/*
  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![4, 3, -5, -2, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {16, 16, w^3 - 2*w^2 - 4*w}>;

primesArray := [
[2, 2, w^3 - 3*w^2 - w + 2],
[5, 5, w^2 - 2*w - 3],
[5, 5, w^3 - 3*w^2 - 3*w + 5],
[8, 2, w^3 - 2*w^2 - 3*w + 3],
[13, 13, -w^2 + w + 3],
[17, 17, -w^2 + 3*w + 1],
[23, 23, w^2 - 3],
[25, 5, -w^2 + 2*w + 1],
[37, 37, w^3 - 4*w^2 - 2*w + 9],
[41, 41, w^2 - w - 5],
[49, 7, -w^3 + 2*w^2 + 2*w - 1],
[49, 7, w^2 + w - 3],
[59, 59, w^3 - 3*w^2 - 4*w + 5],
[67, 67, -w^3 + 3*w^2 + w - 5],
[71, 71, 2*w - 3],
[71, 71, w^3 - 4*w^2 + 2*w + 5],
[79, 79, -w^3 + 5*w^2 - 4*w - 5],
[79, 79, -2*w^3 + 7*w^2 + 5*w - 15],
[81, 3, -3],
[83, 83, -2*w^3 + 6*w^2 + 2*w - 5],
[97, 97, w^3 - w^2 - 7*w - 3],
[101, 101, -w^3 + 3*w^2 + 3*w - 3],
[107, 107, -w^3 + 4*w^2 - 7],
[113, 113, w^3 - 3*w^2 - 5*w + 3],
[121, 11, -w^3 + 3*w^2 + 4*w - 3],
[121, 11, 2*w^3 - 6*w^2 - 3*w + 5],
[127, 127, -2*w^3 + 6*w^2 + 5*w - 9],
[137, 137, -w^3 + 5*w^2 + 3*w - 9],
[139, 139, w^3 - w^2 - 4*w + 1],
[149, 149, w^3 - 3*w^2 - w + 9],
[157, 157, w^2 - 4*w + 1],
[163, 163, -3*w^3 + 7*w^2 + 9*w - 13],
[163, 163, w^3 - 3*w^2 - w + 7],
[179, 179, -2*w^3 + 3*w^2 + 5*w - 5],
[179, 179, -2*w^3 + 4*w^2 + 5*w - 9],
[181, 181, -w^2 + 3*w - 3],
[181, 181, 2*w^3 - 4*w^2 - 8*w + 5],
[191, 191, 2*w - 5],
[193, 193, -2*w^3 + 4*w^2 + 7*w + 3],
[193, 193, -2*w^3 + 5*w^2 + 5*w - 7],
[197, 197, -w^3 + 4*w^2 - 11],
[199, 199, -2*w^3 + 8*w^2 + 6*w - 15],
[211, 211, -2*w^3 + 3*w^2 + 6*w - 5],
[211, 211, -4*w^3 + 10*w^2 + 12*w - 19],
[229, 229, w^3 - 3*w^2 - 4*w + 9],
[229, 229, w^2 - 5],
[233, 233, w^2 - 2*w - 7],
[239, 239, -3*w^3 + 10*w^2 + 2*w - 9],
[239, 239, -2*w^3 + 5*w^2 + 5*w - 5],
[257, 257, -2*w^3 + 5*w^2 + 4*w - 9],
[263, 263, w^3 - 2*w^2 - 4*w - 3],
[263, 263, w^3 - 6*w - 7],
[269, 269, w^3 - 3*w^2 - w - 1],
[269, 269, w - 5],
[281, 281, -2*w + 7],
[293, 293, 2*w^2 - w - 7],
[293, 293, -2*w^3 + 6*w^2 + 6*w - 9],
[307, 307, -2*w^3 + 5*w^2 + 4*w - 11],
[311, 311, -w^3 + 3*w^2 + 2*w - 9],
[317, 317, 4*w^3 - 15*w^2 + 3*w + 15],
[317, 317, w^3 - 4*w^2 - 4*w + 9],
[331, 331, -w^3 + 4*w^2 + 2*w - 11],
[331, 331, 2*w^3 - 7*w^2 - 5*w + 13],
[337, 337, 2*w^3 - 4*w^2 - 8*w - 1],
[347, 347, -2*w^3 + 7*w^2 + 9*w - 9],
[349, 349, -w^3 + 5*w^2 - 2*w - 11],
[349, 349, 4*w^3 - 9*w^2 - 14*w + 13],
[353, 353, -3*w^3 + 7*w^2 + 9*w - 11],
[367, 367, 3*w^3 - 5*w^2 - 11*w + 5],
[367, 367, -w^3 + 3*w^2 + 5*w - 1],
[373, 373, w^3 - 2*w^2 - 6*w + 1],
[379, 379, 2*w^3 - 5*w^2 - 7*w + 5],
[383, 383, 2*w^3 - 3*w^2 - 11*w - 1],
[383, 383, -2*w^3 + 5*w^2 + 3*w - 13],
[397, 397, -2*w^3 + 4*w^2 + 7*w - 7],
[397, 397, 2*w^3 - 3*w^2 - 8*w - 5],
[409, 409, w^3 - 3*w^2 + w - 1],
[419, 419, -4*w^3 + 14*w^2 - w - 13],
[421, 421, -3*w^2 + 2*w + 9],
[421, 421, 2*w^2 - 6*w - 3],
[431, 431, -2*w^3 + 4*w^2 + 9*w - 7],
[431, 431, -2*w^3 + 6*w^2 + 3*w - 15],
[431, 431, -3*w^3 + 6*w^2 + 10*w - 9],
[431, 431, -3*w^3 + 11*w^2 + 10*w - 19],
[443, 443, 3*w^3 - 6*w^2 - 8*w + 13],
[443, 443, -2*w^3 + 5*w^2 + 4*w - 5],
[443, 443, -2*w^3 + 5*w^2 + 6*w - 3],
[443, 443, 5*w^3 - 12*w^2 - 16*w + 19],
[461, 461, w^3 - 2*w^2 + 7],
[461, 461, -3*w^2 + w + 7],
[479, 479, w^2 + w - 5],
[487, 487, 2*w^2 - 4*w + 1],
[487, 487, 2*w^3 - 8*w^2 + 4*w + 7],
[491, 491, -3*w^3 + 7*w^2 + 7*w - 1],
[509, 509, -2*w^3 + 9*w^2 + 5*w - 19],
[509, 509, -3*w^3 + 6*w^2 + 10*w + 1],
[521, 521, w^3 - 2*w^2 - 6*w + 3],
[521, 521, 3*w - 5],
[523, 523, -w^3 + 5*w^2 + 4*w - 7],
[523, 523, -2*w^3 + 4*w^2 + 6*w - 1],
[523, 523, w^2 - 7],
[523, 523, -2*w^3 + 8*w^2 + 8*w - 13],
[541, 541, -w^3 + 5*w^2 - w - 13],
[547, 547, -2*w^3 + 7*w^2 + 3*w - 11],
[547, 547, -w^3 + 4*w^2 + 2*w - 13],
[547, 547, -2*w^3 + 2*w^2 + 6*w - 1],
[547, 547, -w^3 + 6*w^2 - 6*w - 7],
[563, 563, 2*w^2 - 5*w - 1],
[563, 563, w^2 - 2*w + 3],
[569, 569, -2*w - 5],
[571, 571, -2*w^3 + 6*w^2 + 3*w - 11],
[577, 577, -w^3 + 3*w^2 - 7],
[593, 593, w^3 - w^2 - 5*w + 3],
[601, 601, 2*w^3 - 6*w^2 - 4*w + 5],
[607, 607, -6*w^2 + 15*w + 13],
[607, 607, -w^3 + 4*w^2 - 2*w - 7],
[613, 613, -2*w^3 + 6*w^2 + 2*w - 9],
[617, 617, -3*w^3 + 9*w^2 + 5*w - 9],
[617, 617, 3*w^3 - 6*w^2 - 12*w + 7],
[619, 619, -2*w^3 + 8*w^2 + 7*w - 13],
[619, 619, 2*w^3 - 4*w^2 - 5*w + 1],
[641, 641, 2*w^3 - w^2 - 6*w + 1],
[641, 641, -2*w^3 + 4*w^2 + 8*w - 7],
[641, 641, -2*w^3 + 7*w^2 + 4*w - 13],
[641, 641, -2*w^3 + 3*w^2 + 7*w - 1],
[647, 647, -2*w^3 + 7*w^2 + 8*w - 11],
[647, 647, -2*w^3 + 4*w^2 + 6*w - 3],
[653, 653, -4*w^2 + 3*w + 13],
[653, 653, 3*w^3 - 7*w^2 - 11*w + 7],
[659, 659, -5*w^3 + 14*w^2 + 16*w - 25],
[659, 659, -4*w^3 + 13*w^2 + 11*w - 25],
[673, 673, 2*w^3 - 7*w^2 - 6*w + 15],
[677, 677, 3*w^3 - 8*w^2 - 6*w + 11],
[677, 677, w^3 - 5*w^2 + 2*w + 9],
[683, 683, 2*w^2 - 5*w - 9],
[683, 683, w^3 - w^2 - 4*w - 7],
[709, 709, -2*w^3 + 6*w^2 + 4*w - 3],
[733, 733, -2*w^3 + 5*w^2 + 3*w + 1],
[733, 733, 2*w^3 - 2*w^2 - 9*w - 1],
[739, 739, -w - 5],
[751, 751, 3*w^3 - 7*w^2 - 7*w + 11],
[757, 757, -2*w^3 + 3*w^2 + 8*w - 3],
[757, 757, 2*w^3 - 3*w^2 - 12*w - 3],
[761, 761, -3*w^3 + 7*w^2 + 12*w - 11],
[761, 761, -3*w^3 + 9*w^2 + 6*w - 11],
[787, 787, -3*w^3 + 5*w^2 + 12*w + 3],
[787, 787, w^2 - w - 9],
[797, 797, -3*w^3 + 7*w^2 + 6*w - 17],
[827, 827, -2*w^3 + 4*w^2 + 9*w - 11],
[827, 827, -6*w^3 + 17*w^2 + 19*w - 29],
[853, 853, -3*w^3 + 10*w^2 + 6*w - 21],
[857, 857, 3*w^3 - 12*w^2 + 4*w + 9],
[857, 857, -3*w^3 + 9*w^2 + 4*w - 13],
[863, 863, -3*w^3 + 10*w^2 + 4*w - 15],
[863, 863, -4*w^2 + w + 11],
[877, 877, -2*w^3 + 4*w^2 + 8*w - 9],
[881, 881, -2*w^3 + 4*w^2 + 5*w - 3],
[907, 907, -w^3 + 5*w^2 - 2*w - 15],
[907, 907, -2*w^3 + 6*w^2 + 7*w - 15],
[929, 929, -w^2 + 11],
[929, 929, -2*w^3 + 6*w^2 + 3*w - 13],
[937, 937, -3*w^3 + 7*w^2 + 8*w - 3],
[941, 941, w^3 + w^2 - 8*w - 5],
[953, 953, 4*w^2 - 8*w - 13],
[953, 953, w^3 - 4*w^2 - 4*w + 3],
[967, 967, 3*w^2 - 5*w - 9],
[971, 971, 3*w^2 - 4*w - 7],
[971, 971, w^3 - 5*w^2 - 4*w + 11],
[977, 977, -3*w^3 + 11*w^2 + 6*w - 25],
[983, 983, -2*w^3 + 8*w^2 + 3*w - 21],
[991, 991, 2*w^3 - 4*w^2 - 10*w + 3],
[991, 991, 2*w^3 - 8*w^2 - 3*w + 19],
[997, 997, -4*w^3 + 10*w^2 + 11*w - 21]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^7 - 4*x^6 - 17*x^5 + 72*x^4 + 29*x^3 - 198*x^2 + 128;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [0, 1/16*e^6 - 1/8*e^5 - 23/16*e^4 + 17/8*e^3 + 125/16*e^2 - 39/8*e - 8, e, 1/16*e^6 - 1/8*e^5 - 21/16*e^4 + 2*e^3 + 99/16*e^2 - 25/8*e - 8, -1/16*e^6 + 1/4*e^5 + 15/16*e^4 - 31/8*e^3 + 5/16*e^2 + 25/8*e - 4, 1/4*e^4 - 1/4*e^3 - 17/4*e^2 + 7/2*e + 8, -1/16*e^6 + 1/4*e^5 + 17/16*e^4 - 9/2*e^3 - 29/16*e^2 + 83/8*e, -1/8*e^6 + 1/2*e^5 + 2*e^4 - 67/8*e^3 - 3/2*e^2 + 27/2*e, 1/8*e^6 - 1/2*e^5 - 2*e^4 + 67/8*e^3 + 3/2*e^2 - 31/2*e + 4, 1/8*e^5 - 3/8*e^4 - 19/8*e^3 + 6*e^2 + 11/2*e - 4, -1/8*e^6 + 3/8*e^5 + 2*e^4 - 49/8*e^3 - 9/8*e^2 + 33/4*e - 4, -1/8*e^5 + 3/8*e^4 + 19/8*e^3 - 6*e^2 - 19/2*e + 8, 1/2*e^3 - 1/2*e^2 - 9/2*e + 5, 1/4*e^6 - 3/4*e^5 - 9/2*e^4 + 49/4*e^3 + 45/4*e^2 - 17*e - 9, -1/8*e^6 + 1/4*e^5 + 23/8*e^4 - 17/4*e^3 - 125/8*e^2 + 31/4*e + 16, -1/4*e^4 - 1/4*e^3 + 15/4*e^2 + 5*e - 3, -1/8*e^6 + 1/2*e^5 + 17/8*e^4 - 9*e^3 - 29/8*e^2 + 91/4*e + 4, 1/2*e^4 - 1/2*e^3 - 17/2*e^2 + 7*e + 12, -1/4*e^6 + 3/4*e^5 + 9/2*e^4 - 49/4*e^3 - 41/4*e^2 + 17*e + 3, -1/2*e^4 + 1/2*e^3 + 17/2*e^2 - 5*e - 20, -1/4*e^6 + 1/2*e^5 + 11/2*e^4 - 33/4*e^3 - 28*e^2 + 13*e + 34, -1/4*e^5 + e^4 + 4*e^3 - 71/4*e^2 - 3*e + 31, -1/8*e^6 + 3/4*e^5 + 9/8*e^4 - 25/2*e^3 + 101/8*e^2 + 85/4*e - 20, 1/4*e^6 - 3/4*e^5 - 19/4*e^4 + 25/2*e^3 + 33/2*e^2 - 43/2*e - 15, 1/4*e^6 - 3/4*e^5 - 21/4*e^4 + 25/2*e^3 + 47/2*e^2 - 20*e - 28, 1/4*e^6 - e^5 - 17/4*e^4 + 17*e^3 + 29/4*e^2 - 61/2*e, -1/4*e^5 + 3/4*e^4 + 17/4*e^3 - 23/2*e^2 - 13/2*e + 11, -1/8*e^6 + 1/4*e^5 + 11/4*e^4 - 29/8*e^3 - 27/2*e^2 + 5/2*e + 20, 5/16*e^6 - e^5 - 93/16*e^4 + 33/2*e^3 + 301/16*e^2 - 211/8*e - 20, 1/4*e^5 - 5/4*e^4 - 17/4*e^3 + 39/2*e^2 + 10*e - 26, 1/2*e^6 - 3/2*e^5 - 19/2*e^4 + 25*e^3 + 30*e^2 - 45*e - 20, 5/16*e^6 - e^5 - 93/16*e^4 + 33/2*e^3 + 301/16*e^2 - 211/8*e - 28, 1/4*e^5 - 1/2*e^4 - 5*e^3 + 35/4*e^2 + 29/2*e - 14, 3/8*e^6 - e^5 - 61/8*e^4 + 65/4*e^3 + 261/8*e^2 - 103/4*e - 40, -1/4*e^6 + 1/2*e^5 + 11/2*e^4 - 35/4*e^3 - 55/2*e^2 + 41/2*e + 33, 1/4*e^5 - e^4 - 9/2*e^3 + 69/4*e^2 + 17/2*e - 22, -3/8*e^5 + 11/8*e^4 + 47/8*e^3 - 89/4*e^2 - 6*e + 36, 1/4*e^6 - 3/4*e^5 - 9/2*e^4 + 51/4*e^3 + 43/4*e^2 - 47/2*e, -1/2*e^5 + 3/2*e^4 + 17/2*e^3 - 25*e^2 - 12*e + 34, 1/8*e^5 - 1/8*e^4 - 13/8*e^3 + 7/4*e^2 - 8*e + 8, -1/4*e^6 + 1/2*e^5 + 23/4*e^4 - 17/2*e^3 - 125/4*e^2 + 39/2*e + 38, -1/16*e^6 + 1/2*e^5 + 9/16*e^4 - 17/2*e^3 + 95/16*e^2 + 127/8*e - 16, -9/16*e^6 + 7/4*e^5 + 169/16*e^4 - 57/2*e^3 - 541/16*e^2 + 339/8*e + 36, 1/8*e^6 + 1/4*e^5 - 33/8*e^4 - 9/2*e^3 + 283/8*e^2 + 27/4*e - 36, -1/8*e^6 + 1/2*e^5 + 2*e^4 - 67/8*e^3 - 7/2*e^2 + 27/2*e + 20, -3/16*e^6 + 3/4*e^5 + 35/16*e^4 - 23/2*e^3 + 201/16*e^2 + 49/8*e - 28, 1/4*e^5 - 3/4*e^4 - 17/4*e^3 + 21/2*e^2 + 13/2*e - 5, -1/4*e^6 + 1/2*e^5 + 6*e^4 - 35/4*e^3 - 73/2*e^2 + 21*e + 54, -7/16*e^6 + 3/2*e^5 + 111/16*e^4 - 49/2*e^3 - 87/16*e^2 + 313/8*e - 4, -1/2*e^6 + 2*e^5 + 33/4*e^4 - 131/4*e^3 - 45/4*e^2 + 97/2*e + 2, 1/4*e^6 - 3/4*e^5 - 9/2*e^4 + 51/4*e^3 + 43/4*e^2 - 55/2*e, 5/16*e^6 - 5/4*e^5 - 81/16*e^4 + 85/4*e^3 + 109/16*e^2 - 347/8*e - 8, 1/2*e^6 - 5/4*e^5 - 41/4*e^4 + 83/4*e^3 + 85/2*e^2 - 67/2*e - 49, -1/2*e^4 + 1/2*e^3 + 19/2*e^2 - 7*e - 18, -1/4*e^6 + e^5 + 15/4*e^4 - 17*e^3 + 3/4*e^2 + 35*e - 7, 1/16*e^6 - 1/4*e^5 - 17/16*e^4 + 7/2*e^3 + 77/16*e^2 + 37/8*e - 24, -1/4*e^6 + 3/2*e^5 + 3*e^4 - 101/4*e^3 + 12*e^2 + 89/2*e - 15, -1/4*e^6 + 1/2*e^5 + 23/4*e^4 - 19/2*e^3 - 129/4*e^2 + 61/2*e + 38, 3/4*e^4 - 3/4*e^3 - 47/4*e^2 + 13/2*e + 22, 1/2*e^3 + 3/2*e^2 - 9/2*e - 17, -1/16*e^6 + 27/16*e^4 + 7/8*e^3 - 219/16*e^2 - 47/8*e + 36, 1/4*e^6 - 1/4*e^5 - 25/4*e^4 + 9/2*e^3 + 39*e^2 - 14*e - 44, -1/8*e^6 + 1/4*e^5 + 23/8*e^4 - 17/4*e^3 - 125/8*e^2 + 39/4*e + 24, -1/2*e^5 + 3/4*e^4 + 41/4*e^3 - 49/4*e^2 - 71/2*e + 14, -3/4*e^6 + 9/4*e^5 + 14*e^4 - 147/4*e^3 - 163/4*e^2 + 115/2*e + 32, -1/8*e^6 + 1/4*e^5 + 11/4*e^4 - 29/8*e^3 - 31/2*e^2 - 7/2*e + 36, -1/4*e^6 + 3/4*e^5 + 39/8*e^4 - 101/8*e^3 - 137/8*e^2 + 101/4*e + 20, -3/16*e^6 + 5/8*e^5 + 45/16*e^4 - 75/8*e^3 + 21/16*e^2 + 65/8*e - 8, -1/4*e^5 + 3/4*e^4 + 19/4*e^3 - 12*e^2 - 15*e + 4, 1/2*e^6 - 3/2*e^5 - 19/2*e^4 + 24*e^3 + 32*e^2 - 32*e - 32, 1/4*e^6 - 3/4*e^5 - 23/4*e^4 + 14*e^3 + 31*e^2 - 40*e - 26, -1/4*e^5 + e^4 + 9/2*e^3 - 61/4*e^2 - 27/2*e + 22, -3/4*e^5 + 2*e^4 + 13*e^3 - 125/4*e^2 - 31*e + 39, 1/4*e^6 - 1/2*e^5 - 5*e^4 + 31/4*e^3 + 35/2*e^2 - 7*e - 6, 3/8*e^6 - e^5 - 15/2*e^4 + 141/8*e^3 + 57/2*e^2 - 77/2*e - 28, -3/8*e^6 + 7/8*e^5 + 17/2*e^4 - 115/8*e^3 - 377/8*e^2 + 85/4*e + 72, -3/16*e^6 + 1/4*e^5 + 73/16*e^4 - 27/8*e^3 - 469/16*e^2 - 1/8*e + 36, -1/4*e^6 + 3/4*e^5 + 9/2*e^4 - 55/4*e^3 - 47/4*e^2 + 85/2*e - 2, -1/16*e^6 + 27/16*e^4 + 7/8*e^3 - 219/16*e^2 - 63/8*e + 40, 1/4*e^6 - 5/4*e^5 - 13/4*e^4 + 41/2*e^3 - 10*e^2 - 28*e + 38, -5/16*e^6 + 5/4*e^5 + 85/16*e^4 - 41/2*e^3 - 145/16*e^2 + 239/8*e + 20, 1/2*e^6 - 5/4*e^5 - 39/4*e^4 + 83/4*e^3 + 67/2*e^2 - 34*e - 32, -1/8*e^6 + 1/2*e^5 + 21/8*e^4 - 19/2*e^3 - 97/8*e^2 + 103/4*e + 8, -1/2*e^6 + 7/4*e^5 + 33/4*e^4 - 117/4*e^3 - 21/2*e^2 + 45*e + 14, -1/4*e^5 + 1/2*e^4 + 9/2*e^3 - 25/4*e^2 - 10*e + 5, -1/16*e^6 + 1/2*e^5 + 13/16*e^4 - 31/4*e^3 - 5/16*e^2 + 51/8*e + 8, -1/4*e^6 + 1/2*e^5 + 11/2*e^4 - 29/4*e^3 - 29*e^2 - e + 44, -3/16*e^6 + 1/2*e^5 + 63/16*e^4 - 35/4*e^3 - 311/16*e^2 + 129/8*e + 32, -1/4*e^6 + 3/4*e^5 + 9/2*e^4 - 55/4*e^3 - 43/4*e^2 + 73/2*e + 8, -1/4*e^6 + 5/4*e^5 + 21/8*e^4 - 163/8*e^3 + 157/8*e^2 + 95/4*e - 48, -5/8*e^6 + 7/4*e^5 + 99/8*e^4 - 113/4*e^3 - 381/8*e^2 + 167/4*e + 60, 5/16*e^6 - e^5 - 93/16*e^4 + 33/2*e^3 + 237/16*e^2 - 179/8*e, 3/4*e^6 - 2*e^5 - 59/4*e^4 + 67/2*e^3 + 221/4*e^2 - 68*e - 53, -1/8*e^6 + 3/4*e^5 + 5/8*e^4 - 12*e^3 + 153/8*e^2 + 65/4*e - 28, 1/4*e^6 - 5/8*e^5 - 41/8*e^4 + 77/8*e^3 + 23*e^2 - 17/2*e - 36, -1/2*e^5 + 3/2*e^4 + 13/2*e^3 - 23*e^2 + 9*e + 32, 1/2*e^6 - 3/2*e^5 - 19/2*e^4 + 51/2*e^3 + 61/2*e^2 - 99/2*e - 11, -3/16*e^6 + 3/4*e^5 + 35/16*e^4 - 23/2*e^3 + 201/16*e^2 + 81/8*e - 32, 1/4*e^6 - 5/4*e^5 - 9/4*e^4 + 20*e^3 - 47/2*e^2 - 55/2*e + 27, -1/8*e^6 + 1/4*e^5 + 15/8*e^4 - 13/4*e^3 + 11/8*e^2 - 9/4*e - 4, -3/8*e^6 + 5/4*e^5 + 47/8*e^4 - 20*e^3 - 13/8*e^2 + 91/4*e - 20, 3/4*e^6 - 5/2*e^5 - 53/4*e^4 + 41*e^3 + 125/4*e^2 - 60*e - 37, 1/16*e^6 - 3/8*e^5 - 11/16*e^4 + 39/8*e^3 - 67/16*e^2 + 65/8*e + 12, -1/8*e^6 - 1/2*e^5 + 39/8*e^4 + 37/4*e^3 - 363/8*e^2 - 79/4*e + 52, 5/16*e^6 - 3/4*e^5 - 109/16*e^4 + 13*e^3 + 497/16*e^2 - 255/8*e - 16, 1/4*e^6 - 3/4*e^5 - 9/2*e^4 + 51/4*e^3 + 51/4*e^2 - 63/2*e - 20, -3/8*e^6 + 5/4*e^5 + 55/8*e^4 - 21*e^3 - 149/8*e^2 + 147/4*e + 12, 1/2*e^6 - 5/4*e^5 - 41/4*e^4 + 77/4*e^3 + 46*e^2 - 19*e - 68, e^6 - 3*e^5 - 77/4*e^4 + 203/4*e^3 + 251/4*e^2 - 94*e - 41, 1/2*e^6 - 3/2*e^5 - 37/4*e^4 + 97/4*e^3 + 97/4*e^2 - 32*e - 17, -1/4*e^6 + 1/2*e^5 + 11/2*e^4 - 37/4*e^3 - 29*e^2 + 31*e + 28, -1/2*e^6 + 2*e^5 + 29/4*e^4 - 131/4*e^3 + 31/4*e^2 + 93/2*e - 38, 1/4*e^6 - 3/4*e^5 - 37/8*e^4 + 91/8*e^3 + 119/8*e^2 - 27/4*e - 32, -1/2*e^6 + 3/2*e^5 + 19/2*e^4 - 26*e^3 - 31*e^2 + 60*e + 22, 1/4*e^6 - 1/2*e^5 - 13/2*e^4 + 33/4*e^3 + 46*e^2 - 17*e - 64, -1/2*e^6 + e^5 + 45/4*e^4 - 63/4*e^3 - 241/4*e^2 + 41/2*e + 76, -2*e^3 + 2*e^2 + 22*e - 22, -3/4*e^6 + 9/4*e^5 + 55/4*e^4 - 37*e^3 - 41*e^2 + 119/2*e + 53, -11/16*e^6 + 11/8*e^5 + 241/16*e^4 - 189/8*e^3 - 1203/16*e^2 + 449/8*e + 76, 1/2*e^6 - 3/4*e^5 - 12*e^4 + 25/2*e^3 + 283/4*e^2 - 33*e - 85, 1/16*e^6 - 1/4*e^5 - 9/16*e^4 + 4*e^3 - 107/16*e^2 - 59/8*e + 32, -9/16*e^6 + 9/4*e^5 + 147/16*e^4 - 293/8*e^3 - 191/16*e^2 + 381/8*e + 4, -5/8*e^6 + 9/8*e^5 + 59/4*e^4 - 151/8*e^3 - 673/8*e^2 + 181/4*e + 88, 5/16*e^6 - 11/8*e^5 - 83/16*e^4 + 193/8*e^3 + 85/16*e^2 - 431/8*e + 8, -3/8*e^6 + 9/8*e^5 + 25/4*e^4 - 141/8*e^3 - 61/8*e^2 + 61/4*e - 4, -1/4*e^6 + 3/4*e^5 + 9/2*e^4 - 49/4*e^3 - 37/4*e^2 + 25*e - 19, 1/16*e^6 - 1/4*e^5 - 25/16*e^4 + 3*e^3 + 165/16*e^2 + 85/8*e - 20, 13/16*e^6 - 3*e^5 - 219/16*e^4 + 395/8*e^3 + 363/16*e^2 - 585/8*e - 8, 1/4*e^6 - 3/2*e^5 - 2*e^4 + 99/4*e^3 - 61/2*e^2 - 34*e + 50, -1/2*e^6 + 7/4*e^5 + 9*e^4 - 30*e^3 - 97/4*e^2 + 123/2*e + 28, -3/8*e^6 + 1/2*e^5 + 75/8*e^4 - 8*e^3 - 471/8*e^2 + 41/4*e + 68, 1/16*e^6 - 1/4*e^5 - 1/16*e^4 + 9/2*e^3 - 227/16*e^2 - 91/8*e + 8, 1/8*e^6 - 9/2*e^4 - 1/8*e^3 + 85/2*e^2 - 11/2*e - 48, -3/16*e^6 + 1/8*e^5 + 85/16*e^4 - 23/8*e^3 - 635/16*e^2 + 193/8*e + 44, -1/2*e^6 + 7/4*e^5 + 9*e^4 - 28*e^3 - 89/4*e^2 + 67/2*e + 4, -9/16*e^6 + 9/4*e^5 + 149/16*e^4 - 149/4*e^3 - 225/16*e^2 + 471/8*e + 24, -3/4*e^6 + 2*e^5 + 29/2*e^4 - 133/4*e^3 - 50*e^2 + 111/2*e + 47, -1/8*e^6 + 1/2*e^5 + 7/4*e^4 - 73/8*e^3 + 11/4*e^2 + 29*e - 12, 9/8*e^6 - 15/4*e^5 - 21*e^4 + 503/8*e^3 + 257/4*e^2 - 114*e - 52, -1/4*e^6 + 7*e^4 + 5/4*e^3 - 52*e^2 - 8*e + 70, 1/8*e^6 - 3/4*e^5 - 17/8*e^4 + 27/2*e^3 + 19/8*e^2 - 125/4*e, -1/8*e^6 + 1/2*e^5 + 13/8*e^4 - 19/2*e^3 + 15/8*e^2 + 123/4*e + 8, e^6 - 3*e^5 - 75/4*e^4 + 201/4*e^3 + 233/4*e^2 - 94*e - 33, -3/4*e^6 + 2*e^5 + 63/4*e^4 - 33*e^3 - 291/4*e^2 + 109/2*e + 90, 1/2*e^6 - 3/2*e^5 - 39/4*e^4 + 103/4*e^3 + 139/4*e^2 - 57*e - 17, -7/8*e^6 + 9/4*e^5 + 137/8*e^4 - 151/4*e^3 - 479/8*e^2 + 261/4*e + 52, -1/4*e^5 + 3/4*e^4 + 15/4*e^3 - 15*e^2 + 34, 1/2*e^6 - 7/4*e^5 - 37/4*e^4 + 125/4*e^3 + 51/2*e^2 - 74*e - 14, 9/16*e^6 - 9/4*e^5 - 149/16*e^4 + 149/4*e^3 + 193/16*e^2 - 519/8*e + 20, 1/4*e^6 - 3/4*e^5 - 7/2*e^4 + 43/4*e^3 - 21/4*e^2 + 19/2*e + 22, -1/4*e^6 + 1/2*e^5 + 23/4*e^4 - 19/2*e^3 - 113/4*e^2 + 57/2*e + 20, -1/8*e^6 + 7/8*e^5 - 3/4*e^4 - 107/8*e^3 + 369/8*e^2 + 23/4*e - 80, -1/4*e^5 - 1/2*e^4 + 7*e^3 + 41/4*e^2 - 91/2*e - 22, 1/4*e^6 - 5/4*e^5 - 2*e^4 + 77/4*e^3 - 137/4*e^2 - 15/2*e + 80, 5/8*e^6 - 11/4*e^5 - 77/8*e^4 + 93/2*e^3 + 47/8*e^2 - 345/4*e - 4, -1/8*e^6 + 1/2*e^5 + 13/8*e^4 - 15/2*e^3 + 31/8*e^2 + 3/4*e - 36, 5/4*e^6 - 35/8*e^5 - 177/8*e^4 + 585/8*e^3 + 205/4*e^2 - 124*e - 32, -1/2*e^6 + e^5 + 21/2*e^4 - 15*e^3 - 97/2*e^2 + 12*e + 54, 7/16*e^6 - e^5 - 143/16*e^4 + 31/2*e^3 + 543/16*e^2 - 97/8*e - 44, 1/2*e^6 - 2*e^5 - 8*e^4 + 65/2*e^3 + 7*e^2 - 41*e, -1/2*e^4 + 5/2*e^3 + 17/2*e^2 - 33*e - 22, 1/8*e^6 - 1/8*e^5 - 2*e^4 + 13/8*e^3 - 13/8*e^2 + 9/4*e + 48, -1/8*e^6 + 17/8*e^4 + e^3 - 9/8*e^2 - 53/4*e - 16, -3/8*e^6 + 1/2*e^5 + 9*e^4 - 65/8*e^3 - 101/2*e^2 + 25/2*e + 44, -1/4*e^6 + 1/4*e^5 + 6*e^4 - 13/4*e^3 - 131/4*e^2 - 19/2*e + 28, -3/16*e^6 - 1/4*e^5 + 95/16*e^4 + 19/4*e^3 - 787/16*e^2 - 123/8*e + 52, 1/4*e^6 - 5/4*e^5 - 15/4*e^4 + 22*e^3 - 5/2*e^2 - 49*e + 16, 3/8*e^6 - e^5 - 55/8*e^4 + 33/2*e^3 + 143/8*e^2 - 89/4*e - 20, -7/8*e^5 + 17/8*e^4 + 121/8*e^3 - 71/2*e^2 - 55/2*e + 68, -5/16*e^6 + 2*e^5 + 53/16*e^4 - 34*e^3 + 299/16*e^2 + 491/8*e - 12, -1/4*e^6 + 5/4*e^5 + 3*e^4 - 83/4*e^3 + 39/4*e^2 + 38*e - 1, 1/16*e^6 - 1/4*e^5 - 17/16*e^4 + 9/2*e^3 + 29/16*e^2 - 83/8*e - 8, -1/2*e^6 + 9/4*e^5 + 31/4*e^4 - 149/4*e^3 - 5*e^2 + 113/2*e + 9];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

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