8*e^3 + 131/4*e^2 + 65/6*e + 4, 3/4*e^8 - 7/8*e^7 - 12*e^6 + 27/2*e^5 + 55*e^4 - 445/8*e^3 - 525/8*e^2 + 179/4*e - 3/2, -3/8*e^8 + 1/8*e^7 + 8*e^6 - 11/4*e^5 - 441/8*e^4 + 35/2*e^3 + 1013/8*e^2 - 139/4*e - 117/2, 5/8*e^8 + 3/8*e^7 - 10*e^6 - 27/4*e^5 + 379/8*e^4 + 143/4*e^3 - 541/8*e^2 - 185/4*e + 57/2, -1/2*e^8 - 1/8*e^7 + 9*e^6 + e^5 - 203/4*e^4 + 43/8*e^3 + 705/8*e^2 - 155/4*e - 39/2, 5/8*e^8 + 1/4*e^7 - 10*e^6 - 11/4*e^5 + 369/8*e^4 + 45/8*e^3 - 99/2*e^2 - 5*e - 27, -1/4*e^8 - 1/8*e^7 + 9/2*e^6 + 3/2*e^5 - 49/2*e^4 + 1/8*e^3 + 337/8*e^2 - 133/4*e - 33/2, 5/8*e^8 - 9/8*e^7 - 25/2*e^6 + 65/4*e^5 + 627/8*e^4 - 251/4*e^3 - 1245/8*e^2 + 221/4*e + 99/2, 1/4*e^7 - 1/2*e^6 - 4*e^5 + 13/2*e^4 + 61/4*e^3 - 87/4*e^2 - 10*e + 30, -3/4*e^8 + 19/8*e^7 + 14*e^6 - 71/2*e^5 - 83*e^4 + 1153/8*e^3 + 1329/8*e^2 - 507/4*e - 137/2, -7/8*e^8 + 7/8*e^7 + 13*e^6 - 51/4*e^5 - 425/8*e^4 + 195/4*e^3 + 471/8*e^2 - 181/4*e - 59/2];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {2,2,-w^2+w+2}>] := -1;
ALEigenvalues[ideal<ZF | {11,11,-w^2-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;