/*
  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, 0, -6, -2, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {11, 11, w^3 - 2*w^2 - 5*w - 1}>;

primesArray := [
[2, 2, w],
[5, 5, -w^3 + 3*w^2 + 2*w - 1],
[5, 5, w - 1],
[11, 11, w^3 - 2*w^2 - 5*w - 1],
[13, 13, -w^3 + 3*w^2 + 3*w - 1],
[17, 17, 2*w^3 - 5*w^2 - 9*w + 5],
[23, 23, -w^3 + 3*w^2 + 4*w - 5],
[25, 5, -w^2 + 3*w + 1],
[29, 29, -2*w^3 + 6*w^2 + 5*w - 1],
[31, 31, -w^2 + 2*w + 1],
[43, 43, -w^3 + 3*w^2 + 2*w - 3],
[43, 43, -w^3 + 2*w^2 + 6*w - 3],
[59, 59, 2*w^3 - 6*w^2 - 7*w + 7],
[73, 73, 3*w^3 - 6*w^2 - 16*w - 5],
[79, 79, -5*w^3 + 14*w^2 + 20*w - 17],
[81, 3, -3],
[83, 83, -w - 3],
[83, 83, w^2 - 3*w - 7],
[89, 89, w^2 - 2*w - 7],
[101, 101, -2*w^3 + 5*w^2 + 8*w - 3],
[101, 101, 5*w^3 - 14*w^2 - 19*w + 17],
[103, 103, -w^3 + w^2 + 8*w + 3],
[103, 103, 2*w^3 - 5*w^2 - 7*w - 1],
[109, 109, 3*w^3 - 7*w^2 - 14*w + 1],
[109, 109, -w^2 + 4*w + 1],
[127, 127, -3*w^3 + 8*w^2 + 11*w - 7],
[127, 127, -w^3 + 2*w^2 + 7*w + 1],
[137, 137, -2*w^3 + 5*w^2 + 9*w - 1],
[139, 139, -w^3 + 3*w^2 + 2*w - 5],
[149, 149, w^3 - 3*w^2 - w - 1],
[149, 149, -2*w^3 + 4*w^2 + 10*w + 1],
[163, 163, w^3 - 4*w^2 - w + 11],
[163, 163, 2*w^3 - 4*w^2 - 10*w + 1],
[173, 173, -4*w^3 + 11*w^2 + 15*w - 13],
[179, 179, 2*w^2 - 6*w - 3],
[179, 179, -w^3 + 2*w^2 + 7*w - 1],
[181, 181, w^3 - 10*w - 9],
[181, 181, -3*w^3 + 8*w^2 + 14*w - 7],
[199, 199, w^3 - 8*w - 9],
[211, 211, 2*w - 3],
[223, 223, -w^3 + 4*w^2 + w - 5],
[227, 227, w^3 - 2*w^2 - 5*w - 5],
[227, 227, -3*w^3 + 7*w^2 + 14*w - 5],
[227, 227, -4*w^3 + 10*w^2 + 19*w - 7],
[227, 227, w^2 - 4*w - 5],
[239, 239, -2*w^3 + 5*w^2 + 7*w - 3],
[251, 251, -w^3 + 4*w^2 + 2*w - 11],
[251, 251, 2*w^3 - 6*w^2 - 6*w + 1],
[257, 257, -3*w^3 + 6*w^2 + 16*w + 3],
[263, 263, -w^3 + 3*w^2 + 3*w - 7],
[263, 263, w^3 - 4*w^2 + 7],
[269, 269, -2*w^3 + 6*w^2 + 6*w - 11],
[269, 269, -w^3 + 4*w^2 - w - 5],
[271, 271, -3*w^3 + 9*w^2 + 11*w - 11],
[271, 271, w^2 - 5*w - 3],
[277, 277, w^3 - 4*w^2 - 2*w + 7],
[277, 277, -2*w^3 + 7*w^2 + 8*w - 9],
[281, 281, 2*w^2 - 4*w - 5],
[281, 281, 2*w^3 - 4*w^2 - 11*w + 1],
[283, 283, -3*w^3 + 8*w^2 + 14*w - 11],
[293, 293, w^3 - 2*w^2 - 6*w + 5],
[307, 307, -2*w^3 + 5*w^2 + 7*w - 1],
[313, 313, 2*w^3 - 4*w^2 - 9*w + 3],
[313, 313, -2*w^3 + 5*w^2 + 6*w - 3],
[317, 317, -2*w^3 + 5*w^2 + 9*w + 1],
[347, 347, 4*w^3 - 9*w^2 - 21*w + 3],
[347, 347, 4*w^3 - 10*w^2 - 18*w + 5],
[349, 349, -2*w^3 + 6*w^2 + 7*w - 3],
[353, 353, -w^3 + 12*w + 5],
[353, 353, w^2 - w - 7],
[359, 359, -2*w^3 + 5*w^2 + 6*w - 7],
[359, 359, -w^3 + 4*w^2 - 11],
[373, 373, -w^3 + 2*w^2 + 6*w + 5],
[379, 379, -3*w^3 + 8*w^2 + 14*w - 5],
[397, 397, 6*w^3 - 14*w^2 - 30*w + 7],
[401, 401, 3*w^3 - 5*w^2 - 20*w - 7],
[409, 409, 3*w^3 - 7*w^2 - 12*w - 3],
[409, 409, 2*w^2 - 4*w - 7],
[421, 421, w^3 - 11*w - 11],
[421, 421, -w^3 + 3*w^2 + w - 7],
[431, 431, 2*w^3 - 3*w^2 - 14*w - 3],
[431, 431, -4*w^3 + 11*w^2 + 15*w - 15],
[433, 433, 2*w^3 - 5*w^2 - 6*w + 1],
[439, 439, -4*w^3 + 11*w^2 + 13*w - 9],
[439, 439, -2*w^3 + 6*w^2 + 8*w - 11],
[443, 443, 2*w - 5],
[443, 443, -6*w^3 + 17*w^2 + 24*w - 21],
[449, 449, w^3 - 2*w^2 - 8*w + 3],
[449, 449, 4*w^3 - 12*w^2 - 15*w + 17],
[457, 457, -2*w^3 + 4*w^2 + 9*w + 5],
[461, 461, -3*w^3 + 6*w^2 + 15*w - 1],
[461, 461, -3*w^3 + 7*w^2 + 17*w + 1],
[463, 463, -4*w^3 + 10*w^2 + 17*w - 7],
[463, 463, -3*w^3 + 10*w^2 + 9*w - 17],
[467, 467, 4*w^3 - 8*w^2 - 21*w - 7],
[467, 467, w^3 - 6*w^2 + 3*w + 19],
[479, 479, -w^3 + w^2 + 8*w + 1],
[487, 487, -4*w^3 + 11*w^2 + 14*w - 15],
[487, 487, 3*w^3 - 6*w^2 - 15*w - 5],
[491, 491, w^2 - 5],
[503, 503, 2*w^3 - 5*w^2 - 11*w + 3],
[523, 523, -3*w^3 + 7*w^2 + 13*w + 1],
[547, 547, 2*w^3 - 3*w^2 - 12*w - 3],
[547, 547, 2*w^3 - 3*w^2 - 12*w - 9],
[557, 557, -3*w^3 + 9*w^2 + 9*w - 11],
[557, 557, -6*w^3 + 15*w^2 + 27*w - 13],
[563, 563, -3*w^3 + 7*w^2 + 14*w - 7],
[571, 571, -5*w^3 + 11*w^2 + 26*w - 3],
[593, 593, -6*w^3 + 15*w^2 + 26*w - 7],
[599, 599, 7*w^3 - 18*w^2 - 29*w + 17],
[599, 599, -2*w^3 + 6*w^2 + 8*w - 3],
[601, 601, -7*w^3 + 20*w^2 + 28*w - 23],
[601, 601, 3*w^3 - 7*w^2 - 12*w + 5],
[607, 607, 5*w^3 - 12*w^2 - 24*w + 9],
[607, 607, w^3 - 5*w^2 + 15],
[613, 613, -w^3 + 5*w^2 - 2*w - 9],
[617, 617, w^3 + w^2 - 13*w - 15],
[617, 617, 2*w^3 - 4*w^2 - 8*w - 1],
[619, 619, 5*w^3 - 10*w^2 - 27*w - 7],
[619, 619, 2*w^3 - 4*w^2 - 10*w + 3],
[643, 643, w^3 - 5*w^2 + 5*w + 5],
[643, 643, 4*w^3 - 9*w^2 - 20*w - 1],
[643, 643, 3*w^3 - 7*w^2 - 14*w - 3],
[643, 643, -3*w^3 + 7*w^2 + 13*w - 3],
[647, 647, -3*w^3 + 8*w^2 + 12*w - 3],
[653, 653, -w^3 + 9*w + 13],
[653, 653, -w^3 + 4*w^2 + 3*w - 9],
[653, 653, -2*w^3 + 7*w^2 + 4*w - 7],
[653, 653, w^2 - 4*w - 9],
[673, 673, -w^3 + 5*w^2 + w - 13],
[683, 683, w^3 - 3*w^2 - 3*w - 3],
[701, 701, 5*w^3 - 13*w^2 - 23*w + 9],
[709, 709, 2*w^3 - w^2 - 19*w - 15],
[719, 719, -5*w^3 + 15*w^2 + 20*w - 19],
[727, 727, -w - 5],
[751, 751, -4*w^3 + 10*w^2 + 15*w - 9],
[751, 751, 6*w^3 - 17*w^2 - 22*w + 21],
[757, 757, 3*w^3 - 6*w^2 - 18*w - 1],
[757, 757, -w^3 + 4*w^2 + 2*w + 1],
[769, 769, w^2 - 2*w + 3],
[773, 773, 5*w^3 - 12*w^2 - 23*w + 9],
[787, 787, 2*w^3 - 4*w^2 - 5*w + 1],
[797, 797, -3*w^3 + 8*w^2 + 10*w - 3],
[809, 809, 2*w^2 - 5*w - 1],
[809, 809, 4*w^3 - 6*w^2 - 28*w - 11],
[827, 827, 4*w^3 - 8*w^2 - 22*w - 1],
[827, 827, 2*w^2 - 6*w - 9],
[829, 829, -w^3 + w^2 + 10*w + 1],
[839, 839, -2*w^3 + 2*w^2 + 16*w + 7],
[877, 877, 6*w^3 - 16*w^2 - 25*w + 13],
[877, 877, w^3 - w^2 - 7*w + 1],
[881, 881, 2*w^3 - 2*w^2 - 15*w - 7],
[881, 881, -4*w^3 + 7*w^2 + 26*w + 9],
[887, 887, 2*w^3 - 4*w^2 - 13*w - 1],
[907, 907, 5*w^3 - 11*w^2 - 26*w - 1],
[907, 907, 5*w^3 - 13*w^2 - 22*w + 9],
[907, 907, w^3 - 5*w^2 + 2*w + 13],
[907, 907, -2*w^3 + 7*w^2 + 5*w - 7],
[911, 911, -3*w^3 + 8*w^2 + 9*w - 5],
[919, 919, -2*w^3 + 7*w^2 + 6*w - 3],
[919, 919, -3*w^3 + 8*w^2 + 15*w - 7],
[947, 947, -2*w^3 + 5*w^2 + 12*w - 9],
[947, 947, -w^2 + w - 3],
[947, 947, 3*w^3 - 6*w^2 - 14*w - 7],
[947, 947, -w^3 + 5*w^2 - w - 17],
[953, 953, 3*w^3 - 9*w^2 - 9*w + 13],
[953, 953, -4*w^3 + 13*w^2 + 11*w - 23],
[967, 967, 8*w^3 - 20*w^2 - 33*w + 19]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^8 + 4*x^7 - 2*x^6 - 22*x^5 - 14*x^4 + 19*x^3 + 16*x^2 - 1;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, -e^7 - 3*e^6 + 5*e^5 + 17*e^4 - 3*e^3 - 16*e^2, e^7 + 4*e^6 - 3*e^5 - 23*e^4 - 8*e^3 + 23*e^2 + 10*e - 2, 1, -e^7 - 5*e^6 + e^5 + 29*e^4 + 18*e^3 - 31*e^2 - 17*e + 4, 2*e^7 + 6*e^6 - 9*e^5 - 34*e^4 - e^3 + 34*e^2 + 9*e - 4, 2*e^7 + 8*e^6 - 4*e^5 - 44*e^4 - 27*e^3 + 40*e^2 + 28*e - 3, -e^5 - e^4 + 7*e^3 + 6*e^2 - 8*e - 6, e^7 + 4*e^6 - 5*e^5 - 25*e^4 + 4*e^3 + 30*e^2 - 2*e - 4, -4*e^7 - 14*e^6 + 15*e^5 + 79*e^4 + 15*e^3 - 75*e^2 - 21*e + 3, 3*e^7 + 11*e^6 - 9*e^5 - 63*e^4 - 26*e^3 + 64*e^2 + 31*e - 6, -3*e^7 - 9*e^6 + 15*e^5 + 52*e^4 - 8*e^3 - 56*e^2 - 4*e + 5, 3*e^7 + 12*e^6 - 7*e^5 - 65*e^4 - 31*e^3 + 53*e^2 + 23*e, -4*e^7 - 14*e^6 + 17*e^5 + 83*e^4 + 3*e^3 - 97*e^2 - 11*e + 17, 4*e^7 + 12*e^6 - 17*e^5 - 66*e^4 - 6*e^3 + 61*e^2 + 16*e - 8, 3*e^7 + 9*e^6 - 17*e^5 - 57*e^4 + 21*e^3 + 81*e^2 - 12*e - 19, e^7 + 4*e^6 - e^5 - 18*e^4 - 18*e^3 - 3*e^2 + 15*e + 13, -e^7 - 6*e^6 - 3*e^5 + 31*e^4 + 38*e^3 - 20*e^2 - 28*e + 1, -2*e^6 - 2*e^5 + 15*e^4 + 10*e^3 - 27*e^2 - 5*e + 8, -6*e^7 - 19*e^6 + 21*e^5 + 101*e^4 + 35*e^3 - 78*e^2 - 50*e - 3, -e^7 - 4*e^6 + 5*e^5 + 25*e^4 - 7*e^3 - 33*e^2 + 11*e + 7, 6*e^7 + 23*e^6 - 17*e^5 - 127*e^4 - 50*e^3 + 110*e^2 + 47*e - 5, 4*e^7 + 13*e^6 - 10*e^5 - 66*e^4 - 46*e^3 + 42*e^2 + 51*e, -e^7 - e^6 + 9*e^5 + 8*e^4 - 23*e^3 - 21*e^2 + 17*e + 11, -2*e^7 - 7*e^6 + 3*e^5 + 35*e^4 + 32*e^3 - 21*e^2 - 27*e - 8, -3*e^7 - 12*e^6 + 7*e^5 + 64*e^4 + 33*e^3 - 46*e^2 - 32*e - 8, -2*e^7 - 7*e^6 + 7*e^5 + 42*e^4 + 12*e^3 - 49*e^2 - 14*e + 3, -10*e^7 - 37*e^6 + 31*e^5 + 209*e^4 + 75*e^3 - 202*e^2 - 85*e + 11, e^7 + e^6 - 8*e^5 - 5*e^4 + 18*e^3 + 5*e^2 - 5*e + 2, -13*e^7 - 47*e^6 + 43*e^5 + 263*e^4 + 80*e^3 - 247*e^2 - 94*e + 18, 5*e^7 + 13*e^6 - 29*e^5 - 75*e^4 + 38*e^3 + 82*e^2 - 28*e - 14, 7*e^7 + 26*e^6 - 21*e^5 - 145*e^4 - 58*e^3 + 129*e^2 + 66*e - 1, 6*e^7 + 21*e^6 - 23*e^5 - 119*e^4 - 19*e^3 + 114*e^2 + 28*e - 10, e^7 + 5*e^6 - 3*e^5 - 29*e^4 - 8*e^3 + 28*e^2 + 21*e, -3*e^7 - 11*e^6 + 5*e^5 + 59*e^4 + 52*e^3 - 47*e^2 - 64*e - 3, 2*e^7 + 4*e^6 - 14*e^5 - 22*e^4 + 30*e^3 + 17*e^2 - 37*e + 2, 2*e^7 + 3*e^6 - 13*e^5 - 18*e^4 + 15*e^3 + 23*e^2 + e, 8*e^7 + 27*e^6 - 34*e^5 - 156*e^4 - 7*e^3 + 164*e^2 + 17*e - 24, -13*e^7 - 42*e^6 + 58*e^5 + 243*e^4 + 4*e^3 - 256*e^2 - 46*e + 36, e^7 + 4*e^6 - 10*e^5 - 31*e^4 + 36*e^3 + 57*e^2 - 38*e - 15, -7*e^7 - 25*e^6 + 23*e^5 + 142*e^4 + 52*e^3 - 140*e^2 - 88*e + 12, -e^7 - 7*e^6 - 5*e^5 + 38*e^4 + 54*e^3 - 31*e^2 - 63*e - 3, 7*e^7 + 23*e^6 - 27*e^5 - 129*e^4 - 25*e^3 + 119*e^2 + 39*e + 1, -2*e^7 - 4*e^6 + 12*e^5 + 21*e^4 - 13*e^3 - 15*e^2 - 4*e - 6, e^7 + 6*e^6 + 7*e^5 - 33*e^4 - 67*e^3 + 34*e^2 + 63*e - 5, -10*e^7 - 33*e^6 + 40*e^5 + 189*e^4 + 30*e^3 - 194*e^2 - 63*e + 10, 7*e^7 + 20*e^6 - 35*e^5 - 115*e^4 + 16*e^3 + 121*e^2 + 16*e - 16, e^7 + 4*e^6 - 3*e^5 - 20*e^4 - 4*e^3 + 6*e^2 - 5*e + 7, -9*e^7 - 26*e^6 + 47*e^5 + 155*e^4 - 32*e^3 - 185*e^2 - 5*e + 30, -7*e^6 - 14*e^5 + 42*e^4 + 74*e^3 - 48*e^2 - 56*e + 11, -3*e^7 - 12*e^6 + 6*e^5 + 64*e^4 + 39*e^3 - 45*e^2 - 28*e - 7, 7*e^7 + 20*e^6 - 35*e^5 - 117*e^4 + 15*e^3 + 129*e^2 + 13*e - 13, -7*e^7 - 23*e^6 + 30*e^5 + 129*e^4 + 5*e^3 - 120*e^2 - 21*e + 7, -4*e^6 - 12*e^5 + 14*e^4 + 61*e^3 + 21*e^2 - 44*e - 18, 15*e^7 + 48*e^6 - 67*e^5 - 279*e^4 - 2*e^3 + 303*e^2 + 36*e - 51, -5*e^7 - 18*e^6 + 10*e^5 + 97*e^4 + 71*e^3 - 86*e^2 - 75*e + 5, -7*e^7 - 20*e^6 + 33*e^5 + 114*e^4 - 3*e^3 - 117*e^2 - 27*e + 8, -5*e^7 - 13*e^6 + 30*e^5 + 82*e^4 - 35*e^3 - 115*e^2 + 2*e + 32, -5*e^7 - 23*e^6 + 11*e^5 + 135*e^4 + 53*e^3 - 150*e^2 - 36*e + 24, -8*e^7 - 22*e^6 + 43*e^5 + 129*e^4 - 37*e^3 - 147*e^2 + 18*e + 27, 7*e^7 + 19*e^6 - 38*e^5 - 108*e^4 + 32*e^3 + 107*e^2 - 2*e - 18, 3*e^7 + 19*e^6 + 4*e^5 - 107*e^4 - 88*e^3 + 100*e^2 + 78*e - 16, -2*e^7 - 2*e^6 + 14*e^5 + 11*e^4 - 24*e^3 - 22*e^2 + 10*e + 17, 5*e^7 + 15*e^6 - 28*e^5 - 87*e^4 + 35*e^3 + 93*e^2 - 17*e - 23, -5*e^7 - 17*e^6 + 19*e^5 + 102*e^4 + 21*e^3 - 129*e^2 - 36*e + 30, -4*e^7 - 12*e^6 + 25*e^5 + 74*e^4 - 42*e^3 - 87*e^2 + 33*e + 11, -7*e^7 - 24*e^6 + 23*e^5 + 135*e^4 + 48*e^3 - 132*e^2 - 57*e + 3, -11*e^7 - 36*e^6 + 42*e^5 + 196*e^4 + 43*e^3 - 160*e^2 - 66*e + 7, 9*e^7 + 38*e^6 - 19*e^5 - 215*e^4 - 116*e^3 + 208*e^2 + 124*e - 20, -5*e^7 - 18*e^6 + 18*e^5 + 103*e^4 + 24*e^3 - 99*e^2 - 34*e - 1, 18*e^7 + 60*e^6 - 67*e^5 - 339*e^4 - 82*e^3 + 334*e^2 + 124*e - 31, 10*e^7 + 38*e^6 - 30*e^5 - 213*e^4 - 80*e^3 + 192*e^2 + 99*e, 11*e^7 + 42*e^6 - 35*e^5 - 241*e^4 - 72*e^3 + 250*e^2 + 81*e - 38, 5*e^7 + 22*e^6 - 8*e^5 - 129*e^4 - 81*e^3 + 144*e^2 + 88*e - 16, 4*e^7 + 20*e^6 + e^5 - 109*e^4 - 107*e^3 + 87*e^2 + 107*e + 2, -2*e^7 - 9*e^6 - 2*e^5 + 45*e^4 + 59*e^3 - 28*e^2 - 46*e + 9, -10*e^7 - 29*e^6 + 55*e^5 + 170*e^4 - 54*e^3 - 183*e^2 + 15*e + 29, 9*e^7 + 26*e^6 - 44*e^5 - 148*e^4 + 19*e^3 + 148*e^2 - 6*e - 17, -3*e^7 - 8*e^6 + 18*e^5 + 50*e^4 - 25*e^3 - 76*e^2 + 12*e + 27, -5*e^6 - 16*e^5 + 23*e^4 + 86*e^3 - 7*e^2 - 67*e - 10, 2*e^7 + 4*e^6 - 17*e^5 - 27*e^4 + 44*e^3 + 37*e^2 - 30*e - 6, 15*e^7 + 54*e^6 - 58*e^5 - 310*e^4 - 46*e^3 + 311*e^2 + 78*e - 34, 12*e^7 + 36*e^6 - 55*e^5 - 211*e^4 - e^3 + 242*e^2 + 36*e - 46, 8*e^7 + 30*e^6 - 21*e^5 - 162*e^4 - 81*e^3 + 131*e^2 + 89*e - 19, -10*e^7 - 28*e^6 + 46*e^5 + 149*e^4 - 8*e^3 - 114*e^2 - e - 1, 7*e^6 + 18*e^5 - 36*e^4 - 94*e^3 + 22*e^2 + 67*e + 5, -6*e^7 - 24*e^6 + 9*e^5 + 128*e^4 + 98*e^3 - 91*e^2 - 88*e - 7, e^7 + e^6 - 7*e^5 - 3*e^4 + 14*e^3 - 10*e^2 - 18*e + 18, -5*e^7 - 19*e^6 + 13*e^5 + 103*e^4 + 46*e^3 - 84*e^2 - 34*e + 5, -3*e^7 - 14*e^6 + 8*e^5 + 84*e^4 + 23*e^3 - 100*e^2 - 13*e + 15, -2*e^7 - 9*e^6 + 5*e^5 + 55*e^4 + 28*e^3 - 63*e^2 - 49*e - 5, -21*e^7 - 72*e^6 + 77*e^5 + 408*e^4 + 99*e^3 - 397*e^2 - 149*e + 25, 6*e^7 + 14*e^6 - 39*e^5 - 88*e^4 + 61*e^3 + 124*e^2 - 23*e - 37, -14*e^7 - 44*e^6 + 68*e^5 + 255*e^4 - 31*e^3 - 267*e^2 - 20*e + 26, -6*e^7 - 19*e^6 + 32*e^5 + 110*e^4 - 36*e^3 - 111*e^2 + 41*e + 13, 12*e^7 + 42*e^6 - 40*e^5 - 236*e^4 - 74*e^3 + 225*e^2 + 72*e - 20, 3*e^7 + 15*e^6 - 2*e^5 - 89*e^4 - 69*e^3 + 96*e^2 + 87*e - 18, -20*e^7 - 65*e^6 + 80*e^5 + 363*e^4 + 49*e^3 - 340*e^2 - 73*e + 31, 10*e^7 + 34*e^6 - 38*e^5 - 198*e^4 - 44*e^3 + 214*e^2 + 72*e - 22, 8*e^7 + 26*e^6 - 33*e^5 - 149*e^4 - 21*e^3 + 151*e^2 + 44*e - 20, 14*e^7 + 44*e^6 - 58*e^5 - 247*e^4 - 32*e^3 + 240*e^2 + 77*e - 29, -19*e^7 - 69*e^6 + 60*e^5 + 387*e^4 + 131*e^3 - 370*e^2 - 130*e + 34, 19*e^7 + 68*e^6 - 61*e^5 - 378*e^4 - 131*e^3 + 343*e^2 + 143*e - 24, -14*e^7 - 54*e^6 + 42*e^5 + 305*e^4 + 113*e^3 - 291*e^2 - 145*e + 30, -9*e^7 - 24*e^6 + 45*e^5 + 144*e^4 - 7*e^3 - 182*e^2 - 57*e + 44, 6*e^7 + 24*e^6 - 13*e^5 - 140*e^4 - 82*e^3 + 158*e^2 + 101*e - 25, -9*e^7 - 36*e^6 + 23*e^5 + 197*e^4 + 83*e^3 - 161*e^2 - 68*e - 22, -16*e^7 - 51*e^6 + 65*e^5 + 283*e^4 + 39*e^3 - 251*e^2 - 62*e, -7*e^7 - 27*e^6 + 20*e^5 + 152*e^4 + 59*e^3 - 150*e^2 - 58*e + 23, 4*e^7 + 15*e^6 - 16*e^5 - 86*e^4 - 7*e^3 + 90*e^2 + 21*e - 25, e^7 - 5*e^6 - 17*e^5 + 37*e^4 + 66*e^3 - 64*e^2 - 49*e + 11, 9*e^7 + 32*e^6 - 18*e^5 - 171*e^4 - 124*e^3 + 149*e^2 + 113*e - 20, -2*e^7 - 7*e^6 + 16*e^5 + 45*e^4 - 42*e^3 - 50*e^2 + 21*e - 15, 3*e^7 + 17*e^6 + 11*e^5 - 85*e^4 - 134*e^3 + 49*e^2 + 118*e + 4, 8*e^7 + 26*e^6 - 31*e^5 - 150*e^4 - 34*e^3 + 167*e^2 + 76*e - 35, -2*e^7 - 11*e^6 - 10*e^5 + 56*e^4 + 113*e^3 - 30*e^2 - 119*e - 9, 2*e^7 + 4*e^6 - 14*e^5 - 22*e^4 + 28*e^3 + 24*e^2 - 14*e - 16, -2*e^7 - 2*e^6 + 13*e^5 + 6*e^4 - 14*e^3 + 13*e^2 - 13*e - 17, 10*e^7 + 24*e^6 - 58*e^5 - 137*e^4 + 66*e^3 + 143*e^2 - 14*e - 9, 3*e^7 + 17*e^6 + 3*e^5 - 88*e^4 - 80*e^3 + 46*e^2 + 62*e + 19, 19*e^7 + 68*e^6 - 70*e^5 - 385*e^4 - 75*e^3 + 368*e^2 + 93*e - 41, -19*e^7 - 72*e^6 + 57*e^5 + 411*e^4 + 159*e^3 - 411*e^2 - 206*e + 27, 5*e^7 + 11*e^6 - 35*e^5 - 64*e^4 + 73*e^3 + 72*e^2 - 59*e - 5, -11*e^7 - 40*e^6 + 43*e^5 + 233*e^4 + 34*e^3 - 234*e^2 - 62*e - 4, -18*e^7 - 61*e^6 + 80*e^5 + 362*e^4 + 8*e^3 - 409*e^2 - 65*e + 63, 11*e^7 + 41*e^6 - 49*e^5 - 249*e^4 + e^3 + 298*e^2 + 45*e - 59, -3*e^7 - 13*e^6 + 3*e^5 + 71*e^4 + 53*e^3 - 73*e^2 - 37*e + 25, 4*e^7 + 18*e^6 - 5*e^5 - 102*e^4 - 67*e^3 + 101*e^2 + 47*e - 10, 23*e^7 + 81*e^6 - 76*e^5 - 457*e^4 - 151*e^3 + 447*e^2 + 186*e - 26, 9*e^7 + 17*e^6 - 61*e^5 - 94*e^4 + 105*e^3 + 91*e^2 - 49*e - 13, -8*e^7 - 20*e^6 + 46*e^5 + 109*e^4 - 55*e^3 - 94*e^2 + 32*e + 8, 17*e^7 + 60*e^6 - 62*e^5 - 339*e^4 - 75*e^3 + 334*e^2 + 108*e - 37, 9*e^7 + 23*e^6 - 55*e^5 - 133*e^4 + 84*e^3 + 145*e^2 - 59*e - 14, 3*e^7 + 17*e^6 + 2*e^5 - 92*e^4 - 76*e^3 + 77*e^2 + 62*e - 16, -e^7 - e^6 + 4*e^5 + 2*e^4 + 13*e^3 + 9*e^2 - 39*e + 6, -17*e^7 - 59*e^6 + 52*e^5 + 323*e^4 + 140*e^3 - 278*e^2 - 163*e + 8, -7*e^7 - 18*e^6 + 28*e^5 + 94*e^4 + 22*e^3 - 75*e^2 - 17*e + 2, 4*e^7 + 15*e^6 - 12*e^5 - 79*e^4 - 35*e^3 + 43*e^2 + 44*e + 35, -4*e^7 - 7*e^6 + 35*e^5 + 38*e^4 - 101*e^3 - 31*e^2 + 92*e + 9, -13*e^7 - 50*e^6 + 43*e^5 + 275*e^4 + 73*e^3 - 220*e^2 - 87*e - 15, 8*e^7 + 31*e^6 - 28*e^5 - 185*e^4 - 40*e^3 + 213*e^2 + 40*e - 34, 8*e^7 + 24*e^6 - 36*e^5 - 142*e^4 - 8*e^3 + 174*e^2 + 40*e - 58, 19*e^7 + 56*e^6 - 91*e^5 - 323*e^4 + 25*e^3 + 346*e^2 + 35*e - 50, 9*e^7 + 33*e^6 - 16*e^5 - 172*e^4 - 136*e^3 + 126*e^2 + 137*e + 1, 17*e^7 + 58*e^6 - 61*e^5 - 322*e^4 - 83*e^3 + 291*e^2 + 116*e, 5*e^7 + 21*e^6 - 9*e^5 - 117*e^4 - 79*e^3 + 93*e^2 + 96*e + 18, -18*e^7 - 56*e^6 + 76*e^5 + 323*e^4 + 42*e^3 - 339*e^2 - 109*e + 19, -2*e^7 + 2*e^6 + 31*e^5 - 2*e^4 - 118*e^3 - 34*e^2 + 86*e + 7, 19*e^7 + 55*e^6 - 95*e^5 - 324*e^4 + 43*e^3 + 371*e^2 + 27*e - 72, 9*e^7 + 23*e^6 - 54*e^5 - 141*e^4 + 66*e^3 + 183*e^2 - 6*e - 38, 3*e^7 + 9*e^6 - 18*e^5 - 58*e^4 + 28*e^3 + 83*e^2 - 28*e - 4, 17*e^7 + 66*e^6 - 55*e^5 - 370*e^4 - 103*e^3 + 327*e^2 + 106*e - 18, 23*e^7 + 86*e^6 - 66*e^5 - 478*e^4 - 196*e^3 + 431*e^2 + 181*e - 12, 8*e^7 + 33*e^6 - 14*e^5 - 187*e^4 - 127*e^3 + 183*e^2 + 164*e + 2, -22*e^7 - 75*e^6 + 76*e^5 + 416*e^4 + 127*e^3 - 384*e^2 - 169*e + 13, 14*e^7 + 54*e^6 - 33*e^5 - 293*e^4 - 167*e^3 + 233*e^2 + 180*e - 11, -10*e^7 - 31*e^6 + 45*e^5 + 175*e^4 - 2*e^3 - 168*e^2 - 23*e - 7, 18*e^7 + 62*e^6 - 60*e^5 - 344*e^4 - 109*e^3 + 323*e^2 + 111*e - 23, -11*e^7 - 46*e^6 + 23*e^5 + 260*e^4 + 139*e^3 - 247*e^2 - 140*e - 17, 2*e^7 + 10*e^6 + 5*e^5 - 53*e^4 - 81*e^3 + 45*e^2 + 91*e - 4, 8*e^7 + 29*e^6 - 32*e^5 - 170*e^4 - 32*e^3 + 171*e^2 + 90*e - 8, 15*e^7 + 50*e^6 - 64*e^5 - 292*e^4 - 28*e^3 + 307*e^2 + 90*e - 15, -9*e^7 - 33*e^6 + 36*e^5 + 202*e^4 + 21*e^3 - 251*e^2 - 21*e + 43, -15*e^7 - 49*e^6 + 55*e^5 + 267*e^4 + 75*e^3 - 229*e^2 - 121*e + 24, -2*e^7 - 9*e^6 + 4*e^5 + 52*e^4 + 17*e^3 - 68*e^2 + 21*e + 43, 4*e^6 + 13*e^5 - 17*e^4 - 69*e^3 + 4*e^2 + 63*e - 17, -11*e^7 - 40*e^6 + 50*e^5 + 235*e^4 - 16*e^3 - 249*e^2 + 9*e + 28, -10*e^7 - 51*e^6 + 5*e^5 + 297*e^4 + 216*e^3 - 314*e^2 - 208*e + 29];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {11, 11, w^3 - 2*w^2 - 5*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;