/*
  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 | {11,11,-w^2 - w + 1}>;

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^7 + 4*x^6 - x^5 - 18*x^4 - 11*x^3 + 16*x^2 + 11*x - 1;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, -e^6 - 2*e^5 + 6*e^4 + 9*e^3 - 10*e^2 - 8*e + 3, -4*e^6 - 10*e^5 + 18*e^4 + 43*e^3 - 16*e^2 - 34*e - 1, 1, e^6 + 3*e^5 - 4*e^4 - 14*e^3 + e^2 + 12*e + 3, 4*e^6 + 9*e^5 - 20*e^4 - 38*e^3 + 24*e^2 + 29*e - 6, -4*e^6 - 9*e^5 + 18*e^4 + 35*e^3 - 13*e^2 - 18*e - 5, -e^6 - 4*e^5 + 2*e^4 + 19*e^3 + 5*e^2 - 20*e - 3, 4*e^6 + 7*e^5 - 24*e^4 - 29*e^3 + 37*e^2 + 21*e - 11, 3*e^5 + 6*e^4 - 14*e^3 - 19*e^2 + 14*e + 4, -2*e^6 - 7*e^5 + 7*e^4 + 33*e^3 - 3*e^2 - 28*e + 1, 7*e^6 + 17*e^5 - 30*e^4 - 66*e^3 + 22*e^2 + 37*e + 2, -e^6 + 11*e^4 - 28*e^2 + 4*e + 8, 3*e^6 + 7*e^5 - 14*e^4 - 28*e^3 + 15*e^2 + 14*e - 6, 7*e^6 + 15*e^5 - 36*e^4 - 60*e^3 + 46*e^2 + 39*e - 9, -6*e^6 - 13*e^5 + 31*e^4 + 53*e^3 - 38*e^2 - 34*e + 3, -4*e^6 - 10*e^5 + 16*e^4 + 39*e^3 - 9*e^2 - 24*e - 1, -9*e^6 - 22*e^5 + 39*e^4 + 89*e^3 - 29*e^2 - 58*e - 4, 3*e^6 + 4*e^5 - 23*e^4 - 18*e^3 + 48*e^2 + 16*e - 19, 7*e^6 + 20*e^5 - 26*e^4 - 85*e^3 + 7*e^2 + 61*e + 4, 8*e^6 + 17*e^5 - 41*e^4 - 69*e^3 + 44*e^2 + 43*e - 1, -9*e^6 - 18*e^5 + 49*e^4 + 73*e^3 - 63*e^2 - 45*e + 4, -2*e^4 - 2*e^3 + 8*e^2 + 4*e - 2, -4*e^6 - 6*e^5 + 27*e^4 + 23*e^3 - 51*e^2 - 16*e + 15, -5*e^6 - 12*e^5 + 26*e^4 + 55*e^3 - 33*e^2 - 46*e + 9, 20*e^6 + 47*e^5 - 96*e^4 - 200*e^3 + 99*e^2 + 148*e - 15, 17*e^6 + 40*e^5 - 79*e^4 - 166*e^3 + 72*e^2 + 111*e - 1, 10*e^6 + 22*e^5 - 54*e^4 - 100*e^3 + 69*e^2 + 87*e - 9, -13*e^6 - 31*e^5 + 59*e^4 + 128*e^3 - 49*e^2 - 89*e - 6, -4*e^6 - 12*e^5 + 16*e^4 + 59*e^3 - 8*e^2 - 63*e - 12, 10*e^6 + 18*e^5 - 59*e^4 - 73*e^3 + 89*e^2 + 43*e - 22, -15*e^6 - 40*e^5 + 63*e^4 + 171*e^3 - 42*e^2 - 124*e - 2, -4*e^6 - 11*e^5 + 15*e^4 + 46*e^3 - e^2 - 29*e - 11, 6*e^6 + 16*e^5 - 25*e^4 - 67*e^3 + 20*e^2 + 45*e - 9, -e^6 + e^5 + 10*e^4 - 9*e^3 - 21*e^2 + 10*e - 8, 4*e^6 + 16*e^5 - 5*e^4 - 68*e^3 - 27*e^2 + 54*e + 12, 6*e^6 + 15*e^5 - 26*e^4 - 58*e^3 + 25*e^2 + 31*e - 17, -3*e^6 - 5*e^5 + 18*e^4 + 20*e^3 - 29*e^2 - 20*e + 14, -18*e^6 - 43*e^5 + 84*e^4 + 184*e^3 - 76*e^2 - 142*e - 6, 12*e^6 + 22*e^5 - 70*e^4 - 87*e^3 + 105*e^2 + 46*e - 24, -14*e^6 - 31*e^5 + 71*e^4 + 128*e^3 - 86*e^2 - 92*e + 12, -7*e^6 - 15*e^5 + 35*e^4 + 54*e^3 - 44*e^2 - 14*e + 15, -5*e^5 - 11*e^4 + 19*e^3 + 36*e^2 - 6*e - 14, -11*e^6 - 26*e^5 + 49*e^4 + 101*e^3 - 43*e^2 - 57*e + 3, -8*e^6 - 13*e^5 + 52*e^4 + 58*e^3 - 82*e^2 - 49*e + 13, -7*e^6 - 21*e^5 + 25*e^4 + 91*e^3 - 7*e^2 - 70*e - 12, 15*e^6 + 35*e^5 - 72*e^4 - 145*e^3 + 81*e^2 + 102*e - 31, 12*e^6 + 33*e^5 - 48*e^4 - 149*e^3 + 17*e^2 + 126*e + 16, -2*e^6 - 10*e^5 - 2*e^4 + 45*e^3 + 33*e^2 - 33*e - 16, 5*e^6 + 9*e^5 - 29*e^4 - 34*e^3 + 43*e^2 + 8*e - 20, -14*e^6 - 34*e^5 + 67*e^4 + 155*e^3 - 60*e^2 - 135*e - 11, 7*e^6 + 16*e^5 - 34*e^4 - 71*e^3 + 33*e^2 + 63*e + 10, e^4 - 2*e^3 - 6*e^2 + 14*e + 16, -5*e^5 - 15*e^4 + 14*e^3 + 53*e^2 + e - 24, -21*e^6 - 44*e^5 + 108*e^4 + 178*e^3 - 127*e^2 - 114*e + 9, -3*e^6 - 6*e^5 + 18*e^4 + 20*e^3 - 40*e^2 - 2*e + 21, -16*e^6 - 36*e^5 + 78*e^4 + 149*e^3 - 79*e^2 - 100*e - 8, -10*e^6 - 23*e^5 + 53*e^4 + 102*e^3 - 73*e^2 - 82*e + 25, -11*e^6 - 28*e^5 + 52*e^4 + 125*e^3 - 52*e^2 - 108*e, -3*e^6 - 5*e^5 + 19*e^4 + 15*e^3 - 42*e^2 - 3*e + 27, -6*e^6 - 12*e^5 + 37*e^4 + 60*e^3 - 58*e^2 - 58*e + 11, -7*e^6 - 21*e^5 + 22*e^4 + 89*e^3 + 9*e^2 - 67*e - 14, -17*e^6 - 43*e^5 + 75*e^4 + 186*e^3 - 60*e^2 - 161*e - 7, 2*e^6 + 5*e^5 - 8*e^4 - 19*e^3 - 8*e^2 + e + 31, 24*e^6 + 58*e^5 - 111*e^4 - 241*e^3 + 109*e^2 + 164*e - 25, 22*e^6 + 48*e^5 - 114*e^4 - 202*e^3 + 141*e^2 + 145*e - 36, -14*e^6 - 29*e^5 + 73*e^4 + 118*e^3 - 91*e^2 - 72*e + 32, 5*e^6 + 15*e^5 - 22*e^4 - 76*e^3 + 16*e^2 + 82*e + 3, 14*e^6 + 36*e^5 - 59*e^4 - 156*e^3 + 38*e^2 + 127*e - 5, -19*e^6 - 47*e^5 + 85*e^4 + 195*e^3 - 73*e^2 - 137*e - 1, 5*e^6 + 12*e^5 - 21*e^4 - 46*e^3 + 17*e^2 + 27*e - 19, 4*e^6 + 10*e^5 - 19*e^4 - 52*e^3 + 17*e^2 + 71*e, -10*e^6 - 25*e^5 + 47*e^4 + 105*e^3 - 56*e^2 - 69*e + 28, -28*e^6 - 65*e^5 + 135*e^4 + 273*e^3 - 151*e^2 - 205*e + 37, 6*e^6 + 23*e^5 - 10*e^4 - 100*e^3 - 29*e^2 + 80*e + 12, 17*e^6 + 39*e^5 - 82*e^4 - 158*e^3 + 86*e^2 + 102*e - 2, -15*e^6 - 41*e^5 + 63*e^4 + 182*e^3 - 37*e^2 - 138*e - 14, 4*e^6 + 12*e^5 - 14*e^4 - 53*e^3 - 3*e^2 + 41*e + 1, 14*e^6 + 31*e^5 - 74*e^4 - 135*e^3 + 88*e^2 + 106*e + 6, 23*e^6 + 50*e^5 - 122*e^4 - 218*e^3 + 154*e^2 + 178*e - 20, -3*e^6 - 12*e^5 + 3*e^4 + 50*e^3 + 23*e^2 - 32*e + 1, -3*e^5 - 6*e^4 + 18*e^3 + 22*e^2 - 26*e - 15, 34*e^6 + 84*e^5 - 153*e^4 - 358*e^3 + 124*e^2 + 269*e + 15, 14*e^6 + 35*e^5 - 66*e^4 - 156*e^3 + 63*e^2 + 142*e + 8, 9*e^6 + 20*e^5 - 45*e^4 - 82*e^3 + 55*e^2 + 52*e - 36, 5*e^6 + 10*e^5 - 22*e^4 - 30*e^3 + 16*e^2 - e + 1, -16*e^6 - 39*e^5 + 74*e^4 + 163*e^3 - 73*e^2 - 110*e + 15, 11*e^6 + 32*e^5 - 40*e^4 - 141*e^3 + 9*e^2 + 119*e + 14, 12*e^6 + 26*e^5 - 58*e^4 - 96*e^3 + 62*e^2 + 32*e - 12, -9*e^6 - 27*e^5 + 29*e^4 + 115*e^3 + 8*e^2 - 89*e - 17, 3*e^6 + 4*e^5 - 13*e^4 + 2*e^2 - 34*e + 13, -16*e^6 - 34*e^5 + 77*e^4 + 121*e^3 - 86*e^2 - 46*e + 15, 8*e^6 + 21*e^5 - 35*e^4 - 87*e^3 + 35*e^2 + 56*e - 9, -22*e^6 - 44*e^5 + 118*e^4 + 180*e^3 - 141*e^2 - 116*e + 2, 4*e^5 + 5*e^4 - 27*e^3 - 18*e^2 + 35*e - 5, 21*e^6 + 56*e^5 - 86*e^4 - 238*e^3 + 51*e^2 + 173*e + 3, 9*e^6 + 20*e^5 - 53*e^4 - 95*e^3 + 91*e^2 + 94*e - 34, 6*e^5 + 17*e^4 - 19*e^3 - 61*e^2 + 5*e + 31, -7*e^6 - 19*e^5 + 34*e^4 + 96*e^3 - 38*e^2 - 102*e + 2, -4*e^6 - 3*e^5 + 27*e^4 - e^3 - 48*e^2 + 21*e + 7, 22*e^6 + 54*e^5 - 100*e^4 - 234*e^3 + 79*e^2 + 194*e + 28, -16*e^6 - 33*e^5 + 91*e^4 + 142*e^3 - 145*e^2 - 111*e + 52, 4*e^6 + 11*e^5 - 18*e^4 - 51*e^3 + 13*e^2 + 40*e + 3, 10*e^6 + 34*e^5 - 21*e^4 - 147*e^3 - 52*e^2 + 123*e + 31, -10*e^6 - 20*e^5 + 49*e^4 + 74*e^3 - 45*e^2 - 38*e - 20, 9*e^6 + 12*e^5 - 66*e^4 - 51*e^3 + 131*e^2 + 36*e - 34, -34*e^6 - 78*e^5 + 169*e^4 + 340*e^3 - 184*e^2 - 271*e + 19, -12*e^6 - 24*e^5 + 62*e^4 + 91*e^3 - 78*e^2 - 53*e + 10, -4*e^6 - 2*e^5 + 33*e^4 - 6*e^3 - 83*e^2 + 42*e + 44, -2*e^6 - 10*e^5 + 41*e^3 + 2*e^2 - 32*e + 33, -3*e^6 - 9*e^5 + 15*e^4 + 45*e^3 - 22*e^2 - 47*e + 3, e^6 + 7*e^5 + 6*e^4 - 25*e^3 - 31*e^2 + 15*e + 34, -13*e^6 - 33*e^5 + 57*e^4 + 138*e^3 - 49*e^2 - 77*e + 15, 9*e^6 + 24*e^5 - 35*e^4 - 96*e^3 + 21*e^2 + 69*e - 7, 12*e^6 + 29*e^5 - 50*e^4 - 109*e^3 + 36*e^2 + 53*e + 5, 9*e^6 + 24*e^5 - 34*e^4 - 96*e^3 + 16*e^2 + 55*e - 4, -27*e^6 - 66*e^5 + 126*e^4 + 284*e^3 - 114*e^2 - 215*e - 17, 34*e^6 + 80*e^5 - 156*e^4 - 322*e^3 + 137*e^2 + 200*e + 5, 19*e^6 + 43*e^5 - 90*e^4 - 170*e^3 + 89*e^2 + 110*e - 1, -10*e^6 - 35*e^5 + 33*e^4 + 171*e^3 + 2*e^2 - 178*e - 28, 13*e^6 + 35*e^5 - 59*e^4 - 164*e^3 + 39*e^2 + 151*e + 36, 6*e^6 + 11*e^5 - 34*e^4 - 33*e^3 + 62*e^2 - 3*e - 26, 7*e^6 + 13*e^5 - 38*e^4 - 40*e^3 + 59*e^2 - 8*e - 16, 14*e^6 + 42*e^5 - 49*e^4 - 179*e^3 + 7*e^2 + 143*e + 28, -23*e^6 - 54*e^5 + 107*e^4 + 232*e^3 - 94*e^2 - 180*e + 13, -14*e^6 - 31*e^5 + 70*e^4 + 130*e^3 - 73*e^2 - 95*e - 26, 18*e^6 + 54*e^5 - 63*e^4 - 244*e^3 - 5*e^2 + 212*e + 37, 19*e^6 + 43*e^5 - 92*e^4 - 172*e^3 + 105*e^2 + 108*e - 34, 20*e^6 + 45*e^5 - 100*e^4 - 192*e^3 + 108*e^2 + 135*e - 9, -18*e^6 - 34*e^5 + 101*e^4 + 135*e^3 - 146*e^2 - 84*e + 49, -21*e^6 - 45*e^5 + 107*e^4 + 178*e^3 - 137*e^2 - 108*e + 34, 23*e^6 + 52*e^5 - 114*e^4 - 220*e^3 + 115*e^2 + 149*e + 9, 19*e^6 + 46*e^5 - 85*e^4 - 181*e^3 + 77*e^2 + 105*e - 4, 18*e^6 + 41*e^5 - 91*e^4 - 182*e^3 + 91*e^2 + 161*e + 19, -16*e^6 - 34*e^5 + 77*e^4 + 140*e^3 - 61*e^2 - 93*e - 25, -28*e^6 - 63*e^5 + 145*e^4 + 274*e^3 - 182*e^2 - 213*e + 54, 32*e^6 + 68*e^5 - 168*e^4 - 279*e^3 + 219*e^2 + 197*e - 45, 13*e^6 + 35*e^5 - 53*e^4 - 149*e^3 + 27*e^2 + 108*e + 9, 32*e^6 + 68*e^5 - 174*e^4 - 295*e^3 + 232*e^2 + 239*e - 41, 21*e^6 + 43*e^5 - 120*e^4 - 194*e^3 + 175*e^2 + 174*e - 47, 11*e^6 + 22*e^5 - 66*e^4 - 102*e^3 + 115*e^2 + 103*e - 60, 16*e^6 + 35*e^5 - 77*e^4 - 127*e^3 + 85*e^2 + 39*e - 9, -27*e^6 - 67*e^5 + 118*e^4 + 277*e^3 - 78*e^2 - 181*e - 31, -11*e^6 - 19*e^5 + 58*e^4 + 66*e^3 - 64*e^2 - 35*e - 19, 21*e^6 + 44*e^5 - 116*e^4 - 193*e^3 + 153*e^2 + 150*e - 17, 8*e^6 + 18*e^5 - 46*e^4 - 87*e^3 + 75*e^2 + 90*e - 44, 4*e^6 + 8*e^5 - 28*e^4 - 46*e^3 + 68*e^2 + 71*e - 57, -8*e^6 - 24*e^5 + 30*e^4 + 114*e^3 - 3*e^2 - 133*e - 27, -47*e^6 - 111*e^5 + 220*e^4 + 457*e^3 - 211*e^2 - 307*e + 18, 8*e^6 + 22*e^5 - 32*e^4 - 97*e^3 + 16*e^2 + 81*e - 12, 8*e^6 + 15*e^5 - 49*e^4 - 59*e^3 + 95*e^2 + 43*e - 50, -4*e^6 - 4*e^5 + 23*e^4 + 4*e^3 - 29*e^2 + 25*e + 16, 2*e^6 - e^5 - 27*e^4 - 8*e^3 + 73*e^2 + 41*e - 35, 22*e^6 + 54*e^5 - 98*e^4 - 221*e^3 + 75*e^2 + 123*e + 3, -38*e^6 - 91*e^5 + 174*e^4 + 382*e^3 - 156*e^2 - 278*e + 19, -7*e^6 - 8*e^5 + 44*e^4 + 19*e^3 - 67*e^2 + 9*e + 19, 11*e^6 + 23*e^5 - 61*e^4 - 108*e^3 + 72*e^2 + 111*e + 2, -2*e^6 + 2*e^5 + 21*e^4 - 22*e^3 - 69*e^2 + 45*e + 41, 31*e^6 + 79*e^5 - 137*e^4 - 330*e^3 + 133*e^2 + 239*e - 36, 23*e^6 + 49*e^5 - 122*e^4 - 208*e^3 + 165*e^2 + 155*e - 55, -13*e^6 - 31*e^5 + 70*e^4 + 138*e^3 - 105*e^2 - 118*e + 19, 14*e^6 + 34*e^5 - 69*e^4 - 160*e^3 + 61*e^2 + 153*e - 4, 24*e^6 + 51*e^5 - 123*e^4 - 205*e^3 + 136*e^2 + 125*e + 5, -39*e^6 - 89*e^5 + 187*e^4 + 370*e^3 - 177*e^2 - 263*e - 30];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
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;