/*
  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, 4, 5, -4, -2, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {27, 27, w^3 - 2*w^2 - 2*w + 3}>;

primesArray := [
[3, 3, w - 1],
[5, 5, w^2 - w - 2],
[7, 7, w^4 - 2*w^3 - 3*w^2 + 4*w + 2],
[13, 13, w^3 - 2*w^2 - 2*w + 2],
[29, 29, -w^4 + 3*w^3 + 2*w^2 - 7*w - 1],
[29, 29, -w^2 + 2*w + 3],
[31, 31, w^4 - 2*w^3 - 3*w^2 + 5*w],
[31, 31, w^3 - 2*w^2 - 3*w + 2],
[32, 2, 2],
[43, 43, -w^2 - w + 4],
[53, 53, -w^4 + w^3 + 6*w^2 - 2*w - 5],
[53, 53, -w^4 + 2*w^3 + 4*w^2 - 5*w - 2],
[73, 73, w^4 - w^3 - 6*w^2 + 2*w + 6],
[73, 73, w^3 - w^2 - 4*w + 2],
[81, 3, 2*w^4 - 5*w^3 - 4*w^2 + 9*w + 2],
[83, 83, w^3 - w^2 - 5*w],
[89, 89, -w^4 + 2*w^3 + 4*w^2 - 4*w - 2],
[97, 97, w^4 - 3*w^3 - 2*w^2 + 6*w + 2],
[101, 101, -w^4 + 3*w^3 + 3*w^2 - 7*w - 3],
[103, 103, -w^4 + w^3 + 6*w^2 - w - 7],
[103, 103, 2*w^4 - 4*w^3 - 6*w^2 + 8*w + 1],
[103, 103, w^4 - w^3 - 6*w^2 + 3*w + 4],
[107, 107, -w^4 + 2*w^3 + 3*w^2 - 5*w + 2],
[109, 109, w^3 - 4*w - 1],
[109, 109, -w^2 + 2*w + 4],
[131, 131, w^4 - w^3 - 6*w^2 + w + 6],
[137, 137, w^3 - w^2 - 5*w + 3],
[149, 149, w^4 - 3*w^3 - 3*w^2 + 9*w + 6],
[149, 149, -w^4 + 6*w^2 + 3*w],
[163, 163, 2*w^2 - w - 5],
[167, 167, -2*w^4 + 3*w^3 + 8*w^2 - 4*w - 3],
[169, 13, w^4 - 3*w^3 + 6*w - 3],
[169, 13, w^3 - 4*w - 4],
[173, 173, w^4 - w^3 - 4*w^2 + 2*w + 1],
[173, 173, -w^4 + 3*w^3 + 3*w^2 - 7*w - 5],
[173, 173, w^4 - w^3 - 4*w^2 + w - 1],
[179, 179, w^2 - 3*w - 2],
[191, 191, -w^4 + 3*w^3 + w^2 - 6*w - 1],
[193, 193, -w^3 + 2*w^2 + w - 4],
[193, 193, -w^4 + 3*w^3 + 3*w^2 - 8*w - 1],
[199, 199, w^4 - 3*w^3 - w^2 + 7*w],
[199, 199, -2*w^4 + 5*w^3 + 5*w^2 - 10*w - 2],
[199, 199, w^4 - 3*w^3 + 5*w - 4],
[211, 211, 2*w^3 - 2*w^2 - 7*w - 1],
[211, 211, -2*w^3 + 3*w^2 + 5*w - 2],
[223, 223, -2*w^4 + 3*w^3 + 10*w^2 - 7*w - 9],
[227, 227, w^3 - w^2 - 4*w + 3],
[229, 229, w^4 - 4*w^3 + 10*w - 5],
[229, 229, -2*w^4 + 4*w^3 + 7*w^2 - 9*w - 8],
[239, 239, -2*w^4 + 3*w^3 + 7*w^2 - 3*w - 3],
[241, 241, w^3 - w^2 - 2*w - 2],
[257, 257, -w^4 + w^3 + 6*w^2 - 4*w - 4],
[263, 263, 2*w^3 - 4*w^2 - 5*w + 6],
[269, 269, w^3 - 2*w^2 - 4*w + 4],
[269, 269, -2*w^4 + 2*w^3 + 11*w^2 - 3*w - 10],
[271, 271, w^3 - 5*w],
[281, 281, w^4 - w^3 - 5*w^2 + 3*w + 3],
[281, 281, -2*w^4 + 4*w^3 + 6*w^2 - 6*w - 3],
[281, 281, w^3 - 5*w - 1],
[283, 283, -2*w^4 + 4*w^3 + 7*w^2 - 8*w - 3],
[289, 17, -2*w^4 + 3*w^3 + 8*w^2 - 5*w - 2],
[293, 293, 2*w^4 - 5*w^3 - 7*w^2 + 14*w + 9],
[311, 311, -w^4 + 2*w^3 + 2*w^2 - 4*w + 3],
[313, 313, w^3 - 6*w],
[331, 331, w^4 - 2*w^3 - 2*w^2 + w - 2],
[359, 359, w^4 - w^3 - 5*w^2 + 2*w - 1],
[373, 373, -w^4 + 3*w^3 + 2*w^2 - 8*w - 3],
[373, 373, 2*w^3 - 4*w^2 - 3*w + 3],
[379, 379, -w^4 + 4*w^3 - w^2 - 9*w + 3],
[379, 379, -w^4 + 3*w^3 + w^2 - 6*w - 2],
[379, 379, w^4 + w^3 - 7*w^2 - 6*w + 3],
[383, 383, -w^4 + 2*w^3 + 3*w^2 - 6*w - 2],
[383, 383, w^4 - 5*w^2 - 3*w + 3],
[383, 383, 2*w^4 - 5*w^3 - 3*w^2 + 8*w],
[389, 389, 3*w^4 - 7*w^3 - 8*w^2 + 16*w + 1],
[397, 397, -2*w^4 + 3*w^3 + 8*w^2 - 3*w - 7],
[397, 397, -w^4 + 2*w^3 + 3*w^2 - 4*w - 5],
[397, 397, -2*w^4 + 3*w^3 + 7*w^2 - 4*w - 3],
[409, 409, -w^4 + w^3 + 7*w^2 - 5*w - 9],
[419, 419, w^4 - 2*w^3 - w^2 + 2*w - 2],
[419, 419, 2*w^4 - 3*w^3 - 7*w^2 + 5*w + 2],
[419, 419, -w^4 + 4*w^3 - 9*w - 2],
[433, 433, -w^4 + 3*w^3 - 8*w + 4],
[433, 433, w^4 - 7*w^2 - w + 6],
[439, 439, 3*w^4 - 6*w^3 - 10*w^2 + 11*w + 7],
[449, 449, -2*w^4 + 5*w^3 + 5*w^2 - 11*w - 4],
[457, 457, -2*w^4 + 5*w^3 + 4*w^2 - 10*w - 2],
[463, 463, 2*w^4 - 5*w^3 - 4*w^2 + 8*w + 1],
[467, 467, 2*w^4 - 4*w^3 - 6*w^2 + 6*w + 1],
[479, 479, 3*w^4 - 4*w^3 - 12*w^2 + 4*w + 7],
[479, 479, 2*w^4 - 6*w^3 - 4*w^2 + 14*w + 1],
[491, 491, w^4 - 6*w^2 - w + 4],
[503, 503, 2*w^4 - w^3 - 11*w^2 - 3*w + 8],
[503, 503, 2*w^3 - 5*w^2 - 2*w + 6],
[509, 509, w^4 - 5*w^2 - 6*w],
[509, 509, w^4 - 3*w^3 - 3*w^2 + 9*w + 1],
[523, 523, 3*w^4 - 7*w^3 - 7*w^2 + 12*w + 3],
[523, 523, 3*w^4 - 8*w^3 - 7*w^2 + 17*w + 3],
[523, 523, w^4 - 7*w^2 - 2*w + 4],
[529, 23, -w^4 + w^3 + 7*w^2 - 2*w - 9],
[541, 541, w^4 - 3*w^3 - 3*w^2 + 5*w + 2],
[547, 547, -w^3 + 3*w^2 + 3*w - 4],
[557, 557, 2*w^3 - 4*w^2 - 6*w + 9],
[577, 577, 2*w^4 - 2*w^3 - 11*w^2 + 5*w + 10],
[577, 577, w^4 - 4*w^3 - w^2 + 9*w],
[577, 577, 2*w^4 - 3*w^3 - 9*w^2 + 4*w + 8],
[577, 577, w^4 - 7*w^2 - 3*w + 7],
[577, 577, -3*w^4 + 3*w^3 + 13*w^2 - w - 4],
[593, 593, 3*w - 2],
[601, 601, 2*w^4 - 4*w^3 - 7*w^2 + 5*w + 6],
[601, 601, -w^4 + 3*w^3 + w^2 - 7*w - 1],
[607, 607, w^3 - w^2 - 2*w - 3],
[625, 5, 2*w^4 - 5*w^3 - 8*w^2 + 14*w + 11],
[631, 631, -w^2 + w - 2],
[631, 631, w^4 - 7*w^2 - w + 8],
[643, 643, -2*w^4 + 4*w^3 + 9*w^2 - 10*w - 8],
[647, 647, 3*w^4 - 6*w^3 - 10*w^2 + 12*w + 8],
[653, 653, -3*w^4 + 6*w^3 + 11*w^2 - 12*w - 9],
[653, 653, w^4 - w^3 - 2*w^2 - 3*w - 3],
[653, 653, -2*w^4 + 4*w^3 + 7*w^2 - 6*w - 8],
[701, 701, w^4 - w^3 - 2*w^2 - w - 5],
[719, 719, 2*w^4 - 5*w^3 - 6*w^2 + 14*w + 2],
[727, 727, 2*w^4 - 5*w^3 - 5*w^2 + 9*w + 3],
[733, 733, 2*w^4 - 5*w^3 - 5*w^2 + 13*w + 3],
[739, 739, -2*w^3 + 4*w^2 + 4*w - 5],
[743, 743, w^4 - 2*w^3 - 3*w^2 + 6*w - 3],
[743, 743, 2*w^4 - 2*w^3 - 11*w^2 + 2*w + 10],
[757, 757, -w^4 + 2*w^3 + 5*w^2 - 6*w - 10],
[757, 757, 2*w^4 - 6*w^3 - 2*w^2 + 11*w - 4],
[761, 761, 3*w^3 - 4*w^2 - 9*w + 3],
[769, 769, 3*w^3 - 5*w^2 - 9*w + 7],
[769, 769, -2*w^4 + 4*w^3 + 7*w^2 - 11*w - 2],
[773, 773, 2*w^4 - 5*w^3 - 6*w^2 + 12*w + 1],
[773, 773, 3*w^4 - 4*w^3 - 12*w^2 + 3*w + 5],
[773, 773, 2*w^3 - w^2 - 10*w - 4],
[797, 797, -w^4 + 4*w^3 + 2*w^2 - 12*w - 3],
[811, 811, w^4 - 2*w^3 - 6*w^2 + 8*w + 4],
[823, 823, w^4 - 4*w^3 - w^2 + 10*w + 1],
[823, 823, 2*w^4 - 4*w^3 - 4*w^2 + 4*w - 3],
[827, 827, -w^4 + 2*w^3 + 2*w^2 - w + 3],
[827, 827, 2*w^4 - 4*w^3 - 5*w^2 + 7*w + 2],
[829, 829, -w^4 + 4*w^3 + 2*w^2 - 10*w - 2],
[829, 829, 3*w^3 - 3*w^2 - 11*w - 2],
[829, 829, w^4 - 2*w^3 - 3*w^2 + 2*w - 3],
[839, 839, -w^4 + 5*w^3 - 3*w^2 - 10*w + 5],
[839, 839, -w^4 + w^3 + 5*w^2 - 9],
[853, 853, 2*w^4 - 6*w^3 - 5*w^2 + 15*w + 4],
[859, 859, -3*w^4 + 8*w^3 + 6*w^2 - 18*w + 5],
[863, 863, w^4 - w^3 - 6*w^2 + 5*w + 2],
[877, 877, 3*w^3 - 4*w^2 - 8*w + 1],
[881, 881, 3*w^4 - 7*w^3 - 9*w^2 + 15*w + 3],
[881, 881, -w^3 + 7*w + 1],
[883, 883, -2*w^4 + 6*w^3 + 3*w^2 - 15*w],
[887, 887, w^4 - w^3 - 5*w^2 + w - 1],
[907, 907, 3*w^4 - 5*w^3 - 11*w^2 + 9*w + 5],
[919, 919, w^4 - 2*w^3 - 6*w^2 + 8*w + 7],
[937, 937, 2*w^4 - 2*w^3 - 9*w^2 + 2*w + 3],
[937, 937, -3*w^4 + 8*w^3 + 6*w^2 - 16*w + 1],
[937, 937, 3*w^4 - 5*w^3 - 11*w^2 + 7*w + 4],
[947, 947, w^3 - w^2 - 7*w + 2],
[947, 947, -w^4 + 2*w^3 + 4*w^2 - 7*w - 3],
[967, 967, -w^4 + 4*w^3 - 11*w],
[971, 971, 3*w^4 - 6*w^3 - 8*w^2 + 10*w],
[971, 971, w^4 - 8*w^2 + 2*w + 9],
[971, 971, w^4 - 4*w^3 + 2*w^2 + 7*w - 4],
[977, 977, -w^4 + 4*w^3 - 12*w + 5],
[983, 983, -w^4 + w^3 + 4*w^2 + 3*w - 2],
[983, 983, -2*w^4 + 3*w^3 + 8*w^2 - 6*w - 4],
[997, 997, 2*w^4 - 4*w^3 - 5*w^2 + 5*w - 3],
[997, 997, w^4 + w^3 - 5*w^2 - 9*w - 2]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^10 - 46*x^8 + 786*x^6 - 6301*x^4 + 23875*x^2 - 34347;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [0, 7/178*e^8 - 275/178*e^6 + 3681/178*e^4 - 10090/89*e^2 + 38649/178, e, 379/9523*e^9 - 15080/9523*e^7 + 204055/9523*e^5 - 1117561/9523*e^3 + 2105930/9523*e, 450/9523*e^9 - 17704/9523*e^7 + 236000/9523*e^5 - 1269505/9523*e^3 + 2344785/9523*e, 127/19046*e^9 - 4023/19046*e^7 + 33803/19046*e^5 - 15716/9523*e^3 - 290867/19046*e, -154/9523*e^9 + 6228/9523*e^7 - 86055/9523*e^5 + 478047/9523*e^3 - 881161/9523*e, 9/178*e^8 - 379/178*e^6 + 5521/178*e^4 - 16393/89*e^2 + 66805/178, -71/9523*e^9 + 2624/9523*e^7 - 31945/9523*e^5 + 151944/9523*e^3 - 219809/9523*e, -592/9523*e^9 + 22952/9523*e^7 - 299890/9523*e^5 + 1573393/9523*e^3 - 2803449/9523*e, 7/89*e^8 - 275/89*e^6 + 3681/89*e^4 - 20180/89*e^2 + 38649/89, 7/89*e^8 - 275/89*e^6 + 3681/89*e^4 - 20180/89*e^2 + 38649/89, -142/9523*e^9 + 5248/9523*e^7 - 63890/9523*e^5 + 303888/9523*e^3 - 458664/9523*e, 157/19046*e^9 - 6473/19046*e^7 + 93977/19046*e^5 - 288172/9523*e^3 + 1246287/19046*e, -21/178*e^8 + 825/178*e^6 - 11043/178*e^4 + 30181/89*e^2 - 113455/178, -631/19046*e^9 + 26137/19046*e^7 - 374307/19046*e^5 + 1101845/9523*e^3 - 4483681/19046*e, -225/9523*e^9 + 8852/9523*e^7 - 118000/9523*e^5 + 639514/9523*e^3 - 1234292/9523*e, -12/9523*e^9 + 980/9523*e^7 - 22165/9523*e^5 + 174159/9523*e^3 - 412974/9523*e, 35/178*e^8 - 1375/178*e^6 + 18227/178*e^4 - 48225/89*e^2 + 171885/178, e^2 - 8, -631/19046*e^9 + 26137/19046*e^7 - 374307/19046*e^5 + 1101845/9523*e^3 - 4464635/19046*e, -127/19046*e^9 + 4023/19046*e^7 - 33803/19046*e^5 + 15716/9523*e^3 + 252775/19046*e, -7/89*e^8 + 275/89*e^6 - 3681/89*e^4 + 20180/89*e^2 - 38649/89, -9/178*e^8 + 379/178*e^6 - 5521/178*e^4 + 16571/89*e^2 - 69653/178, -49/178*e^8 + 1925/178*e^6 - 25589/178*e^4 + 68494/89*e^2 - 250607/178, -1/89*e^8 + 52/89*e^6 - 920/89*e^4 + 6125/89*e^2 - 12120/89, 601/19046*e^9 - 23687/19046*e^7 + 314133/19046*e^5 - 829389/9523*e^3 + 2965573/19046*e, -355/9523*e^9 + 13120/9523*e^7 - 159725/9523*e^5 + 769243/9523*e^3 - 1279982/9523*e, -631/19046*e^9 + 26137/19046*e^7 - 374307/19046*e^5 + 1101845/9523*e^3 - 4540819/19046*e, 293/9523*e^9 - 11231/9523*e^7 + 142023/9523*e^5 - 683638/9523*e^3 + 1031837/9523*e, -127/19046*e^9 + 4023/19046*e^7 - 33803/19046*e^5 + 15716/9523*e^3 + 290867/19046*e, -12/9523*e^9 + 980/9523*e^7 - 22165/9523*e^5 + 174159/9523*e^3 - 412974/9523*e, 7/89*e^8 - 275/89*e^6 + 3681/89*e^4 - 20180/89*e^2 + 38293/89, 8/89*e^8 - 327/89*e^6 + 4601/89*e^4 - 26572/89*e^2 + 53439/89, -1217/19046*e^9 + 48599/19046*e^7 - 658353/19046*e^5 + 1785483/9523*e^3 - 6528309/19046*e, -7/178*e^8 + 275/178*e^6 - 3503/178*e^4 + 8132/89*e^2 - 21561/178, 37/178*e^8 - 1479/178*e^6 + 20245/178*e^4 - 56575/89*e^2 + 216951/178, -293/19046*e^9 + 11231/19046*e^7 - 142023/19046*e^5 + 351342/9523*e^3 - 1241343/19046*e, -21/178*e^8 + 825/178*e^6 - 11043/178*e^4 + 30359/89*e^2 - 116303/178, -20/89*e^8 + 773/89*e^6 - 10034/89*e^4 + 51923/89*e^2 - 90833/89, -367/9523*e^9 + 14100/9523*e^7 - 181890/9523*e^5 + 943402/9523*e^3 - 1664387/9523*e, -9/89*e^8 + 379/89*e^6 - 5521/89*e^4 + 32786/89*e^2 - 66271/89, 1646/9523*e^9 - 64588/9523*e^7 + 857945/9523*e^5 - 4599973/9523*e^3 + 8440841/9523*e, 21/178*e^8 - 825/178*e^6 + 10865/178*e^4 - 28312/89*e^2 + 100283/178, 743/19046*e^9 - 28935/19046*e^7 + 378023/19046*e^5 - 971810/9523*e^3 + 3233777/19046*e, 9/178*e^8 - 379/178*e^6 + 5521/178*e^4 - 16393/89*e^2 + 64669/178, 13/89*e^8 - 498/89*e^6 + 6353/89*e^4 - 32010/89*e^2 + 54498/89, -151/9523*e^9 + 5983/9523*e^7 - 78133/9523*e^5 + 379750/9523*e^3 - 516035/9523*e, -35/89*e^8 + 1375/89*e^6 - 18316/89*e^4 + 98497/89*e^2 - 180607/89, -63/178*e^8 + 2475/178*e^6 - 32951/178*e^4 + 88852/89*e^2 - 330753/178, 13/89*e^8 - 498/89*e^6 + 6353/89*e^4 - 31921/89*e^2 + 53786/89, 9/178*e^8 - 379/178*e^6 + 5521/178*e^4 - 16215/89*e^2 + 62889/178, -613/19046*e^9 + 24667/19046*e^7 - 345821/19046*e^5 + 1025983/9523*e^3 - 4426077/19046*e, -36/89*e^8 + 1427/89*e^6 - 19147/89*e^4 + 103109/89*e^2 - 188277/89, -450/9523*e^9 + 17704/9523*e^7 - 236000/9523*e^5 + 1269505/9523*e^3 - 2344785/9523*e, -20/89*e^8 + 773/89*e^6 - 10034/89*e^4 + 52190/89*e^2 - 92969/89, 3171/19046*e^9 - 125643/19046*e^7 + 1697931/19046*e^5 - 4682554/9523*e^3 + 17940893/19046*e, -35/178*e^8 + 1375/178*e^6 - 18227/178*e^4 + 48492/89*e^2 - 176157/178, 675/9523*e^9 - 26556/9523*e^7 + 354000/9523*e^5 - 1899496/9523*e^3 + 3483847/9523*e, -4, -2455/19046*e^9 + 98913/19046*e^7 - 1362637/19046*e^5 + 3815014/9523*e^3 - 14745907/19046*e, 20/89*e^8 - 773/89*e^6 + 10034/89*e^4 - 51923/89*e^2 + 91278/89, 364/9523*e^9 - 13855/9523*e^7 + 173968/9523*e^5 - 845105/9523*e^3 + 1337353/9523*e, -657/9523*e^9 + 25086/9523*e^7 - 315991/9523*e^5 + 1528743/9523*e^3 - 2350144/9523*e, -2437/19046*e^9 + 97443/19046*e^7 - 1334151/19046*e^5 + 3739152/9523*e^3 - 14631165/19046*e, 33/178*e^8 - 1271/178*e^6 + 16387/178*e^4 - 41833/89*e^2 + 143373/178, -79/19046*e^9 + 103/19046*e^7 + 54857/19046*e^5 - 332602/9523*e^3 + 1885625/19046*e, -2/89*e^8 + 104/89*e^6 - 1840/89*e^4 + 12784/89*e^2 - 29936/89, -1341/19046*e^9 + 52377/19046*e^7 - 693757/19046*e^5 + 1871088/9523*e^3 - 7024599/19046*e, -15/178*e^8 + 513/178*e^6 - 5345/178*e^4 + 9225/89*e^2 - 11543/178, -14/89*e^8 + 550/89*e^6 - 7362/89*e^4 + 40360/89*e^2 - 76586/89, -45/178*e^8 + 1717/178*e^6 - 21909/178*e^4 + 55888/89*e^2 - 194295/178, 50/89*e^8 - 1977/89*e^6 + 26598/89*e^4 - 145427/89*e^2 + 274653/89, 136/9523*e^9 - 4758/9523*e^7 + 48046/9523*e^5 - 107294/9523*e^3 - 233496/9523*e, -7/178*e^8 + 275/178*e^6 - 3503/178*e^4 + 8132/89*e^2 - 21561/178, -4237/19046*e^9 + 168259/19046*e^7 - 2278151/19046*e^5 + 6278162/9523*e^3 - 23953167/19046*e, 583/19046*e^9 - 22217/19046*e^7 + 285647/19046*e^5 - 753527/9523*e^3 + 2927015/19046*e, -1649/9523*e^9 + 64833/9523*e^7 - 865867/9523*e^5 + 4707793/9523*e^3 - 8910720/9523*e, -1664/9523*e^9 + 66058/9523*e^7 - 895954/9523*e^5 + 4970726/9523*e^3 - 9603113/9523*e, -39/178*e^8 + 1583/178*e^6 - 22085/178*e^4 + 62967/89*e^2 - 247065/178, 49/178*e^8 - 1925/178*e^6 + 25589/178*e^4 - 68672/89*e^2 + 251853/178, -8/89*e^8 + 327/89*e^6 - 4512/89*e^4 + 24347/89*e^2 - 42759/89, -305/19046*e^9 + 12211/19046*e^7 - 173711/19046*e^5 + 547936/9523*e^3 - 2625663/19046*e, 758/9523*e^9 - 30160/9523*e^7 + 408110/9523*e^5 - 2235122/9523*e^3 + 4173768/9523*e, 1661/19046*e^9 - 65813/19046*e^7 + 878509/19046*e^5 - 2317177/9523*e^3 + 8038089/19046*e, -580/9523*e^9 + 21972/9523*e^7 - 277725/9523*e^5 + 1408757/9523*e^3 - 2514274/9523*e, -53/178*e^8 + 2133/178*e^6 - 29269/178*e^4 + 80922/89*e^2 - 303715/178, 284/9523*e^9 - 10496/9523*e^7 + 127780/9523*e^5 - 607776/9523*e^3 + 917328/9523*e, 1945/9523*e^9 - 76309/9523*e^7 + 1015812/9523*e^5 - 5480205/9523*e^3 + 10155315/9523*e, -30/89*e^8 + 1204/89*e^6 - 16475/89*e^4 + 91279/89*e^2 - 174030/89, -225/9523*e^9 + 8852/9523*e^7 - 118000/9523*e^5 + 639514/9523*e^3 - 1262861/9523*e, -7/89*e^8 + 275/89*e^6 - 3681/89*e^4 + 20180/89*e^2 - 37581/89, 459/9523*e^9 - 18439/9523*e^7 + 250243/9523*e^5 - 1335844/9523*e^3 + 2278357/9523*e, 59/178*e^8 - 2267/178*e^6 + 29271/178*e^4 - 75534/89*e^2 + 261981/178, -51/178*e^8 + 2029/178*e^6 - 27251/178*e^4 + 72839/89*e^2 - 260607/178, -2253/9523*e^9 + 88765/9523*e^7 - 1187922/9523*e^5 + 6455345/9523*e^3 - 12127143/9523*e, -35/178*e^8 + 1375/178*e^6 - 18227/178*e^4 + 47958/89*e^2 - 167257/178, 65/178*e^8 - 2579/178*e^6 + 34791/178*e^4 - 94888/89*e^2 + 354637/178, -1942/9523*e^9 + 76064/9523*e^7 - 1007890/9523*e^5 + 5381908/9523*e^3 - 9780666/9523*e, 9/89*e^8 - 379/89*e^6 + 5521/89*e^4 - 32964/89*e^2 + 68941/89, 2793/19046*e^9 - 113819/19046*e^7 + 1594921/19046*e^5 - 4565517/9523*e^3 + 18026337/19046*e, -1309/9523*e^9 + 52938/9523*e^7 - 736229/9523*e^5 + 4201483/9523*e^3 - 8294562/9523*e, 1152/9523*e^9 - 46465/9523*e^7 + 642252/9523*e^5 - 3625139/9523*e^3 + 7019706/9523*e, 11/178*e^8 - 483/178*e^6 + 7183/178*e^4 - 20916/89*e^2 + 78941/178, -36/89*e^8 + 1427/89*e^6 - 19236/89*e^4 + 105156/89*e^2 - 197533/89, -79/178*e^8 + 3129/178*e^6 - 41975/178*e^4 + 113021/89*e^2 - 412355/178, 4799/19046*e^9 - 188761/19046*e^7 + 2517867/19046*e^5 - 6787641/9523*e^3 + 25133755/19046*e, -441/9523*e^9 + 16969/9523*e^7 - 221757/9523*e^5 + 1203166/9523*e^3 - 2430259/9523*e, -1/178*e^8 - 37/178*e^6 + 2017/178*e^4 - 10777/89*e^2 + 62907/178, -21/89*e^8 + 825/89*e^6 - 11043/89*e^4 + 60540/89*e^2 - 116837/89, 45/89*e^8 - 1806/89*e^6 + 24757/89*e^4 - 138209/89*e^2 + 267008/89, -2733/19046*e^9 + 108919/19046*e^7 - 1474573/19046*e^5 + 4020605/9523*e^3 - 14952029/19046*e, 773/19046*e^9 - 31385/19046*e^7 + 438197/19046*e^5 - 1244266/9523*e^3 + 4694747/19046*e, 604/9523*e^9 - 23932/9523*e^7 + 322055/9523*e^5 - 1757075/9523*e^3 + 3397360/9523*e, 1/178*e^8 + 37/178*e^6 - 1839/178*e^4 + 8641/89*e^2 - 40835/178, 237/9523*e^9 - 9832/9523*e^7 + 140165/9523*e^5 - 813673/9523*e^3 + 1694881/9523*e, -3/89*e^8 + 156/89*e^6 - 2849/89*e^4 + 21134/89*e^2 - 51312/89, 65/178*e^8 - 2579/178*e^6 + 34791/178*e^4 - 94710/89*e^2 + 352857/178, -56/89*e^8 + 2200/89*e^6 - 29270/89*e^4 + 156990/89*e^2 - 286764/89, 17/178*e^8 - 617/178*e^6 + 7185/178*e^4 - 15528/89*e^2 + 40767/178, 51/178*e^8 - 2029/178*e^6 + 27607/178*e^4 - 76755/89*e^2 + 295317/178, 31/178*e^8 - 1167/178*e^6 + 14725/178*e^4 - 37399/89*e^2 + 128745/178, 3/178*e^8 - 67/178*e^6 + 1/178*e^4 + 2694/89*e^2 - 20155/178, 272/9523*e^9 - 9516/9523*e^7 + 105615/9523*e^5 - 433617/9523*e^3 + 523400/9523*e, -325/9523*e^9 + 10670/9523*e^7 - 99551/9523*e^5 + 205285/9523*e^3 + 438109/9523*e, -4/89*e^8 + 208/89*e^6 - 3769/89*e^4 + 27793/89*e^2 - 68772/89, 20/89*e^8 - 773/89*e^6 + 9945/89*e^4 - 49965/89*e^2 + 84069/89, -54/89*e^8 + 2096/89*e^6 - 27430/89*e^4 + 144028/89*e^2 - 255760/89, -14/89*e^8 + 550/89*e^6 - 7273/89*e^4 + 38491/89*e^2 - 71068/89, 3141/19046*e^9 - 123193/19046*e^7 + 1637757/19046*e^5 - 4410098/9523*e^3 + 16479923/19046*e, -15/214*e^9 + 583/214*e^7 - 7617/214*e^5 + 19919/107*e^3 - 70723/214*e, -2383/19046*e^9 + 93033/19046*e^7 - 1229647/19046*e^5 + 3292537/9523*e^3 - 12153787/19046*e, 2005/19046*e^9 - 81209/19046*e^7 + 1126637/19046*e^5 - 3185023/9523*e^3 + 12372553/19046*e, -41/89*e^8 + 1598/89*e^6 - 21077/89*e^4 + 112463/89*e^2 - 205890/89, 21/89*e^8 - 825/89*e^6 + 10865/89*e^4 - 56090/89*e^2 + 95655/89, 36/89*e^8 - 1427/89*e^6 + 19236/89*e^4 - 105601/89*e^2 + 202695/89, 1009/19046*e^9 - 37961/19046*e^7 + 477317/19046*e^5 - 1218882/9523*e^3 + 4436329/19046*e, 80/9523*e^9 - 3359/9523*e^7 + 46188/9523*e^5 - 237329/9523*e^3 + 410502/9523*e, -19/178*e^8 + 721/178*e^6 - 8847/178*e^4 + 19695/89*e^2 - 48097/178, 1/89*e^8 - 52/89*e^6 + 920/89*e^4 - 5858/89*e^2 + 8382/89, -820/9523*e^9 + 32049/9523*e^7 - 425812/9523*e^5 + 2320727/9523*e^3 - 4536189/9523*e, -737/9523*e^9 + 28445/9523*e^7 - 371702/9523*e^5 + 1985101/9523*e^3 - 3817699/9523*e, -2857/19046*e^9 + 112697/19046*e^7 - 1509977/19046*e^5 + 4096687/9523*e^3 - 15162629/19046*e, 151/9523*e^9 - 5983/9523*e^7 + 78133/9523*e^5 - 379750/9523*e^3 + 592219/9523*e, -1531/9523*e^9 + 61545/9523*e^7 - 846307/9523*e^5 + 4742700/9523*e^3 - 9173251/9523*e, 803/9523*e^9 - 33835/9523*e^7 + 498371/9523*e^5 - 3052490/9523*e^3 + 6469976/9523*e, 915/19046*e^9 - 36633/19046*e^7 + 502087/19046*e^5 - 1386687/9523*e^3 + 4886767/19046*e, 28/89*e^8 - 1100/89*e^6 + 14546/89*e^4 - 76181/89*e^2 + 132524/89, -3527/19046*e^9 + 142019/19046*e^7 - 1958701/19046*e^5 + 5508919/9523*e^3 - 21412249/19046*e, 135/178*e^8 - 5329/178*e^6 + 71423/178*e^4 - 193385/89*e^2 + 716207/178, -33/178*e^8 + 1271/178*e^6 - 16387/178*e^4 + 42367/89*e^2 - 154053/178, -54/89*e^8 + 2096/89*e^6 - 27430/89*e^4 + 144740/89*e^2 - 261990/89, -91/178*e^8 + 3575/178*e^6 - 47675/178*e^4 + 128767/89*e^2 - 479297/178, -767/19046*e^9 + 30895/19046*e^7 - 422353/19046*e^5 + 1165015/9523*e^3 - 4497783/19046*e, 160/9523*e^9 - 6718/9523*e^7 + 101899/9523*e^5 - 693687/9523*e^3 + 1792350/9523*e, 1208/9523*e^9 - 47864/9523*e^7 + 644110/9523*e^5 - 3514150/9523*e^3 + 6699490/9523*e, 137/178*e^8 - 5433/178*e^6 + 73263/178*e^4 - 200044/89*e^2 + 748991/178, 45/89*e^8 - 1806/89*e^6 + 24757/89*e^4 - 138031/89*e^2 + 266830/89, 23/178*e^8 - 929/178*e^6 + 12883/178*e^4 - 36929/89*e^2 + 152825/178, -1269/19046*e^9 + 46497/19046*e^7 - 560767/19046*e^5 + 1348611/9523*e^3 - 4622939/19046*e, -1737/19046*e^9 + 65671/19046*e^7 - 825253/19046*e^5 + 2044941/9523*e^3 - 6596793/19046*e, 64/89*e^8 - 2527/89*e^6 + 33960/89*e^4 - 186232/89*e^2 + 354443/89, -2899/19046*e^9 + 116127/19046*e^7 - 1582793/19046*e^5 + 4337185/9523*e^3 - 16008089/19046*e, 29/178*e^8 - 1063/178*e^6 + 12529/178*e^4 - 27358/89*e^2 + 72465/178, -66/89*e^8 + 2631/89*e^6 - 35711/89*e^4 + 196079/89*e^2 - 369249/89, -71/178*e^8 + 2891/178*e^6 - 40311/178*e^4 + 113886/89*e^2 - 437325/178, -1437/19046*e^9 + 60217/19046*e^7 - 871077/19046*e^5 + 2558201/9523*e^3 - 10118885/19046*e, 187/19046*e^9 - 8923/19046*e^7 + 154151/19046*e^5 - 570151/9523*e^3 + 3107223/19046*e, -33/89*e^8 + 1271/89*e^6 - 16387/89*e^4 + 83933/89*e^2 - 144797/89, -939/9523*e^9 + 38593/9523*e^7 - 546417/9523*e^5 + 3159784/9523*e^3 - 6169819/9523*e];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {3, 3, 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;