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

NN := ideal<ZF | {23, 23, 2*w - 3}>;

primesArray := [
[2, 2, -w],
[2, 2, w - 1],
[11, 11, w^2 - w - 1],
[17, 17, -w^2 - w + 3],
[19, 19, w^2 - w + 1],
[23, 23, 2*w - 3],
[27, 3, 3],
[29, 29, 2*w + 1],
[31, 31, 2*w^2 - 2*w - 9],
[37, 37, 2*w^2 - 2*w - 5],
[41, 41, 2*w^2 - 9],
[43, 43, w^2 + w - 5],
[43, 43, -3*w^2 + w + 15],
[43, 43, -2*w^2 + 2*w + 11],
[53, 53, w^2 - w - 7],
[61, 61, 4*w^2 - 2*w - 15],
[67, 67, -5*w^2 + 3*w + 23],
[73, 73, 2*w^2 - 3],
[73, 73, -3*w^2 - w + 7],
[73, 73, -6*w^2 + 4*w + 25],
[79, 79, w^2 - 5*w + 1],
[79, 79, 3*w^2 - 3*w - 11],
[83, 83, 2*w^2 - 2*w - 3],
[109, 109, w^2 - 3*w - 3],
[113, 113, 3*w^2 - 3*w - 13],
[121, 11, 3*w^2 - w - 9],
[125, 5, -5],
[131, 131, -6*w^2 + 2*w + 25],
[137, 137, -w^2 + w - 3],
[149, 149, 2*w - 7],
[151, 151, w^2 - 3*w - 5],
[157, 157, 3*w^2 - 3*w - 7],
[163, 163, 2*w^2 - 13],
[167, 167, -4*w + 5],
[173, 173, -7*w^2 + 3*w + 27],
[179, 179, 4*w^2 - 4*w - 11],
[181, 181, 4*w^2 - 15],
[181, 181, w^2 - 3*w + 5],
[181, 181, w^2 + 3*w - 5],
[193, 193, 2*w^2 - 4*w - 5],
[197, 197, 2*w^2 + 2*w - 7],
[211, 211, 4*w^2 - 2*w - 13],
[211, 211, -7*w^2 + 5*w + 29],
[211, 211, w^2 - w - 9],
[223, 223, 5*w^2 - w - 23],
[227, 227, -3*w^2 + 3*w + 17],
[227, 227, 3*w^2 - w - 7],
[227, 227, 3*w^2 + w - 11],
[229, 229, 5*w^2 - w - 21],
[233, 233, 3*w^2 - 5*w - 7],
[239, 239, -w^2 + 7*w - 5],
[257, 257, -4*w - 3],
[257, 257, w^2 - 5*w - 1],
[257, 257, -3*w^2 + w + 17],
[263, 263, 3*w^2 - 3*w - 5],
[271, 271, 4*w^2 - 4*w - 17],
[271, 271, 3*w^2 - w - 3],
[271, 271, w^2 + 3*w - 7],
[283, 283, 6*w - 1],
[289, 17, w^2 + 3*w - 9],
[293, 293, 3*w^2 - w - 5],
[307, 307, 2*w^2 + 2*w - 9],
[307, 307, -2*w^2 + 2*w - 3],
[307, 307, 4*w^2 - 17],
[311, 311, w^2 + w - 11],
[311, 311, 2*w^2 - 4*w - 7],
[311, 311, -2*w^2 - 2*w + 13],
[313, 313, -2*w - 7],
[313, 313, 4*w^2 - 19],
[313, 313, -5*w^2 - w + 13],
[331, 331, 2*w^2 - 4*w - 11],
[331, 331, 6*w^2 - 4*w - 21],
[331, 331, -4*w^2 + 2*w + 21],
[343, 7, -7],
[347, 347, -2*w^2 + 8*w - 1],
[349, 349, w^2 - 7*w + 1],
[353, 353, -6*w^2 + 2*w + 21],
[361, 19, -5*w^2 + 3*w + 25],
[367, 367, 3*w^2 + w - 15],
[373, 373, 2*w^2 - 4*w - 9],
[383, 383, -9*w^2 + 3*w + 37],
[401, 401, 4*w^2 - 2*w - 11],
[409, 409, 4*w^2 - 4*w - 9],
[431, 431, -5*w^2 + 5*w + 19],
[439, 439, 2*w - 9],
[443, 443, w^2 - 5*w - 3],
[443, 443, 5*w^2 - 5*w - 23],
[443, 443, 3*w^2 - 5*w - 17],
[449, 449, -4*w^2 + 4*w + 1],
[461, 461, 5*w^2 - 7*w - 9],
[467, 467, 5*w^2 + w - 17],
[479, 479, 2*w^2 + 4*w - 7],
[487, 487, 5*w^2 - 5*w - 13],
[499, 499, -w^2 + 3*w - 7],
[509, 509, -7*w^2 + w + 27],
[521, 521, w^2 - w - 11],
[523, 523, 4*w^2 - 6*w - 11],
[529, 23, 4*w^2 + 2*w - 13],
[547, 547, -11*w^2 + 5*w + 43],
[563, 563, 4*w^2 - 7],
[569, 569, -6*w - 1],
[569, 569, 4*w^2 - 5],
[569, 569, w^2 - 5*w - 9],
[571, 571, 5*w^2 - 3*w - 15],
[577, 577, w^2 + 5*w - 7],
[599, 599, 4*w^2 - 4*w - 7],
[601, 601, -12*w^2 + 8*w + 51],
[613, 613, 4*w^2 + 1],
[643, 643, 6*w^2 - 2*w - 31],
[647, 647, 2*w^2 - 2*w - 15],
[647, 647, 4*w^2 - 4*w - 23],
[647, 647, w^2 - 5*w - 7],
[661, 661, -8*w^2 + 6*w + 39],
[673, 673, 3*w^2 - 5*w - 15],
[683, 683, -w^2 + 9*w - 7],
[701, 701, 4*w^2 - 4*w - 5],
[709, 709, 4*w^2 - 2*w - 7],
[719, 719, 3*w^2 - 5*w - 13],
[727, 727, w^2 - 7*w - 1],
[733, 733, 2*w^2 + 4*w - 9],
[733, 733, 3*w^2 + 3*w - 11],
[733, 733, w^2 + w - 13],
[743, 743, 2*w^2 - 6*w - 5],
[751, 751, 6*w^2 - 2*w - 19],
[761, 761, -12*w^2 + 6*w + 47],
[761, 761, 3*w^2 - 5*w - 21],
[761, 761, 2*w^2 + 6*w - 7],
[769, 769, -6*w^2 + 6*w + 23],
[773, 773, -2*w^2 + 2*w - 5],
[773, 773, -13*w^2 + 5*w + 57],
[773, 773, -11*w^2 + 7*w + 51],
[787, 787, 5*w^2 - w - 13],
[797, 797, -2*w^2 + 8*w - 11],
[811, 811, -5*w^2 + 5*w + 1],
[821, 821, -7*w^2 + 3*w + 35],
[821, 821, -4*w^2 + 2*w + 23],
[821, 821, 5*w^2 + w - 19],
[823, 823, w^2 + 5*w - 13],
[823, 823, -12*w^2 + 4*w + 49],
[823, 823, -7*w^2 + w + 29],
[829, 829, -w^2 + w - 7],
[839, 839, 4*w^2 + 2*w - 15],
[841, 29, 4*w^2 - 6*w - 13],
[853, 853, 7*w^2 - 7*w - 25],
[863, 863, 7*w^2 - 5*w - 35],
[877, 877, -6*w - 5],
[877, 877, w^2 + 5*w - 11],
[877, 877, 5*w^2 - 3*w - 13],
[881, 881, w^2 - 3*w - 13],
[887, 887, -9*w^2 + 5*w + 33],
[907, 907, w^2 - 9*w + 1],
[911, 911, -6*w^2 + 6*w + 25],
[919, 919, 5*w^2 - 3*w - 27],
[929, 929, 2*w - 11],
[937, 937, -7*w^2 + w + 31],
[941, 941, -6*w^2 + 4*w + 31],
[941, 941, -8*w^2 + 2*w + 27],
[941, 941, 3*w^2 + 5*w - 9],
[947, 947, -10*w + 7],
[947, 947, 2*w^2 + 8*w - 7],
[947, 947, 2*w^2 + 4*w - 11],
[961, 31, -14*w^2 + 8*w + 55],
[967, 967, -w^2 - w - 7],
[971, 971, -10*w^2 + 8*w + 39],
[977, 977, -10*w^2 + 6*w + 37],
[983, 983, 3*w^2 + 3*w - 13],
[983, 983, -4*w - 11],
[983, 983, -10*w^2 + 4*w + 37],
[997, 997, w^2 + 9*w - 7],
[997, 997, 5*w^2 - 7*w - 27],
[997, 997, 6*w^2 - 8*w - 11]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^9 - x^8 - 16*x^7 + 16*x^6 + 82*x^5 - 76*x^4 - 148*x^3 + 108*x^2 + 80*x - 32;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, -1/8*e^8 - 1/8*e^7 + 7/4*e^6 + 3/2*e^5 - 31/4*e^4 - 11/2*e^3 + 23/2*e^2 + 11/2*e - 3, 1/4*e^8 - 13/4*e^6 + e^5 + 25/2*e^4 - 15/2*e^3 - 14*e^2 + 10*e + 4, -1/4*e^8 + 13/4*e^6 - e^5 - 25/2*e^4 + 13/2*e^3 + 13*e^2 - 4*e, -1/2*e^7 - 1/2*e^6 + 6*e^5 + 4*e^4 - 20*e^3 - 6*e^2 + 16*e, 1, 1/4*e^8 + 1/2*e^7 - 11/4*e^6 - 5*e^5 + 15/2*e^4 + 27/2*e^3 - 12*e - 4, 1/4*e^8 + 1/4*e^7 - 7/2*e^6 - 3*e^5 + 29/2*e^4 + 12*e^3 - 15*e^2 - 17*e - 2, -1/4*e^8 - 1/4*e^7 + 7/2*e^6 + 2*e^5 - 31/2*e^4 - 2*e^3 + 21*e^2 - 5*e - 2, 1/4*e^8 + 1/4*e^7 - 7/2*e^6 - 2*e^5 + 33/2*e^4 + 3*e^3 - 29*e^2 - e + 10, -1/2*e^8 + 15/2*e^6 - e^5 - 35*e^4 + 7*e^3 + 52*e^2 - 6*e - 14, -1/4*e^7 + 1/4*e^6 + 4*e^5 - 3*e^4 - 39/2*e^3 + 8*e^2 + 27*e - 2, -3/4*e^7 - 5/4*e^6 + 8*e^5 + 11*e^4 - 45/2*e^3 - 22*e^2 + 13*e + 10, 1/2*e^8 + 1/2*e^7 - 7*e^6 - 5*e^5 + 31*e^4 + 13*e^3 - 44*e^2 - 8*e + 8, 1/4*e^8 - 17/4*e^6 - e^5 + 43/2*e^4 + 19/2*e^3 - 31*e^2 - 20*e + 8, 1/2*e^7 + 1/2*e^6 - 6*e^5 - 3*e^4 + 21*e^3 - 2*e^2 - 18*e + 8, -1/2*e^7 + 1/2*e^6 + 7*e^5 - 6*e^4 - 28*e^3 + 18*e^2 + 26*e - 12, 1/4*e^8 + 1/4*e^7 - 9/2*e^6 - 4*e^5 + 53/2*e^4 + 18*e^3 - 55*e^2 - 19*e + 26, 1/4*e^8 + 1/2*e^7 - 11/4*e^6 - 5*e^5 + 17/2*e^4 + 25/2*e^3 - 10*e^2 - 4*e + 10, 1/4*e^8 + 3/4*e^7 - 2*e^6 - 8*e^5 + 1/2*e^4 + 24*e^3 + 15*e^2 - 19*e - 10, 1/4*e^8 - 17/4*e^6 - e^5 + 43/2*e^4 + 17/2*e^3 - 32*e^2 - 14*e + 8, 1/2*e^8 - 15/2*e^6 + 35*e^4 + 2*e^3 - 54*e^2 - 8*e + 20, -1/2*e^8 + 1/4*e^7 + 29/4*e^6 - 4*e^5 - 31*e^4 + 33/2*e^3 + 36*e^2 - 15*e - 6, 3/4*e^8 + 1/2*e^7 - 41/4*e^6 - 5*e^5 + 85/2*e^4 + 31/2*e^3 - 54*e^2 - 20*e + 18, 1/4*e^8 - 1/2*e^7 - 15/4*e^6 + 7*e^5 + 33/2*e^4 - 55/2*e^3 - 22*e^2 + 28*e + 10, -e^5 + 9*e^3 - 2*e^2 - 14*e + 6, -1/2*e^8 - e^7 + 13/2*e^6 + 12*e^5 - 27*e^4 - 42*e^3 + 42*e^2 + 36*e - 18, -3/2*e^8 - 1/2*e^7 + 21*e^6 + 3*e^5 - 89*e^4 - 2*e^3 + 110*e^2 + 8*e - 20, 1/4*e^8 + 1/4*e^7 - 7/2*e^6 - 3*e^5 + 33/2*e^4 + 10*e^3 - 31*e^2 - 5*e + 18, -1/4*e^8 + 1/2*e^7 + 19/4*e^6 - 5*e^5 - 51/2*e^4 + 23/2*e^3 + 36*e^2 - 4*e + 2, -e^6 + 12*e^4 + e^3 - 38*e^2 - 8*e + 28, -e^8 - 3/2*e^7 + 25/2*e^6 + 16*e^5 - 48*e^4 - 51*e^3 + 62*e^2 + 54*e - 18, 1/2*e^7 - 1/2*e^6 - 7*e^5 + 8*e^4 + 30*e^3 - 32*e^2 - 40*e + 20, 1/2*e^8 + 1/2*e^7 - 7*e^6 - 6*e^5 + 31*e^4 + 22*e^3 - 46*e^2 - 22*e + 16, 1/2*e^8 + 1/2*e^7 - 8*e^6 - 6*e^5 + 41*e^4 + 21*e^3 - 68*e^2 - 18*e + 22, 1/2*e^8 + 3/4*e^7 - 25/4*e^6 - 8*e^5 + 23*e^4 + 51/2*e^3 - 22*e^2 - 25*e - 2, -1/2*e^8 - e^7 + 11/2*e^6 + 11*e^5 - 17*e^4 - 38*e^3 + 14*e^2 + 46*e + 6, -3/4*e^8 - 1/4*e^7 + 10*e^6 + e^5 - 75/2*e^4 + 3*e^3 + 29*e^2 + e + 6, -1/4*e^8 - 3/4*e^7 + 3*e^6 + 8*e^5 - 25/2*e^4 - 23*e^3 + 27*e^2 + 17*e - 22, -5/4*e^8 - 3/4*e^7 + 17*e^6 + 6*e^5 - 145/2*e^4 - 12*e^3 + 105*e^2 + 11*e - 38, -1/4*e^8 - 3/2*e^7 + 7/4*e^6 + 17*e^5 - 1/2*e^4 - 109/2*e^3 - 4*e^2 + 44*e + 6, 1/2*e^7 + 3/2*e^6 - 5*e^5 - 12*e^4 + 14*e^3 + 18*e^2 - 10*e - 12, -1/4*e^8 - 1/2*e^7 + 7/4*e^6 + 3*e^5 + 7/2*e^4 + 9/2*e^3 - 30*e^2 - 24*e + 20, e^6 - 12*e^4 + e^3 + 38*e^2 - 6*e - 28, -1/4*e^8 - 1/4*e^7 + 3/2*e^6 + 13/2*e^4 + 12*e^3 - 39*e^2 - 17*e + 22, -3/4*e^8 - 3/4*e^7 + 21/2*e^6 + 8*e^5 - 93/2*e^4 - 24*e^3 + 71*e^2 + 13*e - 30, 1/2*e^7 + 1/2*e^6 - 7*e^5 - 6*e^4 + 31*e^3 + 18*e^2 - 48*e - 8, -3/4*e^8 - 3/4*e^7 + 21/2*e^6 + 8*e^5 - 89/2*e^4 - 22*e^3 + 55*e^2 + 13*e - 14, 1/2*e^7 + 1/2*e^6 - 6*e^5 - 3*e^4 + 21*e^3 - 2*e^2 - 22*e + 12, -e^8 + 1/2*e^7 + 31/2*e^6 - 8*e^5 - 72*e^4 + 32*e^3 + 92*e^2 - 18*e - 10, -e^8 - 1/4*e^7 + 57/4*e^6 - 62*e^4 + 21/2*e^3 + 80*e^2 - 9*e - 18, 1/4*e^8 + 1/4*e^7 - 5/2*e^6 - 2*e^5 + 9/2*e^4 + 4*e^3 + 9*e^2 - 7*e - 10, 1/2*e^7 + 1/2*e^6 - 5*e^5 - 2*e^4 + 13*e^3 - 10*e^2 - 16*e + 30, 1/4*e^8 + e^7 - 9/4*e^6 - 11*e^5 + 5/2*e^4 + 67/2*e^3 + 13*e^2 - 24*e - 12, -1/2*e^8 - 1/2*e^7 + 8*e^6 + 7*e^5 - 41*e^4 - 30*e^3 + 74*e^2 + 36*e - 44, -1/2*e^8 - 3/4*e^7 + 29/4*e^6 + 10*e^5 - 34*e^4 - 83/2*e^3 + 58*e^2 + 53*e - 22, 1/2*e^7 + 1/2*e^6 - 5*e^5 - 4*e^4 + 11*e^3 + 2*e^2 + 16, -1/2*e^8 + 15/2*e^6 - e^5 - 35*e^4 + 5*e^3 + 50*e^2 + 6*e - 12, 1/2*e^8 - 15/2*e^6 + e^5 + 33*e^4 - 7*e^3 - 34*e^2 + 6*e - 12, 1/4*e^8 - 1/4*e^7 - 5*e^6 + 3*e^5 + 61/2*e^4 - 13*e^3 - 59*e^2 + 21*e + 10, 1/2*e^8 + 3/2*e^7 - 7*e^6 - 18*e^5 + 35*e^4 + 63*e^3 - 72*e^2 - 54*e + 34, -1/2*e^8 - 1/2*e^7 + 7*e^6 + 5*e^5 - 31*e^4 - 13*e^3 + 44*e^2 + 8*e - 8, 3/4*e^8 - e^7 - 47/4*e^6 + 13*e^5 + 113/2*e^4 - 89/2*e^3 - 84*e^2 + 26*e + 20, -1/2*e^7 + 1/2*e^6 + 7*e^5 - 6*e^4 - 28*e^3 + 12*e^2 + 28*e + 12, 5/4*e^8 + 5/4*e^7 - 35/2*e^6 - 14*e^5 + 155/2*e^4 + 50*e^3 - 117*e^2 - 63*e + 42, -1/4*e^7 - 3/4*e^6 + 2*e^5 + 6*e^4 - 7/2*e^3 - 10*e^2 + 7*e + 14, -5/4*e^8 - 1/2*e^7 + 71/4*e^6 + 5*e^5 - 151/2*e^4 - 31/2*e^3 + 88*e^2 + 24*e, -1/2*e^8 + 1/2*e^7 + 8*e^6 - 6*e^5 - 37*e^4 + 20*e^3 + 46*e^2 - 22*e - 6, 1/4*e^8 + 1/2*e^7 - 7/4*e^6 - 5*e^5 - 5/2*e^4 + 33/2*e^3 + 24*e^2 - 28*e - 18, 1/4*e^8 - 1/2*e^7 - 15/4*e^6 + 9*e^5 + 33/2*e^4 - 95/2*e^3 - 16*e^2 + 68*e + 2, -1/2*e^8 + 3/2*e^7 + 8*e^6 - 21*e^5 - 37*e^4 + 84*e^3 + 46*e^2 - 76*e - 8, -1/2*e^8 - 1/4*e^7 + 23/4*e^6 + 2*e^5 - 15*e^4 - 5/2*e^3 - 10*e^2 - 5*e + 22, 1/2*e^8 + 1/2*e^7 - 6*e^6 - 3*e^5 + 21*e^4 - 4*e^3 - 18*e^2 + 20*e - 12, 1/2*e^7 + 1/2*e^6 - 7*e^5 - 4*e^4 + 29*e^3 + 2*e^2 - 32*e + 8, 1/2*e^7 + 1/2*e^6 - 5*e^5 - 2*e^4 + 11*e^3 - 8*e^2 + 4*e + 8, 1/2*e^8 - 1/2*e^7 - 6*e^6 + 9*e^5 + 19*e^4 - 43*e^3 - 12*e^2 + 48*e + 2, -1/4*e^8 - 5/4*e^7 + 7/2*e^6 + 17*e^5 - 33/2*e^4 - 67*e^3 + 33*e^2 + 67*e - 22, -5/4*e^8 - 1/2*e^7 + 71/4*e^6 + 5*e^5 - 159/2*e^4 - 41/2*e^3 + 119*e^2 + 42*e - 44, 3/2*e^8 + e^7 - 41/2*e^6 - 10*e^5 + 83*e^4 + 31*e^3 - 90*e^2 - 40*e + 8, 3/4*e^8 + 1/2*e^7 - 45/4*e^6 - 5*e^5 + 105/2*e^4 + 23/2*e^3 - 75*e^2 - 6*e + 16, -1/2*e^8 + 1/2*e^7 + 6*e^6 - 11*e^5 - 19*e^4 + 59*e^3 + 8*e^2 - 66*e - 4, 1/4*e^8 + 1/4*e^7 - 11/2*e^6 - 5*e^5 + 69/2*e^4 + 24*e^3 - 63*e^2 - 17*e + 10, -1/4*e^8 + 1/2*e^7 + 15/4*e^6 - 9*e^5 - 35/2*e^4 + 83/2*e^3 + 27*e^2 - 34*e - 28, 1/4*e^8 - 13/4*e^6 + e^5 + 21/2*e^4 - 11/2*e^3 + 2*e^2 - 2*e - 16, -1/2*e^8 - 3/2*e^7 + 6*e^6 + 18*e^5 - 27*e^4 - 66*e^3 + 62*e^2 + 74*e - 32, -3/4*e^8 + 43/4*e^6 - 3*e^5 - 93/2*e^4 + 41/2*e^3 + 56*e^2 - 14*e - 4, -1/2*e^8 - e^7 + 15/2*e^6 + 14*e^5 - 37*e^4 - 59*e^3 + 68*e^2 + 72*e - 32, -e^8 + 16*e^6 + 2*e^5 - 78*e^4 - 19*e^3 + 114*e^2 + 36*e - 20, 5/4*e^8 + 1/4*e^7 - 33/2*e^6 + 2*e^5 + 129/2*e^4 - 29*e^3 - 73*e^2 + 51*e + 22, -e^8 + 14*e^6 - 4*e^5 - 60*e^4 + 31*e^3 + 78*e^2 - 48*e - 22, -1/2*e^8 + 1/4*e^7 + 37/4*e^6 - 2*e^5 - 55*e^4 + 1/2*e^3 + 112*e^2 + 13*e - 46, 1/2*e^8 + e^7 - 13/2*e^6 - 12*e^5 + 27*e^4 + 46*e^3 - 38*e^2 - 52*e, 1/2*e^7 + 1/2*e^6 - 9*e^5 - 6*e^4 + 45*e^3 + 22*e^2 - 48*e - 28, 3/2*e^7 + 3/2*e^6 - 17*e^5 - 10*e^4 + 55*e^3 - 52*e + 24, -1/4*e^8 - 3/2*e^7 + 11/4*e^6 + 19*e^5 - 21/2*e^4 - 133/2*e^3 + 23*e^2 + 50*e - 24, -3/2*e^8 - e^7 + 43/2*e^6 + 10*e^5 - 98*e^4 - 29*e^3 + 148*e^2 + 24*e - 40, 1/2*e^8 + 5/4*e^7 - 27/4*e^6 - 12*e^5 + 33*e^4 + 57/2*e^3 - 70*e^2 - 19*e + 42, e^6 + 4*e^5 - 8*e^4 - 33*e^3 + 10*e^2 + 44*e + 14, -2*e^5 - 4*e^4 + 18*e^3 + 32*e^2 - 28*e - 36, -e^8 - 2*e^7 + 13*e^6 + 21*e^5 - 54*e^4 - 59*e^3 + 78*e^2 + 32*e - 24, -1/4*e^8 + 2*e^7 + 17/4*e^6 - 27*e^5 - 39/2*e^4 + 205/2*e^3 + 23*e^2 - 88*e - 8, 5/4*e^8 + 9/4*e^7 - 29/2*e^6 - 23*e^5 + 93/2*e^4 + 60*e^3 - 31*e^2 - 21*e - 18, -3/4*e^8 + 1/2*e^7 + 45/4*e^6 - 7*e^5 - 101/2*e^4 + 49/2*e^3 + 66*e^2 - 12*e - 2, -1/2*e^8 - 1/2*e^7 + 6*e^6 + 3*e^5 - 25*e^4 + 46*e^2 - 20, -1/4*e^8 - 3/4*e^7 + 3*e^6 + 7*e^5 - 25/2*e^4 - 16*e^3 + 17*e^2 + 13*e + 2, 1/2*e^8 + 3/4*e^7 - 37/4*e^6 - 12*e^5 + 52*e^4 + 107/2*e^3 - 88*e^2 - 53*e + 6, 5/4*e^8 + 9/4*e^7 - 31/2*e^6 - 21*e^5 + 121/2*e^4 + 43*e^3 - 81*e^2 - 3*e + 22, -1/2*e^7 - 3/2*e^6 + 5*e^5 + 14*e^4 - 8*e^3 - 28*e^2 - 28*e + 14, 3/2*e^8 + 5/2*e^7 - 20*e^6 - 26*e^5 + 87*e^4 + 72*e^3 - 134*e^2 - 50*e + 28, -1/2*e^8 - 1/2*e^7 + 7*e^6 + 6*e^5 - 27*e^4 - 22*e^3 + 18*e^2 + 30*e, -e^8 - 3/2*e^7 + 25/2*e^6 + 15*e^5 - 48*e^4 - 38*e^3 + 60*e^2 + 12*e - 20, -1/2*e^7 + 1/2*e^6 + 7*e^5 - 10*e^4 - 28*e^3 + 52*e^2 + 36*e - 60, -1/2*e^8 + e^7 + 17/2*e^6 - 13*e^5 - 41*e^4 + 49*e^3 + 44*e^2 - 46*e + 18, -e^7 - e^6 + 14*e^5 + 8*e^4 - 58*e^3 - 12*e^2 + 64*e + 10, -e^6 + 2*e^5 + 14*e^4 - 21*e^3 - 54*e^2 + 50*e + 36, 1/2*e^8 - 11/2*e^6 + 4*e^5 + 14*e^4 - 33*e^3 + 8*e^2 + 56*e - 24, -5/4*e^8 + 77/4*e^6 + e^5 - 183/2*e^4 - 31/2*e^3 + 137*e^2 + 44*e - 44, -3/2*e^8 + e^7 + 45/2*e^6 - 16*e^5 - 101*e^4 + 71*e^3 + 132*e^2 - 72*e - 36, 1/2*e^8 + 3/2*e^7 - 6*e^6 - 16*e^5 + 27*e^4 + 48*e^3 - 66*e^2 - 30*e + 48, 3/4*e^8 + 3/2*e^7 - 33/4*e^6 - 13*e^5 + 51/2*e^4 + 47/2*e^3 - 26*e^2 - 4*e + 22, 7/4*e^8 + 3/2*e^7 - 97/4*e^6 - 13*e^5 + 215/2*e^4 + 47/2*e^3 - 161*e^2 + 2*e + 32, 1/4*e^8 - 5/4*e^7 - 3*e^6 + 15*e^5 + 9/2*e^4 - 44*e^3 + 35*e^2 + 15*e - 58, 1/4*e^7 - 5/4*e^6 - 6*e^5 + 10*e^4 + 71/2*e^3 - 4*e^2 - 43*e - 38, 1/2*e^8 - 15/2*e^6 + 2*e^5 + 33*e^4 - 22*e^3 - 40*e^2 + 50*e + 12, 5/4*e^8 + 1/2*e^7 - 67/4*e^6 - 3*e^5 + 133/2*e^4 + 5/2*e^3 - 75*e^2 - 2*e + 8, -7/4*e^8 - 3/2*e^7 + 89/4*e^6 + 13*e^5 - 167/2*e^4 - 51/2*e^3 + 81*e^2 + 14*e + 32, 1/4*e^8 + 1/2*e^7 - 15/4*e^6 - 9*e^5 + 35/2*e^4 + 97/2*e^3 - 22*e^2 - 60*e + 6, -e^8 + 1/2*e^7 + 27/2*e^6 - 12*e^5 - 53*e^4 + 69*e^3 + 54*e^2 - 98*e - 12, 1/4*e^8 + 1/2*e^7 - 15/4*e^6 - 7*e^5 + 27/2*e^4 + 65/2*e^3 + 4*e^2 - 64*e - 22, 5/4*e^8 + 1/4*e^7 - 33/2*e^6 + 2*e^5 + 133/2*e^4 - 27*e^3 - 85*e^2 + 35*e + 22, -1/2*e^7 - 1/2*e^6 + 6*e^5 + 7*e^4 - 17*e^3 - 30*e^2 + 6*e + 16, 3/2*e^8 - 1/2*e^7 - 23*e^6 + 8*e^5 + 111*e^4 - 30*e^3 - 178*e^2 + 6*e + 68, -2*e^8 - e^7 + 27*e^6 + 8*e^5 - 112*e^4 - 18*e^3 + 148*e^2 + 20*e - 38, 3/4*e^8 + 3*e^7 - 31/4*e^6 - 35*e^5 + 41/2*e^4 + 239/2*e^3 - 8*e^2 - 122*e + 4, -1/4*e^8 + 21/4*e^6 + e^5 - 69/2*e^4 - 11/2*e^3 + 75*e^2 - 44, e^8 + e^7 - 12*e^6 - 11*e^5 + 40*e^4 + 39*e^3 - 30*e^2 - 46*e + 6, -1/4*e^8 + 1/4*e^7 + 4*e^6 - 33/2*e^4 - 15*e^3 + 11*e^2 + 29*e - 6, e^8 + 3/2*e^7 - 25/2*e^6 - 13*e^5 + 48*e^4 + 24*e^3 - 54*e^2 - 10*e + 4, 3/4*e^8 - 35/4*e^6 + 7*e^5 + 57/2*e^4 - 105/2*e^3 - 20*e^2 + 62*e - 8, e^8 + e^7 - 13*e^6 - 10*e^5 + 48*e^4 + 25*e^3 - 42*e^2 - 10*e + 4, -1/2*e^7 - 1/2*e^6 + 6*e^5 + 7*e^4 - 17*e^3 - 22*e^2 + 6*e, -1/2*e^7 + 5/2*e^6 + 9*e^5 - 26*e^4 - 40*e^3 + 68*e^2 + 28*e - 48, 5/4*e^8 + 3/4*e^7 - 16*e^6 - 3*e^5 + 121/2*e^4 - 16*e^3 - 69*e^2 + 35*e + 46, -2*e^7 - 4*e^6 + 20*e^5 + 36*e^4 - 50*e^3 - 72*e^2 + 20*e + 18, -3/2*e^8 - 2*e^7 + 43/2*e^6 + 25*e^5 - 99*e^4 - 94*e^3 + 164*e^2 + 92*e - 64, -3/4*e^8 - e^7 + 35/4*e^6 + 11*e^5 - 55/2*e^4 - 73/2*e^3 + 15*e^2 + 40*e, e^8 + 3/2*e^7 - 23/2*e^6 - 18*e^5 + 33*e^4 + 69*e^3 - 6*e^2 - 86*e - 12, e^8 + e^7 - 14*e^6 - 12*e^5 + 62*e^4 + 44*e^3 - 92*e^2 - 40*e + 34, 7/4*e^8 - 1/4*e^7 - 49/2*e^6 + 9*e^5 + 203/2*e^4 - 56*e^3 - 117*e^2 + 61*e + 34, 5/4*e^8 + e^7 - 69/4*e^6 - 7*e^5 + 149/2*e^4 + 9/2*e^3 - 106*e^2 - 6*e + 32, -e^8 - 1/2*e^7 + 21/2*e^6 - e^5 - 26*e^4 + 31*e^3 + 6*e^2 - 48*e - 20, -1/2*e^8 - 1/2*e^7 + 5*e^6 + 3*e^5 - 13*e^4 - e^3 + 16*e^2 + 2*e - 28, 1/2*e^8 + 5/4*e^7 - 23/4*e^6 - 14*e^5 + 20*e^4 + 93/2*e^3 - 22*e^2 - 59*e + 10, -e^8 + 13*e^6 - 4*e^5 - 50*e^4 + 26*e^3 + 52*e^2 - 20*e - 10, 5/4*e^8 + 1/2*e^7 - 59/4*e^6 - e^5 + 89/2*e^4 - 31/2*e^3 - 3*e^2 + 26*e - 48, -3/4*e^8 - 5/2*e^7 + 33/4*e^6 + 27*e^5 - 57/2*e^4 - 151/2*e^3 + 42*e^2 + 28*e - 22, 3/2*e^8 + e^7 - 35/2*e^6 - 2*e^5 + 60*e^4 - 33*e^3 - 72*e^2 + 52*e + 44, 9/4*e^8 - 1/4*e^7 - 31*e^6 + 8*e^5 + 249/2*e^4 - 42*e^3 - 129*e^2 + 37*e - 2, -e^8 - e^7 + 16*e^6 + 12*e^5 - 84*e^4 - 44*e^3 + 156*e^2 + 48*e - 68, e^8 - 15*e^6 + 2*e^5 + 70*e^4 - 18*e^3 - 100*e^2 + 32*e + 16, 1/2*e^8 + 5/2*e^7 - 5*e^6 - 29*e^5 + 13*e^4 + 95*e^3 - 80*e - 24, -1/2*e^8 - e^7 + 13/2*e^6 + 12*e^5 - 24*e^4 - 43*e^3 + 24*e^2 + 44*e - 8, -3/2*e^8 - e^7 + 39/2*e^6 + 7*e^5 - 75*e^4 + e^3 + 78*e^2 - 38*e - 12, 3/4*e^8 + 7/4*e^7 - 19/2*e^6 - 20*e^5 + 77/2*e^4 + 64*e^3 - 59*e^2 - 45*e + 38, 5/4*e^8 + 2*e^7 - 61/4*e^6 - 21*e^5 + 115/2*e^4 + 119/2*e^3 - 81*e^2 - 28*e + 52, 1/2*e^8 + 13/4*e^7 - 23/4*e^6 - 40*e^5 + 24*e^4 + 281/2*e^3 - 44*e^2 - 131*e + 10, -e^8 - 5/2*e^7 + 23/2*e^6 + 27*e^5 - 38*e^4 - 80*e^3 + 32*e^2 + 64*e - 12, -e^8 + 1/4*e^7 + 63/4*e^6 - 6*e^5 - 78*e^4 + 63/2*e^3 + 120*e^2 - 19*e - 22, -3/4*e^8 - 2*e^7 + 27/4*e^6 + 21*e^5 - 21/2*e^4 - 121/2*e^3 - 15*e^2 + 48*e + 16, 3/2*e^8 + 2*e^7 - 43/2*e^6 - 22*e^5 + 99*e^4 + 67*e^3 - 152*e^2 - 56*e + 38, -5/4*e^8 - 15/4*e^7 + 17*e^6 + 47*e^5 - 153/2*e^4 - 176*e^3 + 121*e^2 + 177*e - 22];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {23, 23, 2*w - 3}>] := -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;