/*
  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, 1, 8, -1, -6, 0, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {13, 13, -w^2 + 3}>;

primesArray := [
[2, 2, -w],
[13, 13, -w^2 + 3],
[17, 17, -w^5 + w^4 + 5*w^3 - 3*w^2 - 5*w + 1],
[23, 23, w^4 - w^3 - 4*w^2 + 2*w + 1],
[31, 31, w^5 - w^4 - 6*w^3 + 5*w^2 + 7*w - 5],
[32, 2, w^5 - 6*w^3 - w^2 + 8*w + 1],
[43, 43, w^4 - 5*w^2 + 3],
[47, 47, -w^4 + w^3 + 5*w^2 - 2*w - 5],
[49, 7, -w^5 + w^4 + 5*w^3 - 3*w^2 - 4*w - 1],
[53, 53, w^5 - 2*w^4 - 5*w^3 + 9*w^2 + 5*w - 7],
[59, 59, w^5 - w^4 - 6*w^3 + 4*w^2 + 7*w - 3],
[71, 71, w^5 - w^4 - 5*w^3 + 4*w^2 + 4*w - 5],
[79, 79, w^3 - w^2 - 4*w + 1],
[83, 83, w^5 - 6*w^3 - w^2 + 7*w - 1],
[83, 83, 2*w^5 - w^4 - 11*w^3 + 2*w^2 + 12*w + 1],
[89, 89, -w^5 + w^4 + 4*w^3 - 2*w^2 - w - 1],
[89, 89, -w^5 + w^4 + 5*w^3 - 3*w^2 - 4*w + 3],
[89, 89, w^5 + w^4 - 6*w^3 - 6*w^2 + 6*w + 3],
[89, 89, 2*w^4 - w^3 - 9*w^2 + w + 5],
[101, 101, w^5 - 5*w^3 - w^2 + 5*w - 1],
[107, 107, w^4 - w^3 - 3*w^2 + 2*w - 1],
[107, 107, -w^5 + w^4 + 6*w^3 - 3*w^2 - 7*w + 1],
[107, 107, -w^5 + 4*w^3 + 2*w^2 + 1],
[113, 113, 2*w^5 - w^4 - 10*w^3 + 2*w^2 + 8*w - 3],
[121, 11, 2*w^5 - w^4 - 10*w^3 + 2*w^2 + 8*w - 1],
[125, 5, -w^3 + 4*w - 1],
[125, 5, -2*w^5 + w^4 + 10*w^3 - 3*w^2 - 9*w + 3],
[131, 131, w^5 + 2*w^4 - 7*w^3 - 10*w^2 + 9*w + 5],
[137, 137, w^5 - 2*w^4 - 5*w^3 + 9*w^2 + 6*w - 7],
[137, 137, w^4 - 3*w^2 - 2*w + 1],
[139, 139, -w^5 + w^4 + 5*w^3 - 4*w^2 - 5*w + 1],
[139, 139, w^5 - w^4 - 5*w^3 + 4*w^2 + 3*w - 3],
[149, 149, -w^5 + 2*w^4 + 4*w^3 - 9*w^2 - w + 7],
[151, 151, w^4 - 6*w^2 - w + 5],
[163, 163, -w^5 + w^4 + 4*w^3 - 3*w^2 - w + 3],
[167, 167, -w^5 + w^4 + 5*w^3 - 2*w^2 - 6*w + 1],
[169, 13, -2*w^5 + 12*w^3 + w^2 - 13*w + 1],
[179, 179, -w^5 + 6*w^3 + 2*w^2 - 9*w - 3],
[179, 179, -w^3 + 5*w - 1],
[191, 191, w^4 - 3*w^2 - 1],
[193, 193, -w^5 - 3*w^4 + 6*w^3 + 17*w^2 - 5*w - 15],
[197, 197, 2*w^5 - 11*w^3 - 2*w^2 + 10*w + 1],
[197, 197, w^5 - 5*w^3 - 2*w^2 + 3*w - 1],
[227, 227, w^5 - w^4 - 5*w^3 + 5*w^2 + 4*w - 5],
[229, 229, -2*w^4 + w^3 + 10*w^2 - w - 9],
[241, 241, 3*w^5 - 2*w^4 - 16*w^3 + 6*w^2 + 17*w - 3],
[263, 263, -2*w^5 + 11*w^3 + w^2 - 11*w - 1],
[269, 269, -w^5 + w^4 + 5*w^3 - 4*w^2 - 7*w + 3],
[277, 277, 2*w^5 - w^4 - 10*w^3 + w^2 + 7*w + 1],
[277, 277, 2*w^5 - 2*w^4 - 10*w^3 + 7*w^2 + 11*w - 5],
[283, 283, -w^4 + w^3 + 4*w^2 - 3],
[289, 17, -w^5 - w^4 + 7*w^3 + 4*w^2 - 10*w - 1],
[293, 293, -3*w^5 + 16*w^3 + 3*w^2 - 15*w - 1],
[293, 293, -2*w^5 - w^4 + 10*w^3 + 7*w^2 - 6*w - 5],
[307, 307, -2*w^5 + w^4 + 9*w^3 - w^2 - 4*w - 1],
[311, 311, w^4 - w^3 - 5*w^2 + 4*w + 1],
[311, 311, -3*w^5 + 17*w^3 + 4*w^2 - 18*w - 7],
[311, 311, -w^5 - w^4 + 5*w^3 + 6*w^2 - 2*w - 5],
[311, 311, 2*w^4 - w^3 - 10*w^2 + w + 5],
[313, 313, w^5 - 5*w^3 - 3*w^2 + 3*w + 3],
[313, 313, -w^5 - w^4 + 5*w^3 + 7*w^2 - 3*w - 3],
[317, 317, -w^5 + 2*w^4 + 3*w^3 - 8*w^2 + w + 5],
[337, 337, 3*w^5 + w^4 - 16*w^3 - 10*w^2 + 14*w + 11],
[337, 337, -w^5 - 2*w^4 + 6*w^3 + 10*w^2 - 6*w - 7],
[337, 337, 2*w^5 + w^4 - 11*w^3 - 7*w^2 + 10*w + 3],
[337, 337, w^4 - w^3 - 6*w^2 + 4*w + 7],
[347, 347, -2*w^5 + 2*w^4 + 9*w^3 - 7*w^2 - 7*w + 3],
[347, 347, -w^5 - w^4 + 5*w^3 + 7*w^2 - 2*w - 7],
[347, 347, -w^5 - w^4 + 6*w^3 + 7*w^2 - 5*w - 7],
[349, 349, 3*w^5 - w^4 - 17*w^3 + 3*w^2 + 19*w - 3],
[349, 349, w^4 + w^3 - 6*w^2 - 4*w + 7],
[359, 359, -2*w^5 + w^4 + 11*w^3 - 3*w^2 - 13*w + 1],
[373, 373, -2*w^5 + 10*w^3 + 2*w^2 - 9*w - 1],
[379, 379, -w^5 + 2*w^4 + 4*w^3 - 9*w^2 - w + 9],
[389, 389, -2*w^5 + w^4 + 11*w^3 - 2*w^2 - 13*w - 1],
[397, 397, w^5 - 3*w^4 - 5*w^3 + 14*w^2 + 7*w - 11],
[419, 419, w^5 - 6*w^3 + w^2 + 7*w - 1],
[419, 419, -w^5 + w^4 + 3*w^3 - 2*w^2 + w - 1],
[433, 433, -w^5 + w^4 + 4*w^3 - 4*w^2 - w + 5],
[433, 433, w^5 + w^4 - 7*w^3 - 4*w^2 + 9*w - 1],
[439, 439, w^4 - 2*w^3 - 4*w^2 + 7*w + 3],
[443, 443, -w^3 + w^2 + 3*w - 5],
[443, 443, w^5 - w^4 - 5*w^3 + 2*w^2 + 5*w + 3],
[449, 449, 2*w^5 - 2*w^4 - 11*w^3 + 8*w^2 + 12*w - 5],
[457, 457, -2*w^5 - 2*w^4 + 11*w^3 + 13*w^2 - 10*w - 11],
[461, 461, w^5 + 2*w^4 - 6*w^3 - 10*w^2 + 5*w + 7],
[479, 479, -2*w^5 + 10*w^3 + w^2 - 6*w + 1],
[479, 479, w^5 - w^4 - 6*w^3 + 3*w^2 + 9*w - 1],
[491, 491, w^5 - w^4 - 4*w^3 + w^2 + w + 3],
[499, 499, -w^3 + 5*w - 3],
[499, 499, w^5 - 2*w^4 - 5*w^3 + 7*w^2 + 6*w - 1],
[509, 509, -2*w^5 + 2*w^4 + 10*w^3 - 9*w^2 - 9*w + 11],
[521, 521, -w^5 + 4*w^3 + 2*w^2 - w - 3],
[521, 521, 2*w^5 + 2*w^4 - 11*w^3 - 13*w^2 + 9*w + 13],
[521, 521, -w^4 + 2*w^3 + 4*w^2 - 5*w - 3],
[523, 523, -w^5 + w^4 + 6*w^3 - 5*w^2 - 8*w + 3],
[523, 523, w^5 + w^4 - 5*w^3 - 8*w^2 + 3*w + 7],
[541, 541, w^5 + 3*w^4 - 7*w^3 - 16*w^2 + 8*w + 13],
[547, 547, w^5 - 2*w^4 - 5*w^3 + 7*w^2 + 7*w - 3],
[557, 557, 3*w^5 - w^4 - 15*w^3 + w^2 + 11*w - 1],
[557, 557, 2*w^5 - w^4 - 11*w^3 + 2*w^2 + 11*w + 1],
[563, 563, w^5 - 3*w^4 - 4*w^3 + 13*w^2 + 3*w - 9],
[569, 569, -w^5 - w^4 + 5*w^3 + 6*w^2 - 4*w - 1],
[571, 571, -3*w^5 + 2*w^4 + 15*w^3 - 7*w^2 - 10*w + 3],
[577, 577, 2*w^5 - 11*w^3 - 2*w^2 + 9*w + 3],
[587, 587, w^4 - 6*w^2 - 2*w + 7],
[599, 599, -2*w^5 + 2*w^4 + 10*w^3 - 7*w^2 - 10*w + 3],
[599, 599, w^5 - 2*w^4 - 4*w^3 + 9*w^2 + 3*w - 7],
[601, 601, 2*w^2 - 5],
[607, 607, -w^5 - w^4 + 4*w^3 + 7*w^2 - 5],
[613, 613, -w^5 + 5*w^3 + w^2 - 3*w + 3],
[617, 617, -2*w^4 + 11*w^2 + 2*w - 9],
[619, 619, -2*w^5 + 10*w^3 + w^2 - 7*w + 1],
[619, 619, -2*w^5 + 4*w^4 + 8*w^3 - 15*w^2 - 5*w + 7],
[643, 643, -2*w^5 - w^4 + 12*w^3 + 6*w^2 - 12*w - 3],
[647, 647, 3*w^5 - w^4 - 18*w^3 + w^2 + 23*w + 3],
[647, 647, 2*w^5 + w^4 - 10*w^3 - 9*w^2 + 9*w + 9],
[653, 653, -w^4 + 7*w^2 + w - 7],
[673, 673, -2*w^4 + w^3 + 8*w^2 - w - 5],
[673, 673, -2*w^5 + 11*w^3 + w^2 - 9*w + 1],
[677, 677, 4*w^5 - w^4 - 20*w^3 + w^2 + 15*w - 3],
[683, 683, 3*w^5 - 2*w^4 - 14*w^3 + 6*w^2 + 11*w - 5],
[683, 683, 2*w^5 - w^4 - 11*w^3 + 3*w^2 + 10*w - 3],
[683, 683, -w^5 + 5*w^3 + w^2 - 4*w + 3],
[691, 691, w^5 + w^4 - 6*w^3 - 6*w^2 + 8*w + 3],
[709, 709, -2*w^5 + 2*w^4 + 11*w^3 - 7*w^2 - 13*w + 3],
[719, 719, 2*w^5 - 2*w^4 - 10*w^3 + 8*w^2 + 9*w - 9],
[719, 719, -2*w^5 + w^4 + 11*w^3 - 4*w^2 - 12*w + 3],
[727, 727, w^5 - w^4 - 6*w^3 + 4*w^2 + 7*w - 1],
[729, 3, -3],
[739, 739, 2*w^4 - w^3 - 9*w^2 + 5],
[739, 739, -w^5 + w^4 + 4*w^3 - 3*w^2 - 3*w - 1],
[751, 751, -2*w^4 + w^3 + 9*w^2 - 3*w - 3],
[761, 761, -w^5 - 2*w^4 + 7*w^3 + 12*w^2 - 9*w - 13],
[773, 773, -w^5 + 5*w^3 + 2*w^2 - 2*w - 3],
[787, 787, 2*w^5 - 12*w^3 - 2*w^2 + 13*w + 3],
[787, 787, -2*w^5 + w^4 + 10*w^3 - 3*w^2 - 10*w + 1],
[797, 797, -w^5 + 2*w^4 + 5*w^3 - 8*w^2 - 7*w + 3],
[811, 811, -2*w^5 + 12*w^3 + w^2 - 15*w - 1],
[821, 821, 4*w^5 + w^4 - 22*w^3 - 8*w^2 + 21*w + 3],
[821, 821, w^5 - 2*w^4 - 3*w^3 + 8*w^2 - 3*w - 3],
[821, 821, -w^5 + 7*w^3 - 9*w - 1],
[827, 827, w^5 - 7*w^3 - w^2 + 10*w + 1],
[829, 829, -2*w^4 + 10*w^2 + w - 7],
[839, 839, -w^5 + 2*w^4 + 6*w^3 - 10*w^2 - 8*w + 9],
[853, 853, -w^5 + w^4 + 3*w^3 - 3*w^2 + 2*w + 1],
[859, 859, -w^5 + 6*w^3 - 9*w + 1],
[859, 859, -2*w^5 + 3*w^4 + 8*w^3 - 10*w^2 - 6*w + 5],
[859, 859, w^5 - 2*w^4 - 4*w^3 + 8*w^2 + 2*w - 7],
[859, 859, 2*w^5 - 2*w^4 - 10*w^3 + 6*w^2 + 10*w - 3],
[863, 863, 3*w^5 - 4*w^4 - 14*w^3 + 14*w^2 + 13*w - 7],
[877, 877, w^3 + w^2 - 4*w - 1],
[881, 881, -w^5 + w^4 + 4*w^3 - 4*w^2 - 2*w + 5],
[881, 881, -2*w^5 + 10*w^3 + 4*w^2 - 7*w - 5],
[907, 907, -w^4 + w^3 + 4*w^2 - w + 1],
[937, 937, w^5 - 7*w^3 + w^2 + 11*w - 3],
[941, 941, -2*w^5 + w^4 + 11*w^3 - w^2 - 12*w - 1],
[947, 947, -w^5 + 5*w^3 + 3*w^2 - 5*w - 7],
[953, 953, 2*w^5 - w^4 - 11*w^3 + 4*w^2 + 10*w - 7],
[971, 971, w^5 - w^4 - 5*w^3 + 3*w^2 + 7*w + 1],
[983, 983, -2*w^5 + 9*w^3 + 3*w^2 - 5*w - 3],
[997, 997, 3*w^5 - 18*w^3 - 2*w^2 + 22*w + 1]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^10 + 5*x^9 - 2*x^8 - 40*x^7 - 27*x^6 + 97*x^5 + 89*x^4 - 73*x^3 - 51*x^2 + 24*x - 2;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, 1, -e^8 - 3*e^7 + 8*e^6 + 25*e^5 - 21*e^4 - 64*e^3 + 22*e^2 + 47*e - 12, -e^9 - 2*e^8 + 12*e^7 + 18*e^6 - 56*e^5 - 49*e^4 + 117*e^3 + 31*e^2 - 86*e + 17, e^9 + 5*e^8 - 2*e^7 - 39*e^6 - 25*e^5 + 89*e^4 + 77*e^3 - 52*e^2 - 35*e + 5, 4*e^9 + 16*e^8 - 22*e^7 - 131*e^6 + 10*e^5 + 322*e^4 + 56*e^3 - 212*e^2 - e + 5, -3*e^9 - 12*e^8 + 16*e^7 + 97*e^6 - 4*e^5 - 233*e^4 - 49*e^3 + 144*e^2 + 4*e - 4, -e^9 - 3*e^8 + 10*e^7 + 28*e^6 - 41*e^5 - 85*e^4 + 84*e^3 + 80*e^2 - 62*e + 2, 3*e^9 + 12*e^8 - 16*e^7 - 97*e^6 + 3*e^5 + 232*e^4 + 53*e^3 - 144*e^2 - 6*e + 4, -2*e^8 - 9*e^7 + 9*e^6 + 74*e^5 + 6*e^4 - 185*e^3 - 34*e^2 + 123*e - 21, 4*e^9 + 15*e^8 - 26*e^7 - 126*e^6 + 41*e^5 + 322*e^4 - 13*e^3 - 227*e^2 + 34*e + 1, -3*e^9 - 15*e^8 + 5*e^7 + 116*e^6 + 86*e^5 - 258*e^4 - 264*e^3 + 139*e^2 + 131*e - 24, -e^9 - 4*e^8 + 7*e^7 + 36*e^6 - 12*e^5 - 96*e^4 + 5*e^3 + 64*e^2 - 13*e + 3, -3*e^9 - 13*e^8 + 14*e^7 + 106*e^6 + 11*e^5 - 258*e^4 - 78*e^3 + 165*e^2 + 9*e - 13, -e^9 - 3*e^8 + 9*e^7 + 27*e^6 - 28*e^5 - 75*e^4 + 39*e^3 + 62*e^2 - 31*e - 5, -e^9 - 4*e^8 + 6*e^7 + 33*e^6 - 7*e^5 - 78*e^4 + 2*e^3 + 39*e^2 - 20*e + 4, e^6 - 12*e^4 - 3*e^3 + 35*e^2 + 10*e - 19, e^9 + 6*e^8 + 3*e^7 - 45*e^6 - 71*e^5 + 97*e^4 + 206*e^3 - 62*e^2 - 134*e + 33, 2*e^8 + 7*e^7 - 12*e^6 - 53*e^5 + 15*e^4 + 117*e^3 + 7*e^2 - 66*e + 4, 2*e^9 + 10*e^8 - 3*e^7 - 77*e^6 - 62*e^5 + 171*e^4 + 198*e^3 - 98*e^2 - 118*e + 23, -5*e^9 - 21*e^8 + 24*e^7 + 170*e^6 + 16*e^5 - 407*e^4 - 131*e^3 + 250*e^2 + 18*e - 12, e^9 + 3*e^8 - 13*e^7 - 32*e^6 + 71*e^5 + 110*e^4 - 176*e^3 - 108*e^2 + 132*e - 15, -3*e^9 - 14*e^8 + 11*e^7 + 116*e^6 + 41*e^5 - 289*e^4 - 166*e^3 + 198*e^2 + 84*e - 24, -e^9 - 8*e^8 - 10*e^7 + 59*e^6 + 130*e^5 - 122*e^4 - 356*e^3 + 67*e^2 + 240*e - 45, -e^9 - 3*e^8 + 10*e^7 + 29*e^6 - 36*e^5 - 85*e^4 + 60*e^3 + 70*e^2 - 40*e - 8, -6*e^9 - 28*e^8 + 21*e^7 + 225*e^6 + 77*e^5 - 535*e^4 - 282*e^3 + 322*e^2 + 97*e - 19, -e^9 - 5*e^8 - e^7 + 36*e^6 + 58*e^5 - 72*e^4 - 191*e^3 + 45*e^2 + 144*e - 37, 6*e^9 + 27*e^8 - 22*e^7 - 216*e^6 - 77*e^5 + 513*e^4 + 315*e^3 - 329*e^2 - 155*e + 38, e^9 + 4*e^8 - 7*e^7 - 36*e^6 + 14*e^5 + 95*e^4 - 23*e^3 - 57*e^2 + 49*e - 8, -6*e^9 - 26*e^8 + 23*e^7 + 203*e^6 + 66*e^5 - 461*e^4 - 282*e^3 + 263*e^2 + 131*e - 30, -2*e^9 - 9*e^8 + 8*e^7 + 73*e^6 + 18*e^5 - 178*e^4 - 76*e^3 + 115*e^2 + 17*e - 2, -3*e^9 - 12*e^8 + 19*e^7 + 102*e^6 - 32*e^5 - 265*e^4 + 35*e^3 + 189*e^2 - 69*e - 4, -e^9 - 8*e^8 - 11*e^7 + 53*e^6 + 128*e^5 - 87*e^4 - 321*e^3 + 19*e^2 + 188*e - 36, 7*e^9 + 31*e^8 - 26*e^7 - 246*e^6 - 88*e^5 + 574*e^4 + 366*e^3 - 351*e^2 - 182*e + 49, -2*e^9 - 5*e^8 + 24*e^7 + 50*e^6 - 111*e^5 - 154*e^4 + 232*e^3 + 129*e^2 - 170*e + 15, 5*e^9 + 24*e^8 - 11*e^7 - 187*e^6 - 128*e^5 + 424*e^4 + 433*e^3 - 251*e^2 - 253*e + 50, e^8 + e^7 - 11*e^6 - 5*e^5 + 43*e^4 + 3*e^3 - 58*e^2 + 7*e + 10, 7*e^9 + 25*e^8 - 49*e^7 - 211*e^6 + 98*e^5 + 541*e^4 - 82*e^3 - 385*e^2 + 100*e + 5, -2*e^8 - 11*e^7 + 8*e^6 + 98*e^5 + 11*e^4 - 271*e^3 - 30*e^2 + 204*e - 36, -3*e^9 - 11*e^8 + 20*e^7 + 92*e^6 - 36*e^5 - 237*e^4 + 25*e^3 + 184*e^2 - 33*e - 21, e^9 + 2*e^8 - 11*e^7 - 18*e^6 + 43*e^5 + 51*e^4 - 66*e^3 - 44*e^2 + 32*e - 9, -3*e^9 - 13*e^8 + 10*e^7 + 99*e^6 + 46*e^5 - 216*e^4 - 180*e^3 + 110*e^2 + 103*e - 16, e^9 + 8*e^8 + 8*e^7 - 62*e^6 - 110*e^5 + 142*e^4 + 294*e^3 - 97*e^2 - 191*e + 44, 2*e^9 + 12*e^8 + 2*e^7 - 92*e^6 - 95*e^5 + 204*e^4 + 246*e^3 - 113*e^2 - 110*e + 23, -6*e^9 - 19*e^8 + 51*e^7 + 165*e^6 - 160*e^5 - 436*e^4 + 270*e^3 + 306*e^2 - 238*e + 30, -e^9 + 2*e^8 + 30*e^7 - 203*e^5 - 61*e^4 + 479*e^3 + 103*e^2 - 313*e + 48, -2*e^9 - 8*e^8 + 10*e^7 + 62*e^6 - 3*e^5 - 143*e^4 - 12*e^3 + 89*e^2 - 34*e - 7, 4*e^9 + 8*e^8 - 50*e^7 - 80*e^6 + 234*e^5 + 255*e^4 - 470*e^3 - 248*e^2 + 309*e - 25, -e^9 - 5*e^8 + 4*e^7 + 44*e^6 + 9*e^5 - 126*e^4 - 44*e^3 + 117*e^2 + 36*e - 14, -2*e^8 - 4*e^7 + 17*e^6 + 23*e^5 - 51*e^4 - 23*e^3 + 55*e^2 - 15*e - 11, -2*e^9 - 14*e^8 - 11*e^7 + 102*e^6 + 171*e^5 - 205*e^4 - 447*e^3 + 89*e^2 + 263*e - 49, e^9 - e^8 - 25*e^7 + 169*e^5 + 29*e^4 - 414*e^3 - 38*e^2 + 292*e - 54, -e^9 - 4*e^8 + 5*e^7 + 29*e^6 - 8*e^5 - 57*e^4 + 37*e^3 + 10*e^2 - 79*e + 22, -3*e^9 - 13*e^8 + 12*e^7 + 102*e^6 + 28*e^5 - 234*e^4 - 125*e^3 + 140*e^2 + 48*e - 29, 3*e^9 + 12*e^8 - 19*e^7 - 101*e^6 + 32*e^5 + 253*e^4 - 39*e^3 - 150*e^2 + 83*e - 30, -e^9 - 4*e^8 + 13*e^7 + 47*e^6 - 70*e^5 - 176*e^4 + 176*e^3 + 194*e^2 - 144*e - 1, -2*e^9 - 9*e^8 + 5*e^7 + 67*e^6 + 45*e^5 - 137*e^4 - 155*e^3 + 54*e^2 + 84*e - 6, 2*e^9 + 4*e^8 - 26*e^7 - 43*e^6 + 121*e^5 + 141*e^4 - 238*e^3 - 133*e^2 + 166*e - 14, -3*e^8 - 10*e^7 + 23*e^6 + 82*e^5 - 60*e^4 - 201*e^3 + 71*e^2 + 132*e - 37, -5*e^9 - 21*e^8 + 20*e^7 + 162*e^6 + 48*e^5 - 362*e^4 - 220*e^3 + 196*e^2 + 96*e - 2, 2*e^9 + 9*e^8 - 9*e^7 - 76*e^6 - 16*e^5 + 190*e^4 + 95*e^3 - 116*e^2 - 64*e - 4, -e^9 - e^8 + 19*e^7 + 17*e^6 - 118*e^5 - 75*e^4 + 287*e^3 + 87*e^2 - 204*e + 14, e^9 + 3*e^8 - 8*e^7 - 25*e^6 + 21*e^5 + 65*e^4 - 23*e^3 - 53*e^2 + 14*e - 2, -8*e^9 - 30*e^8 + 49*e^7 + 244*e^6 - 62*e^5 - 593*e^4 - 4*e^3 + 387*e^2 - 81*e - 18, -2*e^9 - 7*e^8 + 14*e^7 + 57*e^6 - 28*e^5 - 129*e^4 + 30*e^3 + 46*e^2 - 56*e + 29, 8*e^9 + 37*e^8 - 26*e^7 - 294*e^6 - 126*e^5 + 688*e^4 + 458*e^3 - 420*e^2 - 208*e + 50, -6*e^9 - 21*e^8 + 46*e^7 + 178*e^6 - 131*e^5 - 458*e^4 + 238*e^3 + 307*e^2 - 248*e + 32, -2*e^8 - 4*e^7 + 20*e^6 + 35*e^5 - 60*e^4 - 89*e^3 + 49*e^2 + 70*e - 9, 6*e^9 + 21*e^8 - 47*e^7 - 183*e^6 + 130*e^5 + 491*e^4 - 198*e^3 - 364*e^2 + 179*e - 20, -e^9 - 9*e^8 - 10*e^7 + 69*e^6 + 120*e^5 - 149*e^4 - 289*e^3 + 61*e^2 + 139*e - 8, e^9 + 12*e^8 + 25*e^7 - 82*e^6 - 249*e^5 + 144*e^4 + 638*e^3 - 55*e^2 - 419*e + 82, e^8 + 2*e^7 - 10*e^6 - 16*e^5 + 42*e^4 + 46*e^3 - 81*e^2 - 42*e + 40, 8*e^9 + 28*e^8 - 61*e^7 - 241*e^6 + 158*e^5 + 627*e^4 - 238*e^3 - 433*e^2 + 260*e - 19, -2*e^9 - 9*e^8 + 12*e^7 + 76*e^6 - 26*e^5 - 196*e^4 + 64*e^3 + 127*e^2 - 87*e + 15, 3*e^9 + 20*e^8 + 14*e^7 - 148*e^6 - 249*e^5 + 305*e^4 + 700*e^3 - 157*e^2 - 473*e + 86, -4*e^9 - 22*e^8 + 169*e^6 + 171*e^5 - 372*e^4 - 499*e^3 + 202*e^2 + 286*e - 53, 8*e^9 + 40*e^8 - 19*e^7 - 320*e^6 - 179*e^5 + 758*e^4 + 563*e^3 - 470*e^2 - 238*e + 55, 6*e^9 + 30*e^8 - 8*e^7 - 227*e^6 - 188*e^5 + 488*e^4 + 576*e^3 - 256*e^2 - 307*e + 61, 3*e^9 + 4*e^8 - 47*e^7 - 53*e^6 + 248*e^5 + 201*e^4 - 525*e^3 - 204*e^2 + 358*e - 50, -2*e^9 - 4*e^8 + 24*e^7 + 40*e^6 - 103*e^5 - 129*e^4 + 176*e^3 + 136*e^2 - 87*e - 8, -6*e^9 - 28*e^8 + 20*e^7 + 225*e^6 + 89*e^5 - 538*e^4 - 324*e^3 + 338*e^2 + 128*e - 28, -3*e^9 - 14*e^8 + 10*e^7 + 114*e^6 + 51*e^5 - 270*e^4 - 193*e^3 + 157*e^2 + 101*e - 24, -5*e^9 - 11*e^8 + 57*e^7 + 105*e^6 - 239*e^5 - 310*e^4 + 439*e^3 + 266*e^2 - 299*e + 24, -4*e^9 - 17*e^8 + 18*e^7 + 139*e^6 + 32*e^5 - 336*e^4 - 188*e^3 + 212*e^2 + 106*e - 20, 10*e^9 + 42*e^8 - 46*e^7 - 338*e^6 - 53*e^5 + 805*e^4 + 338*e^3 - 500*e^2 - 111*e + 34, e^9 + 8*e^8 + 12*e^7 - 50*e^6 - 139*e^5 + 54*e^4 + 355*e^3 + 76*e^2 - 206*e, -4*e^9 - 20*e^8 + 6*e^7 + 152*e^6 + 117*e^5 - 331*e^4 - 359*e^3 + 172*e^2 + 199*e - 22, -5*e^9 - 20*e^8 + 35*e^7 + 178*e^6 - 77*e^5 - 494*e^4 + 103*e^3 + 392*e^2 - 128*e + 2, 8*e^9 + 29*e^8 - 54*e^7 - 242*e^6 + 101*e^5 + 614*e^4 - 84*e^3 - 432*e^2 + 125*e + 4, 3*e^9 + 13*e^8 - 13*e^7 - 100*e^6 - 8*e^5 + 218*e^4 + 32*e^3 - 85*e^2 + 56*e - 31, -13*e^9 - 58*e^8 + 50*e^7 + 465*e^6 + 143*e^5 - 1106*e^4 - 602*e^3 + 694*e^2 + 255*e - 63, e^9 - 20*e^7 - 9*e^6 + 118*e^5 + 57*e^4 - 263*e^3 - 77*e^2 + 184*e - 22, 2*e^9 + 8*e^8 - 7*e^7 - 59*e^6 - 35*e^5 + 121*e^4 + 158*e^3 - 63*e^2 - 119*e + 18, -8*e^9 - 35*e^8 + 36*e^7 + 288*e^6 + 47*e^5 - 716*e^4 - 280*e^3 + 495*e^2 + 117*e - 61, -4*e^9 - 15*e^8 + 24*e^7 + 119*e^6 - 32*e^5 - 284*e^4 + 186*e^2 - 15*e - 3, 9*e^9 + 32*e^8 - 65*e^7 - 267*e^6 + 161*e^5 + 678*e^4 - 248*e^3 - 465*e^2 + 273*e - 36, -9*e^9 - 32*e^8 + 63*e^7 + 263*e^6 - 143*e^5 - 652*e^4 + 193*e^3 + 427*e^2 - 220*e + 30, -7*e^9 - 22*e^8 + 62*e^7 + 195*e^6 - 206*e^5 - 534*e^4 + 350*e^3 + 414*e^2 - 276*e + 20, -4*e^9 - 21*e^8 + 6*e^7 + 165*e^6 + 115*e^5 - 381*e^4 - 341*e^3 + 229*e^2 + 174*e - 38, -10*e^9 - 46*e^8 + 30*e^7 + 361*e^6 + 187*e^5 - 826*e^4 - 685*e^3 + 488*e^2 + 380*e - 84, 5*e^9 + 29*e^8 + 3*e^7 - 226*e^6 - 239*e^5 + 513*e^4 + 690*e^3 - 306*e^2 - 413*e + 79, -9*e^9 - 41*e^8 + 33*e^7 + 334*e^6 + 121*e^5 - 818*e^4 - 506*e^3 + 557*e^2 + 288*e - 82, 10*e^9 + 39*e^8 - 59*e^7 - 321*e^6 + 63*e^5 + 798*e^4 + 26*e^3 - 536*e^2 + 95*e + 4, 11*e^9 + 46*e^8 - 54*e^7 - 377*e^6 - 33*e^5 + 920*e^4 + 312*e^3 - 583*e^2 - 100*e + 15, 11*e^9 + 49*e^8 - 47*e^7 - 398*e^6 - 75*e^5 + 961*e^4 + 365*e^3 - 597*e^2 - 88*e + 18, e^9 + 7*e^8 + 9*e^7 - 45*e^6 - 117*e^5 + 71*e^4 + 314*e^3 - 32*e^2 - 197*e + 55, -2*e^9 - 7*e^8 + 14*e^7 + 54*e^6 - 34*e^5 - 107*e^4 + 67*e^3 + 4*e^2 - 95*e + 37, 13*e^9 + 54*e^8 - 63*e^7 - 433*e^6 - 30*e^5 + 1034*e^4 + 319*e^3 - 655*e^2 - 80*e + 30, -3*e^9 - 15*e^8 + 113*e^6 + 149*e^5 - 237*e^4 - 496*e^3 + 130*e^2 + 349*e - 72, -2*e^9 - 11*e^8 + 81*e^6 + 78*e^5 - 172*e^4 - 225*e^3 + 100*e^2 + 144*e - 40, 2*e^9 + 6*e^8 - 24*e^7 - 60*e^6 + 123*e^5 + 185*e^4 - 313*e^3 - 147*e^2 + 281*e - 33, -2*e^9 - 7*e^8 + 12*e^7 + 53*e^6 - 16*e^5 - 115*e^4 + 2*e^3 + 55*e^2 - 19*e - 5, 2*e^9 + 17*e^8 + 19*e^7 - 127*e^6 - 232*e^5 + 271*e^4 + 567*e^3 - 145*e^2 - 284*e + 55, 9*e^9 + 43*e^8 - 27*e^7 - 342*e^6 - 153*e^5 + 796*e^4 + 511*e^3 - 465*e^2 - 197*e + 53, -e^9 + 3*e^8 + 30*e^7 - 13*e^6 - 199*e^5 - 15*e^4 + 438*e^3 + 68*e^2 - 242*e + 26, 15*e^9 + 67*e^8 - 53*e^7 - 526*e^6 - 204*e^5 + 1202*e^4 + 793*e^3 - 691*e^2 - 346*e + 70, -5*e^9 - 16*e^8 + 43*e^7 + 139*e^6 - 142*e^5 - 369*e^4 + 263*e^3 + 262*e^2 - 238*e + 30, -e^9 - 5*e^8 + 3*e^7 + 39*e^6 + 16*e^5 - 76*e^4 - 49*e^3 - 17*e^2 + 8*e + 34, -10*e^9 - 51*e^8 + 20*e^7 + 407*e^6 + 261*e^5 - 951*e^4 - 819*e^3 + 557*e^2 + 409*e - 76, e^9 + e^8 - 20*e^7 - 16*e^6 + 134*e^5 + 61*e^4 - 359*e^3 - 22*e^2 + 290*e - 79, 3*e^9 + 11*e^8 - 20*e^7 - 93*e^6 + 29*e^5 + 233*e^4 + 13*e^3 - 145*e^2 + 1, -11*e^9 - 42*e^8 + 69*e^7 + 355*e^6 - 85*e^5 - 913*e^4 - 48*e^3 + 655*e^2 - 33*e - 30, e^9 - 2*e^8 - 34*e^7 - 8*e^6 + 236*e^5 + 105*e^4 - 567*e^3 - 134*e^2 + 379*e - 87, -e^9 - 8*e^8 - 6*e^7 + 64*e^6 + 90*e^5 - 148*e^4 - 221*e^3 + 76*e^2 + 101*e - 4, 2*e^9 + 7*e^8 - 10*e^7 - 51*e^6 - 9*e^5 + 103*e^4 + 102*e^3 - 48*e^2 - 121*e + 11, -5*e^9 - 27*e^8 + 9*e^7 + 224*e^6 + 140*e^5 - 565*e^4 - 434*e^3 + 410*e^2 + 226*e - 61, 6*e^9 + 32*e^8 - 6*e^7 - 251*e^6 - 204*e^5 + 567*e^4 + 588*e^3 - 300*e^2 - 247*e + 37, -7*e^9 - 29*e^8 + 33*e^7 + 229*e^6 + 19*e^5 - 530*e^4 - 163*e^3 + 291*e^2 + 24*e + 12, e^9 + 10*e^8 + 12*e^7 - 81*e^6 - 140*e^5 + 203*e^4 + 359*e^3 - 161*e^2 - 230*e + 62, -6*e^9 - 26*e^8 + 24*e^7 + 201*e^6 + 52*e^5 - 445*e^4 - 232*e^3 + 228*e^2 + 92*e - 7, -11*e^9 - 45*e^8 + 60*e^7 + 368*e^6 - 36*e^5 - 900*e^4 - 79*e^3 + 581*e^2 - 102*e - 18, 14*e^9 + 55*e^8 - 84*e^7 - 458*e^6 + 95*e^5 + 1166*e^4 + 48*e^3 - 852*e^2 + 86*e + 52, 7*e^9 + 27*e^8 - 45*e^7 - 231*e^6 + 65*e^5 + 604*e^4 - 4*e^3 - 440*e^2 + 51*e + 6, -12*e^9 - 52*e^8 + 55*e^7 + 425*e^6 + 60*e^5 - 1034*e^4 - 374*e^3 + 660*e^2 + 112*e - 48, 9*e^9 + 38*e^8 - 41*e^7 - 304*e^6 - 45*e^5 + 716*e^4 + 281*e^3 - 421*e^2 - 81*e + 10, 2*e^9 + 14*e^8 + 8*e^7 - 112*e^6 - 153*e^5 + 268*e^4 + 411*e^3 - 181*e^2 - 225*e + 41, -11*e^9 - 49*e^8 + 46*e^7 + 394*e^6 + 74*e^5 - 947*e^4 - 341*e^3 + 608*e^2 + 63*e - 46, -9*e^9 - 32*e^8 + 66*e^7 + 274*e^6 - 151*e^5 - 713*e^4 + 168*e^3 + 501*e^2 - 173*e + 34, 13*e^9 + 54*e^8 - 57*e^7 - 423*e^6 - 84*e^5 + 968*e^4 + 463*e^3 - 559*e^2 - 174*e + 25, 20*e^9 + 76*e^8 - 122*e^7 - 625*e^6 + 142*e^5 + 1542*e^4 + 75*e^3 - 1022*e^2 + 114*e + 16, -4*e^9 - 17*e^8 + 20*e^7 + 140*e^6 + 7*e^5 - 345*e^4 - 102*e^3 + 225*e^2 + 33*e - 7, -2*e^9 + 4*e^8 + 57*e^7 - 7*e^6 - 382*e^5 - 80*e^4 + 888*e^3 + 153*e^2 - 562*e + 80, e^9 + 8*e^8 + 11*e^7 - 55*e^6 - 129*e^5 + 107*e^4 + 328*e^3 - 68*e^2 - 198*e + 48, -3*e^9 - 16*e^8 + 2*e^7 + 123*e^6 + 110*e^5 - 256*e^4 - 298*e^3 + 72*e^2 + 88*e + 16, 10*e^9 + 39*e^8 - 58*e^7 - 319*e^6 + 47*e^5 + 771*e^4 + 88*e^3 - 462*e^2 + 42*e - 38, e^9 + 2*e^8 - 20*e^7 - 33*e^6 + 131*e^5 + 146*e^4 - 352*e^3 - 169*e^2 + 303*e - 14, -e^9 - 7*e^8 - 12*e^7 + 44*e^6 + 158*e^5 - 49*e^4 - 453*e^3 - 27*e^2 + 295*e - 59, 4*e^9 + 16*e^8 - 24*e^7 - 137*e^6 + 25*e^5 + 368*e^4 + 24*e^3 - 304*e^2 + 18*e + 47, -2*e^9 - e^8 + 37*e^7 + 23*e^6 - 217*e^5 - 121*e^4 + 484*e^3 + 175*e^2 - 316*e + 32, -3*e^9 - e^8 + 53*e^7 + 23*e^6 - 294*e^5 - 98*e^4 + 620*e^3 + 80*e^2 - 393*e + 71, -8*e^9 - 34*e^8 + 42*e^7 + 283*e^6 - 10*e^5 - 716*e^4 - 113*e^3 + 497*e^2 - 19*e - 14, -e^9 - 6*e^8 - 3*e^7 + 44*e^6 + 68*e^5 - 86*e^4 - 185*e^3 + 23*e^2 + 114*e - 6, 13*e^9 + 55*e^8 - 51*e^7 - 425*e^6 - 137*e^5 + 943*e^4 + 602*e^3 - 499*e^2 - 262*e + 41, 8*e^9 + 30*e^8 - 50*e^7 - 244*e^6 + 77*e^5 + 597*e^4 - 49*e^3 - 384*e^2 + 114*e - 20, -16*e^9 - 70*e^8 + 73*e^7 + 569*e^6 + 63*e^5 - 1387*e^4 - 408*e^3 + 894*e^2 + 71*e - 22, 15*e^9 + 67*e^8 - 61*e^7 - 545*e^6 - 138*e^5 + 1336*e^4 + 635*e^3 - 911*e^2 - 276*e + 96, 2*e^9 - 2*e^8 - 50*e^7 - 5*e^6 + 333*e^5 + 97*e^4 - 800*e^3 - 144*e^2 + 543*e - 100, 2*e^9 + 14*e^8 + 17*e^7 - 93*e^6 - 233*e^5 + 143*e^4 + 648*e^3 - 9*e^2 - 433*e + 91, -4*e^9 - 25*e^8 - 6*e^7 + 193*e^6 + 206*e^5 - 432*e^4 - 531*e^3 + 240*e^2 + 239*e - 59, e^9 + 3*e^8 - 9*e^7 - 31*e^6 + 18*e^5 + 103*e^4 + 20*e^3 - 123*e^2 - 49*e + 28, -2*e^9 - 4*e^8 + 34*e^7 + 59*e^6 - 192*e^5 - 243*e^4 + 442*e^3 + 271*e^2 - 319*e + 17, 10*e^9 + 33*e^8 - 77*e^7 - 271*e^6 + 211*e^5 + 651*e^4 - 341*e^3 - 353*e^2 + 353*e - 85];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {13, 13, -w^2 + 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;