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

NN := ideal<ZF | {29, 29, 2*w^3 - w^2 - 9*w + 5}>;

primesArray := [
[5, 5, w^3 - 4*w],
[7, 7, w + 1],
[7, 7, -w^3 + 4*w - 1],
[13, 13, -w^2 + 3],
[16, 2, 2],
[17, 17, w^3 + w^2 - 4*w - 2],
[17, 17, -w^3 + 5*w - 2],
[19, 19, -w^2 - w + 4],
[19, 19, -w^2 - w + 1],
[29, 29, 2*w^3 - w^2 - 9*w + 5],
[41, 41, -w^3 + w^2 + 5*w - 3],
[43, 43, w^3 - w^2 - 4*w + 2],
[43, 43, w^3 - 6*w],
[47, 47, -w^3 - w^2 + 5*w],
[49, 7, w^2 + 2*w - 2],
[59, 59, 2*w^3 - 8*w + 3],
[67, 67, w^2 - w - 4],
[79, 79, w^3 - w^2 - 4*w + 1],
[81, 3, -3],
[97, 97, w^3 + w^2 - 5*w + 1],
[107, 107, -2*w^3 + 3*w^2 + 10*w - 12],
[109, 109, 2*w^3 - w^2 - 9*w + 3],
[125, 5, -3*w^3 - w^2 + 12*w + 1],
[127, 127, -2*w^3 + w^2 + 10*w - 8],
[137, 137, -w^3 + w^2 + 3*w - 5],
[139, 139, w^3 - w^2 - 5*w + 1],
[149, 149, 2*w^2 - w - 6],
[149, 149, 2*w^3 + w^2 - 9*w],
[163, 163, w^3 - 5*w - 3],
[163, 163, 2*w^3 - 7*w],
[173, 173, -2*w^3 + 11*w - 3],
[173, 173, -w^3 + w^2 + 4*w - 8],
[179, 179, 2*w^3 - w^2 - 8*w + 5],
[181, 181, 2*w^3 - w^2 - 8*w],
[181, 181, 2*w^3 - 9*w - 3],
[191, 191, 2*w^2 - 5],
[191, 191, -2*w^2 - w + 10],
[193, 193, w^3 + w^2 - 3*w - 4],
[193, 193, -w^3 + 2*w^2 + 3*w - 3],
[197, 197, -2*w^3 + 9*w - 4],
[197, 197, w^2 - 7],
[199, 199, 3*w^3 - 13*w + 1],
[199, 199, w^3 + 2*w^2 - 5*w - 7],
[227, 227, -3*w^3 + w^2 + 13*w - 4],
[227, 227, -w^3 + 2*w^2 + 7*w - 6],
[241, 241, -2*w^3 + 7*w - 1],
[257, 257, -2*w^3 + w^2 + 7*w - 5],
[257, 257, 2*w^3 - w^2 - 7*w + 2],
[263, 263, -w^3 + 7*w - 3],
[269, 269, -2*w^3 + 2*w^2 + 11*w - 8],
[271, 271, w^2 + 3*w - 2],
[271, 271, w^2 + 2*w - 7],
[277, 277, -2*w^3 + 11*w - 2],
[281, 281, 2*w^3 - 11*w + 1],
[283, 283, 2*w^3 - 9*w + 5],
[283, 283, -2*w^3 + w^2 + 11*w - 7],
[289, 17, -w^3 + 2*w^2 + 4*w - 4],
[293, 293, -2*w^3 - w^2 + 7*w + 3],
[307, 307, -3*w^3 + 2*w^2 + 13*w - 9],
[311, 311, -w^3 + w^2 + 3*w - 7],
[313, 313, -w^3 + 2*w^2 + 3*w - 2],
[313, 313, w^2 - w - 8],
[317, 317, -2*w^3 - w^2 + 9*w - 1],
[331, 331, w^3 + 2*w^2 - 5*w - 3],
[337, 337, w^3 + 2*w^2 - 6*w - 4],
[337, 337, w^3 - 2*w^2 - 6*w + 5],
[347, 347, 2*w^3 + 2*w^2 - 9*w - 3],
[347, 347, -2*w^3 + w^2 + 7*w - 4],
[349, 349, w^2 - 2*w - 4],
[353, 353, -w^3 + 2*w^2 + 2*w - 6],
[359, 359, -w^3 + w^2 + 7*w - 6],
[359, 359, -w^2 - w - 2],
[361, 19, 2*w^3 - w^2 - 11*w],
[367, 367, -2*w^3 + 2*w^2 + 8*w - 5],
[373, 373, w^3 + w^2 - 2*w - 4],
[379, 379, 2*w^3 - w^2 - 11*w + 4],
[389, 389, 3*w^2 + w - 11],
[397, 397, 3*w^3 - 13*w + 5],
[401, 401, w - 5],
[409, 409, 3*w^3 - 12*w + 4],
[409, 409, 3*w^3 + 2*w^2 - 14*w - 4],
[421, 421, 2*w^3 - w^2 - 10*w + 1],
[421, 421, w^3 + 2*w^2 - 5*w - 4],
[439, 439, -2*w^3 + 10*w - 5],
[443, 443, -2*w^3 + 12*w - 5],
[443, 443, -4*w^3 + w^2 + 18*w - 9],
[443, 443, -2*w^3 + w^2 + 7*w - 9],
[443, 443, 3*w^3 + 2*w^2 - 14*w - 5],
[449, 449, w^3 + 2*w^2 - 2*w - 6],
[449, 449, -2*w^3 + w^2 + 11*w - 5],
[461, 461, 2*w^3 + 2*w^2 - 9*w - 4],
[467, 467, -w^3 + w^2 + 7*w - 5],
[487, 487, 2*w^2 + 4*w - 7],
[491, 491, -w^3 + w^2 + 7*w - 4],
[503, 503, 2*w^3 + w^2 - 9*w + 2],
[521, 521, w^2 - 3*w - 3],
[523, 523, -w^3 + 2*w^2 + 5*w - 4],
[523, 523, -w^3 + 8*w - 5],
[563, 563, w^3 + 3*w^2 - w - 8],
[563, 563, w^3 - 2*w^2 - 8*w + 4],
[563, 563, -w^3 + w^2 + 2*w - 5],
[563, 563, -3*w^3 - w^2 + 10*w - 2],
[577, 577, 3*w^3 - 16*w],
[587, 587, w^3 + 2*w^2 - 6],
[587, 587, -w^3 + 4*w - 6],
[601, 601, -w - 5],
[607, 607, w^3 - 2*w^2 - 4*w + 1],
[613, 613, 4*w^3 - 17*w],
[617, 617, -3*w^3 + w^2 + 14*w - 2],
[617, 617, -4*w^3 + 17*w - 4],
[619, 619, -3*w^3 + 12*w - 2],
[643, 643, 3*w^3 - w^2 - 12*w + 3],
[643, 643, -2*w^3 - w^2 + 11*w - 2],
[647, 647, 2*w^3 + w^2 - 9*w + 3],
[653, 653, w^3 + w^2 - 7*w - 4],
[661, 661, -w^3 + 4*w^2 + 6*w - 15],
[661, 661, 3*w^3 - 2*w^2 - 12*w + 10],
[677, 677, -3*w^2 - 2*w + 12],
[691, 691, -2*w^3 + 3*w^2 + 11*w - 10],
[691, 691, -3*w^3 + 16*w - 3],
[701, 701, 3*w^3 - 13*w + 6],
[719, 719, -3*w^2 + w + 10],
[727, 727, w^3 - 3*w^2 - 4*w + 6],
[743, 743, -w^3 + 2*w^2 + 5*w - 3],
[761, 761, -w^3 + 3*w^2 + 3*w - 12],
[761, 761, 3*w^3 - 2*w^2 - 14*w + 7],
[769, 769, -3*w^3 + 3*w^2 + 16*w - 13],
[769, 769, w^3 + 2*w^2 - 7*w - 6],
[773, 773, w^3 - 8*w + 4],
[773, 773, 4*w^2 + w - 16],
[787, 787, w^2 + 4*w - 2],
[797, 797, -2*w^3 + 3*w^2 + 10*w - 10],
[797, 797, -w^3 - w^2 + 4*w - 4],
[797, 797, w^3 + 2*w^2 - 3*w - 9],
[797, 797, -3*w^3 + 3*w^2 + 11*w - 10],
[821, 821, w^2 + w - 9],
[827, 827, w^3 - 8*w + 1],
[827, 827, -2*w^3 + w^2 + 9*w - 11],
[827, 827, 3*w^3 - 14*w + 5],
[827, 827, 2*w^3 + w^2 - 7*w - 5],
[829, 829, -2*w^3 - 2*w^2 + 9*w - 1],
[839, 839, -3*w^3 + 15*w - 5],
[853, 853, -4*w^3 + 15*w - 8],
[853, 853, -w^3 + 2*w^2 + 6*w - 3],
[859, 859, -w^3 + w^2 + w - 4],
[859, 859, w^3 + w^2 - w - 5],
[881, 881, -w^3 + w^2 + 2*w - 6],
[881, 881, w^3 + w^2 - 2*w - 6],
[883, 883, w^3 + 3*w^2 - 3*w - 6],
[887, 887, w^3 + 3*w^2 - 3*w - 8],
[919, 919, 2*w^3 - w^2 - 13*w],
[929, 929, -w^3 - 3*w^2 + 2*w + 10],
[929, 929, w^3 + w^2 - 5*w - 8],
[941, 941, w^3 + 2*w^2 - 3*w - 10],
[953, 953, -w^3 + 5*w - 7],
[953, 953, -2*w^3 + 3*w^2 + 9*w - 9],
[953, 953, -2*w^3 - w^2 + 10*w - 3],
[953, 953, -3*w^3 + 11*w - 1],
[961, 31, -w^3 + 3*w^2 + 2*w - 10],
[961, 31, -3*w^2 + 2*w + 7],
[967, 967, w^3 + 3*w^2 - 3*w - 7],
[967, 967, 4*w^3 + w^2 - 20*w + 1],
[983, 983, 2*w^3 - w^2 - 12*w + 8],
[983, 983, 2*w^3 + w^2 - 6*w - 4],
[991, 991, 2*w^3 - 12*w + 3]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^6 + 11*x^5 + 40*x^4 + 40*x^3 - 62*x^2 - 133*x - 51;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, e^5 + 8*e^4 + 17*e^3 - 5*e^2 - 41*e - 24, -e^4 - 5*e^3 - 2*e^2 + 11*e + 6, -e^5 - 8*e^4 - 16*e^3 + 9*e^2 + 39*e + 12, e^5 + 7*e^4 + 12*e^3 - 8*e^2 - 32*e - 11, e^5 + 9*e^4 + 22*e^3 - 4*e^2 - 55*e - 30, e^2 + 3*e, -e^3 - 4*e^2 + 2*e + 10, -e^5 - 8*e^4 - 17*e^3 + 6*e^2 + 44*e + 18, 1, -3*e^5 - 24*e^4 - 51*e^3 + 16*e^2 + 123*e + 60, e^5 + 9*e^4 + 20*e^3 - 12*e^2 - 52*e - 12, e^4 + 5*e^3 - 17*e - 4, e^5 + 11*e^4 + 33*e^3 + 5*e^2 - 74*e - 48, e^5 + 11*e^4 + 34*e^3 + 7*e^2 - 80*e - 53, 2*e^5 + 16*e^4 + 32*e^3 - 18*e^2 - 77*e - 21, -4*e^5 - 33*e^4 - 73*e^3 + 20*e^2 + 176*e + 84, -e^5 - 5*e^4 - e^3 + 14*e^2 + 3*e - 3, -3*e^5 - 24*e^4 - 50*e^3 + 18*e^2 + 119*e + 64, -e^4 - 7*e^3 - 8*e^2 + 15*e + 10, -4*e^5 - 32*e^4 - 67*e^3 + 26*e^2 + 165*e + 72, 2*e^5 + 12*e^4 + 10*e^3 - 34*e^2 - 32*e + 12, e^5 + 8*e^4 + 19*e^3 + 3*e^2 - 41*e - 33, -6*e^4 - 29*e^3 - 6*e^2 + 68*e + 19, -e^3 - 3*e^2 + 7*e + 6, 4*e^4 + 19*e^3 + 2*e^2 - 48*e - 15, 4*e^4 + 19*e^3 + 4*e^2 - 45*e - 24, -3*e^5 - 24*e^4 - 49*e^3 + 24*e^2 + 118*e + 36, -2*e^5 - 18*e^4 - 45*e^3 + 3*e^2 + 107*e + 59, e^5 + 8*e^4 + 17*e^3 - 6*e^2 - 41*e - 24, 5*e^5 + 41*e^4 + 88*e^3 - 34*e^2 - 222*e - 102, e^5 + 6*e^4 + 3*e^3 - 26*e^2 - 14*e + 24, -3*e^5 - 20*e^4 - 31*e^3 + 20*e^2 + 69*e + 45, 6*e^5 + 47*e^4 + 98*e^3 - 29*e^2 - 236*e - 132, -9*e^5 - 74*e^4 - 162*e^3 + 51*e^2 + 401*e + 190, -e^5 - 3*e^4 + 10*e^3 + 20*e^2 - 29*e - 24, e^5 + 8*e^4 + 15*e^3 - 14*e^2 - 38*e - 9, e^5 + 5*e^4 + 4*e^3 - 2*e^2 - e - 10, -5*e^5 - 40*e^4 - 84*e^3 + 30*e^2 + 202*e + 99, -7*e^5 - 51*e^4 - 89*e^3 + 66*e^2 + 219*e + 66, 5*e^5 + 37*e^4 + 68*e^3 - 39*e^2 - 168*e - 75, 4*e^5 + 33*e^4 + 71*e^3 - 28*e^2 - 179*e - 84, -2*e^5 - 15*e^4 - 29*e^3 + 16*e^2 + 79*e + 27, 2*e^5 + 10*e^4 - 40*e^2 - 14*e + 30, -2*e^5 - 15*e^4 - 29*e^3 + 14*e^2 + 75*e + 33, 5*e^5 + 40*e^4 + 82*e^3 - 38*e^2 - 205*e - 87, 7*e^5 + 56*e^4 + 116*e^3 - 49*e^2 - 279*e - 111, 9*e^5 + 69*e^4 + 138*e^3 - 54*e^2 - 340*e - 171, 4*e^5 + 36*e^4 + 89*e^3 - 7*e^2 - 212*e - 129, 5*e^5 + 44*e^4 + 102*e^3 - 35*e^2 - 258*e - 90, e^5 - e^4 - 31*e^3 - 31*e^2 + 81*e + 65, -4*e^5 - 33*e^4 - 72*e^3 + 29*e^2 + 186*e + 61, -3*e^5 - 21*e^4 - 31*e^3 + 38*e^2 + 69*e + 1, 4*e^5 + 29*e^4 + 51*e^3 - 35*e^2 - 130*e - 57, -6*e^5 - 41*e^4 - 61*e^3 + 71*e^2 + 166*e + 38, e^5 + 9*e^4 + 24*e^3 + 4*e^2 - 56*e - 35, 2*e^5 + 17*e^4 + 38*e^3 - 12*e^2 - 94*e - 39, -6*e^5 - 53*e^4 - 125*e^3 + 32*e^2 + 307*e + 141, -e^5 - 11*e^4 - 33*e^3 - 2*e^2 + 80*e + 21, 3*e^5 + 29*e^4 + 77*e^3 - 4*e^2 - 182*e - 99, -3*e^5 - 23*e^4 - 41*e^3 + 36*e^2 + 97*e + 6, -5*e^5 - 36*e^4 - 65*e^3 + 39*e^2 + 177*e + 73, 7*e^4 + 32*e^3 - e^2 - 81*e - 12, e^5 + 12*e^4 + 38*e^3 + 6*e^2 - 87*e - 48, -e^5 - 15*e^4 - 55*e^3 - 18*e^2 + 126*e + 75, -e^5 - 8*e^4 - 16*e^3 + 9*e^2 + 40*e + 9, 3*e^5 + 28*e^4 + 73*e^3 - e^2 - 170*e - 99, e^5 + 7*e^4 + 10*e^3 - 17*e^2 - 37*e - 9, -3*e^5 - 30*e^4 - 80*e^3 + 11*e^2 + 206*e + 102, 7*e^4 + 37*e^3 + 19*e^2 - 93*e - 66, 5*e^5 + 36*e^4 + 63*e^3 - 42*e^2 - 151*e - 42, -10*e^5 - 78*e^4 - 158*e^3 + 63*e^2 + 382*e + 189, e^5 + 7*e^4 + 12*e^3 - 4*e^2 - 33*e - 45, -6*e^5 - 51*e^4 - 118*e^3 + 21*e^2 + 281*e + 159, 3*e^4 + 16*e^3 + 15*e^2 - 19*e - 37, 11*e^5 + 89*e^4 + 193*e^3 - 52*e^2 - 461*e - 244, -7*e^5 - 55*e^4 - 109*e^3 + 63*e^2 + 278*e + 84, 3*e^5 + 14*e^4 - 36*e^2 + 2*e + 10, 4*e^5 + 36*e^4 + 93*e^3 + 5*e^2 - 222*e - 144, -3*e^5 - 29*e^4 - 74*e^3 + 18*e^2 + 191*e + 90, -5*e^5 - 40*e^4 - 87*e^3 + 20*e^2 + 218*e + 126, -2*e^5 - 14*e^4 - 17*e^3 + 41*e^2 + 47*e - 21, 9*e^5 + 77*e^4 + 179*e^3 - 35*e^2 - 424*e - 210, e^5 + 5*e^4 - e^3 - 20*e^2 + 3*e + 8, e^5 + 14*e^4 + 42*e^3 - 14*e^2 - 105*e - 18, 5*e^5 + 41*e^4 + 88*e^3 - 30*e^2 - 212*e - 111, 2*e^5 + 16*e^4 + 30*e^3 - 32*e^2 - 90*e - 6, -7*e^5 - 63*e^4 - 155*e^3 + 19*e^2 + 370*e + 198, -6*e^5 - 51*e^4 - 116*e^3 + 30*e^2 + 291*e + 150, 2*e^5 + 20*e^4 + 53*e^3 - 10*e^2 - 134*e - 54, 2*e^5 + 11*e^4 + 11*e^3 - 6*e^2 - 12*e - 36, -7*e^5 - 52*e^4 - 101*e^3 + 37*e^2 + 245*e + 120, -e^5 - 6*e^4 - e^3 + 28*e^2 - 3*e - 29, -e^5 - 9*e^4 - 22*e^3 + 5*e^2 + 57*e + 30, -11*e^5 - 80*e^4 - 141*e^3 + 95*e^2 + 341*e + 123, -3*e^5 - 12*e^4 + 13*e^3 + 58*e^2 - 4*e - 42, -13*e^5 - 104*e^4 - 220*e^3 + 77*e^2 + 547*e + 252, -5*e^5 - 41*e^4 - 90*e^3 + 24*e^2 + 214*e + 130, 7*e^5 + 53*e^4 + 103*e^3 - 46*e^2 - 262*e - 150, 6*e^5 + 52*e^4 + 123*e^3 - 24*e^2 - 303*e - 168, 3*e^5 + 24*e^4 + 53*e^3 - 6*e^2 - 132*e - 108, -4*e^5 - 33*e^4 - 73*e^3 + 19*e^2 + 182*e + 90, -2*e^5 - 13*e^4 - 21*e^3 + 9*e^2 + 54*e + 43, 5*e^5 + 42*e^4 + 91*e^3 - 45*e^2 - 238*e - 81, 6*e^5 + 43*e^4 + 81*e^3 - 23*e^2 - 187*e - 117, 3*e^5 + 25*e^4 + 52*e^3 - 27*e^2 - 119*e - 40, 3*e^5 + 17*e^4 + 20*e^3 - 12*e^2 - 49*e - 69, e^5 + 6*e^4 + 5*e^3 - 12*e^2 - 6*e - 18, 6*e^4 + 29*e^3 + 4*e^2 - 72*e - 6, -9*e^5 - 68*e^4 - 129*e^3 + 72*e^2 + 321*e + 141, 8*e^5 + 59*e^4 + 105*e^3 - 70*e^2 - 244*e - 90, 4*e^5 + 32*e^4 + 69*e^3 - 23*e^2 - 189*e - 84, 8*e^5 + 70*e^4 + 167*e^3 - 28*e^2 - 397*e - 200, 3*e^5 + 20*e^4 + 30*e^3 - 30*e^2 - 83*e - 42, -e^5 + e^4 + 29*e^3 + 24*e^2 - 73*e - 54, 4*e^5 + 36*e^4 + 84*e^3 - 25*e^2 - 193*e - 78, -2*e^5 - 4*e^4 + 30*e^3 + 46*e^2 - 76*e - 87, e^5 + 6*e^4 + 6*e^3 - 9*e^2 + 4*e + 12, 12*e^5 + 98*e^4 + 215*e^3 - 58*e^2 - 530*e - 285, -5*e^5 - 41*e^4 - 86*e^3 + 44*e^2 + 225*e + 53, -e^5 - 8*e^4 - 22*e^3 - 15*e^2 + 39*e + 51, e^5 + 4*e^4 + 6*e^2 + 19*e - 45, 6*e^5 + 39*e^4 + 51*e^3 - 67*e^2 - 118*e - 36, 8*e^5 + 64*e^4 + 134*e^3 - 48*e^2 - 327*e - 165, -7*e^5 - 61*e^4 - 143*e^3 + 35*e^2 + 353*e + 162, 10*e^5 + 73*e^4 + 126*e^3 - 106*e^2 - 330*e - 93, 4*e^4 + 20*e^3 + e^2 - 62*e - 17, 22*e^5 + 178*e^4 + 386*e^3 - 103*e^2 - 920*e - 477, -17*e^5 - 138*e^4 - 293*e^3 + 115*e^2 + 734*e + 321, -9*e^5 - 74*e^4 - 158*e^3 + 59*e^2 + 380*e + 174, 4*e^4 + 22*e^3 + 15*e^2 - 40*e - 29, 5*e^5 + 42*e^4 + 97*e^3 - 25*e^2 - 258*e - 111, -e^5 - 3*e^4 + 3*e^3 - 10*e^2 - 30*e + 33, -6*e^5 - 43*e^4 - 68*e^3 + 78*e^2 + 179*e + 18, -9*e^5 - 61*e^4 - 87*e^3 + 111*e^2 + 230*e + 51, 2*e^5 + 18*e^4 + 38*e^3 - 32*e^2 - 89*e + 12, 2*e^5 + 13*e^4 + 17*e^3 - 24*e^2 - 53*e - 30, e^5 - 6*e^4 - 62*e^3 - 62*e^2 + 149*e + 144, 6*e^5 + 52*e^4 + 118*e^3 - 43*e^2 - 294*e - 96, -7*e^5 - 65*e^4 - 165*e^3 + 18*e^2 + 395*e + 198, -2*e^5 - 9*e^4 + 9*e^3 + 50*e^2 - 22*e - 71, -11*e^5 - 101*e^4 - 254*e^3 + 29*e^2 + 623*e + 345, 5*e^5 + 34*e^4 + 55*e^3 - 39*e^2 - 134*e - 33, -7*e^5 - 56*e^4 - 119*e^3 + 38*e^2 + 304*e + 162, -3*e^5 - 21*e^4 - 34*e^3 + 28*e^2 + 78*e + 39, -5*e^5 - 31*e^4 - 38*e^3 + 49*e^2 + 84*e + 23, 12*e^5 + 106*e^4 + 252*e^3 - 51*e^2 - 604*e - 297, 16*e^5 + 127*e^4 + 269*e^3 - 83*e^2 - 667*e - 354, -12*e^5 - 94*e^4 - 191*e^3 + 83*e^2 + 488*e + 237, 7*e^5 + 57*e^4 + 125*e^3 - 28*e^2 - 304*e - 189, -3*e^5 - 26*e^4 - 62*e^3 + 7*e^2 + 142*e + 71, 2*e^5 + 22*e^4 + 68*e^3 + 13*e^2 - 150*e - 75, 12*e^5 + 95*e^4 + 204*e^3 - 45*e^2 - 483*e - 267, -10*e^5 - 82*e^4 - 176*e^3 + 70*e^2 + 441*e + 174, 7*e^5 + 54*e^4 + 105*e^3 - 59*e^2 - 269*e - 93, 7*e^5 + 55*e^4 + 118*e^3 - 23*e^2 - 279*e - 150, 3*e^5 + 34*e^4 + 97*e^3 - 19*e^2 - 253*e - 99, 9*e^5 + 69*e^4 + 141*e^3 - 36*e^2 - 319*e - 201, -e^5 - 21*e^4 - 87*e^3 - 29*e^2 + 231*e + 141, e^5 + 11*e^4 + 36*e^3 + 16*e^2 - 89*e - 96, 11*e^5 + 83*e^4 + 150*e^3 - 118*e^2 - 385*e - 102, 6*e^5 + 44*e^4 + 77*e^3 - 57*e^2 - 184*e - 55, -12*e^5 - 88*e^4 - 160*e^3 + 87*e^2 + 368*e + 171, -10*e^5 - 81*e^4 - 174*e^3 + 52*e^2 + 426*e + 240, 4*e^4 + 22*e^3 + 15*e^2 - 62*e - 77];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {29, 29, 2*w^3 - w^2 - 9*w + 5}>] := -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;