/*
  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, 7, 12, -1, -8, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {44,22,3*w^5 - 3*w^4 - 25*w^3 + 40*w + 12}>;

primesArray := [
[4, 2, -2*w^5 + 2*w^4 + 17*w^3 - w^2 - 27*w - 8],
[11, 11, -w^5 + 2*w^4 + 6*w^3 - 5*w^2 - 7*w + 1],
[11, 11, -w^5 + w^4 + 8*w^3 + w^2 - 12*w - 5],
[11, 11, w - 1],
[31, 31, -4*w^5 + 6*w^4 + 28*w^3 - 8*w^2 - 39*w - 11],
[41, 41, -4*w^5 + 5*w^4 + 31*w^3 - 4*w^2 - 49*w - 16],
[41, 41, w^4 - 3*w^3 - 3*w^2 + 8*w + 1],
[41, 41, -w^2 + w + 2],
[59, 59, w^2 - w - 4],
[59, 59, 4*w^5 - 5*w^4 - 31*w^3 + 4*w^2 + 49*w + 14],
[59, 59, -w^4 + 3*w^3 + 3*w^2 - 8*w - 3],
[71, 71, 5*w^5 - 7*w^4 - 37*w^3 + 9*w^2 + 55*w + 16],
[71, 71, -w^5 + 11*w^3 + 4*w^2 - 19*w - 7],
[71, 71, -w^4 + 2*w^3 + 5*w^2 - 4*w - 4],
[79, 79, 3*w^5 - 6*w^4 - 17*w^3 + 11*w^2 + 21*w + 7],
[79, 79, -5*w^5 + 8*w^4 + 35*w^3 - 14*w^2 - 52*w - 11],
[79, 79, -w^5 + 3*w^4 + 3*w^3 - 7*w^2 - 2*w + 2],
[101, 101, -2*w^5 + 3*w^4 + 14*w^3 - 3*w^2 - 21*w - 9],
[101, 101, -2*w^5 + 2*w^4 + 17*w^3 - 29*w - 9],
[101, 101, -w^5 + 2*w^4 + 6*w^3 - 5*w^2 - 9*w + 1],
[101, 101, -2*w^5 + 3*w^4 + 15*w^3 - 6*w^2 - 23*w - 4],
[101, 101, -3*w^5 + 3*w^4 + 26*w^3 - 2*w^2 - 43*w - 13],
[101, 101, -12*w^5 + 17*w^4 + 89*w^3 - 25*w^2 - 133*w - 30],
[125, 5, -2*w^5 + 3*w^4 + 14*w^3 - 4*w^2 - 20*w - 8],
[131, 131, -5*w^5 + 7*w^4 + 37*w^3 - 9*w^2 - 56*w - 17],
[131, 131, w^4 - 3*w^3 - 3*w^2 + 7*w],
[131, 131, w^5 - w^4 - 8*w^3 + 11*w + 5],
[149, 149, w^5 - w^4 - 9*w^3 + 2*w^2 + 16*w + 3],
[149, 149, -13*w^5 + 19*w^4 + 95*w^3 - 30*w^2 - 141*w - 28],
[149, 149, 2*w^5 - 2*w^4 - 17*w^3 + 30*w + 8],
[151, 151, -4*w^5 + 5*w^4 + 31*w^3 - 4*w^2 - 48*w - 14],
[151, 151, -w^5 + 11*w^3 + 4*w^2 - 20*w - 9],
[151, 151, 3*w^5 - 3*w^4 - 25*w^3 + 39*w + 12],
[151, 151, 3*w^5 - 4*w^4 - 23*w^3 + 5*w^2 + 35*w + 11],
[151, 151, 4*w^5 - 6*w^4 - 28*w^3 + 8*w^2 + 39*w + 10],
[151, 151, -2*w^5 + 4*w^4 + 12*w^3 - 9*w^2 - 16*w - 1],
[179, 179, w^5 - w^4 - 9*w^3 + w^2 + 18*w + 5],
[179, 179, 4*w^5 - 5*w^4 - 31*w^3 + 6*w^2 + 46*w + 11],
[179, 179, -7*w^5 + 10*w^4 + 51*w^3 - 12*w^2 - 78*w - 24],
[191, 191, -3*w^5 + 5*w^4 + 20*w^3 - 8*w^2 - 29*w - 11],
[191, 191, -w^5 + 3*w^4 + 4*w^3 - 10*w^2 - 4*w + 2],
[191, 191, 12*w^5 - 16*w^4 - 91*w^3 + 19*w^2 + 138*w + 36],
[211, 211, 5*w^5 - 7*w^4 - 37*w^3 + 10*w^2 + 55*w + 10],
[211, 211, -7*w^5 + 9*w^4 + 54*w^3 - 10*w^2 - 83*w - 24],
[211, 211, -w^5 + 11*w^3 + 4*w^2 - 21*w - 9],
[229, 229, 7*w^5 - 10*w^4 - 51*w^3 + 14*w^2 + 75*w + 19],
[229, 229, 6*w^5 - 7*w^4 - 48*w^3 + 5*w^2 + 75*w + 22],
[229, 229, 6*w^5 - 9*w^4 - 42*w^3 + 11*w^2 + 62*w + 20],
[239, 239, -12*w^5 + 16*w^4 + 91*w^3 - 19*w^2 - 139*w - 35],
[239, 239, -8*w^5 + 12*w^4 + 58*w^3 - 20*w^2 - 87*w - 16],
[239, 239, 5*w^5 - 7*w^4 - 37*w^3 + 10*w^2 + 55*w + 14],
[269, 269, -5*w^5 + 7*w^4 + 37*w^3 - 10*w^2 - 53*w - 12],
[269, 269, -7*w^5 + 9*w^4 + 54*w^3 - 10*w^2 - 82*w - 24],
[269, 269, 5*w^5 - 7*w^4 - 37*w^3 + 10*w^2 + 54*w + 15],
[331, 331, 7*w^5 - 10*w^4 - 51*w^3 + 12*w^2 + 77*w + 25],
[331, 331, 8*w^5 - 11*w^4 - 59*w^3 + 13*w^2 + 87*w + 24],
[331, 331, 3*w^5 - 3*w^4 - 25*w^3 + w^2 + 39*w + 12],
[359, 359, -w^5 + 2*w^4 + 7*w^3 - 7*w^2 - 11*w + 3],
[359, 359, -2*w^5 + 4*w^4 + 12*w^3 - 9*w^2 - 18*w - 3],
[359, 359, 5*w^5 - 7*w^4 - 37*w^3 + 10*w^2 + 53*w + 14],
[389, 389, -4*w^5 + 6*w^4 + 28*w^3 - 8*w^2 - 38*w - 13],
[389, 389, 6*w^5 - 7*w^4 - 48*w^3 + 6*w^2 + 74*w + 21],
[389, 389, -4*w^5 + 5*w^4 + 32*w^3 - 6*w^2 - 52*w - 13],
[421, 421, -4*w^5 + 6*w^4 + 29*w^3 - 11*w^2 - 42*w - 8],
[421, 421, -15*w^5 + 20*w^4 + 114*w^3 - 25*w^2 - 174*w - 43],
[421, 421, 8*w^5 - 11*w^4 - 60*w^3 + 14*w^2 + 92*w + 22],
[431, 431, 8*w^5 - 11*w^4 - 59*w^3 + 13*w^2 + 87*w + 23],
[431, 431, 3*w^5 - 5*w^4 - 19*w^3 + 7*w^2 + 24*w + 6],
[431, 431, 7*w^5 - 11*w^4 - 48*w^3 + 16*w^2 + 69*w + 18],
[439, 439, -6*w^5 + 7*w^4 + 48*w^3 - 6*w^2 - 74*w - 19],
[439, 439, 4*w^5 - 7*w^4 - 25*w^3 + 11*w^2 + 33*w + 8],
[439, 439, 5*w^5 - 6*w^4 - 39*w^3 + 5*w^2 + 58*w + 16],
[449, 449, -w^5 + 2*w^4 + 6*w^3 - 6*w^2 - 6*w + 1],
[449, 449, w^5 - 2*w^4 - 5*w^3 + 2*w^2 + 4*w + 6],
[449, 449, 4*w^5 - 5*w^4 - 31*w^3 + 4*w^2 + 50*w + 17],
[449, 449, 6*w^5 - 9*w^4 - 43*w^3 + 13*w^2 + 65*w + 17],
[449, 449, -3*w^5 + 3*w^4 + 25*w^3 - w^2 - 38*w - 10],
[449, 449, -3*w^5 + 3*w^4 + 26*w^3 - 2*w^2 - 43*w - 12],
[461, 461, 2*w^5 - 2*w^4 - 16*w^3 - 2*w^2 + 25*w + 9],
[461, 461, 2*w^5 - 2*w^4 - 17*w^3 + w^2 + 26*w + 5],
[461, 461, w^4 - 2*w^3 - 5*w^2 + 3*w + 6],
[461, 461, -4*w^5 + 6*w^4 + 29*w^3 - 10*w^2 - 43*w - 12],
[461, 461, w^5 - 3*w^4 - 4*w^3 + 9*w^2 + 6*w - 1],
[461, 461, 3*w^5 - 5*w^4 - 20*w^3 + 9*w^2 + 26*w + 3],
[479, 479, 15*w^5 - 21*w^4 - 111*w^3 + 28*w^2 + 167*w + 42],
[479, 479, -4*w^5 + 6*w^4 + 29*w^3 - 11*w^2 - 43*w - 7],
[479, 479, -9*w^5 + 12*w^4 + 68*w^3 - 14*w^2 - 104*w - 26],
[491, 491, -8*w^5 + 11*w^4 + 59*w^3 - 13*w^2 - 87*w - 25],
[491, 491, 7*w^5 - 11*w^4 - 48*w^3 + 16*w^2 + 69*w + 20],
[491, 491, 3*w^5 - 5*w^4 - 19*w^3 + 7*w^2 + 24*w + 8],
[499, 499, 3*w^5 - 4*w^4 - 23*w^3 + 5*w^2 + 35*w + 13],
[499, 499, -7*w^5 + 10*w^4 + 51*w^3 - 12*w^2 - 77*w - 23],
[499, 499, -2*w^5 + w^4 + 20*w^3 + 3*w^2 - 36*w - 12],
[509, 509, -2*w^5 + 3*w^4 + 15*w^3 - 7*w^2 - 21*w - 1],
[509, 509, -5*w^5 + 6*w^4 + 40*w^3 - 5*w^2 - 64*w - 19],
[509, 509, 4*w^5 - 7*w^4 - 26*w^3 + 12*w^2 + 37*w + 11],
[521, 521, 2*w^5 - w^4 - 19*w^3 - 4*w^2 + 30*w + 9],
[521, 521, 5*w^5 - 8*w^4 - 35*w^3 + 14*w^2 + 52*w + 10],
[521, 521, 5*w^5 - 6*w^4 - 40*w^3 + 6*w^2 + 62*w + 19],
[529, 23, -6*w^5 + 8*w^4 + 46*w^3 - 11*w^2 - 71*w - 18],
[529, 23, -4*w^5 + 5*w^4 + 30*w^3 - 3*w^2 - 44*w - 14],
[529, 23, -5*w^5 + 6*w^4 + 39*w^3 - 4*w^2 - 58*w - 17],
[541, 541, 2*w^5 - w^4 - 19*w^3 - 5*w^2 + 31*w + 15],
[541, 541, w^5 - 3*w^4 - 4*w^3 + 10*w^2 + 3*w - 2],
[541, 541, 5*w^5 - 7*w^4 - 37*w^3 + 9*w^2 + 54*w + 14],
[569, 569, -3*w^5 + 4*w^4 + 23*w^3 - 6*w^2 - 34*w - 10],
[569, 569, -2*w^5 + 3*w^4 + 15*w^3 - 6*w^2 - 24*w - 1],
[569, 569, 5*w^5 - 6*w^4 - 39*w^3 + 4*w^2 + 60*w + 16],
[571, 571, 5*w^5 - 7*w^4 - 37*w^3 + 9*w^2 + 54*w + 16],
[571, 571, 5*w^5 - 6*w^4 - 40*w^3 + 6*w^2 + 64*w + 17],
[571, 571, 2*w^5 - w^4 - 19*w^3 - 5*w^2 + 31*w + 13],
[599, 599, 5*w^5 - 6*w^4 - 41*w^3 + 8*w^2 + 66*w + 15],
[599, 599, 5*w^5 - 10*w^4 - 29*w^3 + 21*w^2 + 37*w + 5],
[599, 599, 12*w^5 - 16*w^4 - 90*w^3 + 17*w^2 + 134*w + 39],
[601, 601, 5*w^5 - 5*w^4 - 42*w^3 + w^2 + 67*w + 19],
[601, 601, 9*w^5 - 13*w^4 - 65*w^3 + 17*w^2 + 96*w + 26],
[601, 601, 5*w^5 - 6*w^4 - 40*w^3 + 5*w^2 + 64*w + 21],
[631, 631, -8*w^5 + 11*w^4 + 60*w^3 - 14*w^2 - 91*w - 25],
[631, 631, -4*w^5 + 6*w^4 + 28*w^3 - 7*w^2 - 40*w - 12],
[631, 631, w^5 - 11*w^3 - 5*w^2 + 21*w + 12],
[659, 659, -4*w^5 + 5*w^4 + 31*w^3 - 4*w^2 - 47*w - 18],
[659, 659, -6*w^5 + 7*w^4 + 48*w^3 - 5*w^2 - 76*w - 24],
[659, 659, 6*w^5 - 9*w^4 - 43*w^3 + 14*w^2 + 62*w + 17],
[659, 659, 5*w^5 - 7*w^4 - 37*w^3 + 10*w^2 + 53*w + 13],
[659, 659, -2*w^5 + 4*w^4 + 11*w^3 - 8*w^2 - 11*w],
[659, 659, 6*w^5 - 8*w^4 - 45*w^3 + 9*w^2 + 68*w + 17],
[691, 691, -4*w^5 + 7*w^4 + 26*w^3 - 12*w^2 - 38*w - 10],
[691, 691, 4*w^5 - 5*w^4 - 32*w^3 + 6*w^2 + 52*w + 14],
[691, 691, 6*w^5 - 7*w^4 - 48*w^3 + 6*w^2 + 74*w + 20],
[701, 701, w^3 - 3*w^2 - 2*w + 3],
[701, 701, -2*w^5 + 3*w^4 + 14*w^3 - 5*w^2 - 17*w - 2],
[701, 701, -4*w^5 + 6*w^4 + 29*w^3 - 10*w^2 - 45*w - 8],
[709, 709, 4*w^5 - 4*w^4 - 34*w^3 + w^2 + 56*w + 19],
[709, 709, 4*w^5 - 6*w^4 - 28*w^3 + 9*w^2 + 36*w + 8],
[709, 709, -5*w^5 + 7*w^4 + 37*w^3 - 8*w^2 - 57*w - 16],
[709, 709, 7*w^5 - 11*w^4 - 48*w^3 + 17*w^2 + 68*w + 15],
[709, 709, w^5 - 11*w^3 - 3*w^2 + 19*w + 3],
[709, 709, -7*w^5 + 11*w^4 + 50*w^3 - 20*w^2 - 75*w - 16],
[729, 3, -3],
[739, 739, 5*w^5 - 7*w^4 - 36*w^3 + 7*w^2 + 52*w + 14],
[739, 739, w^5 - w^4 - 9*w^3 + 18*w + 5],
[739, 739, -7*w^5 + 9*w^4 + 54*w^3 - 10*w^2 - 84*w - 24],
[769, 769, 5*w^5 - 6*w^4 - 40*w^3 + 6*w^2 + 63*w + 20],
[769, 769, 4*w^5 - 6*w^4 - 29*w^3 + 9*w^2 + 45*w + 9],
[769, 769, w^5 - 11*w^3 - 3*w^2 + 18*w + 2],
[811, 811, 4*w^5 - 5*w^4 - 32*w^3 + 6*w^2 + 51*w + 13],
[811, 811, 5*w^5 - 9*w^4 - 32*w^3 + 17*w^2 + 45*w + 11],
[811, 811, 7*w^5 - 8*w^4 - 56*w^3 + 5*w^2 + 86*w + 27],
[859, 859, -11*w^5 + 16*w^4 + 81*w^3 - 26*w^2 - 121*w - 23],
[859, 859, -12*w^5 + 17*w^4 + 88*w^3 - 23*w^2 - 131*w - 32],
[859, 859, 7*w^5 - 11*w^4 - 49*w^3 + 19*w^2 + 72*w + 13],
[911, 911, -2*w^5 + w^4 + 20*w^3 + 3*w^2 - 37*w - 12],
[911, 911, 2*w^5 - 2*w^4 - 17*w^3 + 2*w^2 + 27*w + 4],
[911, 911, 6*w^5 - 8*w^4 - 45*w^3 + 7*w^2 + 70*w + 23],
[919, 919, w^4 - 2*w^3 - 7*w^2 + 8*w + 6],
[919, 919, 16*w^5 - 22*w^4 - 120*w^3 + 29*w^2 + 182*w + 46],
[919, 919, -5*w^5 + 9*w^4 + 32*w^3 - 18*w^2 - 43*w - 11]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^9 - 10*x^8 + 9*x^7 + 145*x^6 - 283*x^5 - 635*x^4 + 1492*x^3 + 634*x^2 - 1983*x + 745;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [-1, -56660/830269*e^8 + 479652/830269*e^7 + 27939/830269*e^6 - 6345355/830269*e^5 + 4595317/830269*e^4 + 25872120/830269*e^3 - 1700924/75479*e^2 - 29372468/830269*e + 21602374/830269, 1, e, 15469/830269*e^8 - 88867/830269*e^7 - 250891/830269*e^6 + 806637/830269*e^5 + 3120912/830269*e^4 + 845633/830269*e^3 - 1635053/75479*e^2 - 15613049/830269*e + 20262512/830269, -47059/830269*e^8 + 251078/830269*e^7 + 1185977/830269*e^6 - 4832218/830269*e^5 - 9733285/830269*e^4 + 23280819/830269*e^3 + 2689704/75479*e^2 - 13916173/830269*e - 10726212/830269, 138896/830269*e^8 - 1155184/830269*e^7 - 437524/830269*e^6 + 17302840/830269*e^5 - 10440409/830269*e^4 - 81294212/830269*e^3 + 5395928/75479*e^2 + 110863020/830269*e - 82340425/830269, 5568/75479*e^8 - 37379/75479*e^7 - 100061/75479*e^6 + 766499/75479*e^5 + 394410/75479*e^4 - 4444891/75479*e^3 + 455138/75479*e^2 + 6656376/75479*e - 3337161/75479, 17425/830269*e^8 - 42298/830269*e^7 - 720534/830269*e^6 + 571137/830269*e^5 + 9214173/830269*e^4 - 222773/830269*e^3 - 3517815/75479*e^2 - 14345731/830269*e + 26169112/830269, -60879/830269*e^8 + 416134/830269*e^7 + 936941/830269*e^6 - 7553961/830269*e^5 - 4002729/830269*e^4 + 40152700/830269*e^3 + 81303/75479*e^2 - 56643094/830269*e + 23769604/830269, 67983/830269*e^8 - 685056/830269*e^7 + 642782/830269*e^6 + 9739572/830269*e^5 - 17356941/830269*e^4 - 47944988/830269*e^3 + 7336960/75479*e^2 + 88700776/830269*e - 79165685/830269, -491/830269*e^8 + 60895/830269*e^7 - 391901/830269*e^6 - 852652/830269*e^5 + 6458610/830269*e^4 + 5045212/830269*e^3 - 2605276/75479*e^2 - 17100935/830269*e + 20049972/830269, -82963/830269*e^8 + 577998/830269*e^7 + 943669/830269*e^6 - 8114277/830269*e^5 - 6035217/830269*e^4 + 33132815/830269*e^3 + 2167832/75479*e^2 - 25860699/830269*e - 5211428/830269, 79210/830269*e^8 - 729748/830269*e^7 + 114071/830269*e^6 + 11724212/830269*e^5 - 13134782/830269*e^4 - 60689837/830269*e^3 + 6472334/75479*e^2 + 97106541/830269*e - 86489636/830269, 61244/830269*e^8 - 681229/830269*e^7 + 786647/830269*e^6 + 11590049/830269*e^5 - 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250276996/830269*e + 170673394/830269, 41464/75479*e^8 - 376066/75479*e^7 + 67454/75479*e^6 + 5685980/75479*e^5 - 6247255/75479*e^4 - 28168768/75479*e^3 + 32227306/75479*e^2 + 45284877/75479*e - 36492310/75479, -140343/830269*e^8 + 1432721/830269*e^7 - 1381980/830269*e^6 - 20464980/830269*e^5 + 35024017/830269*e^4 + 101151066/830269*e^3 - 14020591/75479*e^2 - 181844522/830269*e + 137463294/830269, -11442/75479*e^8 + 96414/75479*e^7 + 37664/75479*e^6 - 1511437/75479*e^5 + 874931/75479*e^4 + 7756649/75479*e^3 - 5072807/75479*e^2 - 12768263/75479*e + 7174197/75479, -327952/830269*e^8 + 3255488/830269*e^7 - 3126422/830269*e^6 - 42548615/830269*e^5 + 73714445/830269*e^4 + 185631719/830269*e^3 - 27740047/75479*e^2 - 292942383/830269*e + 288671827/830269, -7872/830269*e^8 - 105920/830269*e^7 + 2109125/830269*e^6 - 6820075/830269*e^5 - 12144942/830269*e^4 + 69962405/830269*e^3 - 1002813/75479*e^2 - 143862262/830269*e + 56291034/830269, -423009/830269*e^8 + 3461503/830269*e^7 + 2665997/830269*e^6 - 60473998/830269*e^5 + 32029169/830269*e^4 + 326987632/830269*e^3 - 23425040/75479*e^2 - 520716444/830269*e + 369226982/830269, -7444/75479*e^8 + 33001/75479*e^7 + 260928/75479*e^6 - 903671/75479*e^5 - 2186208/75479*e^4 + 5426226/75479*e^3 + 5850066/75479*e^2 - 5831767/75479*e - 607433/75479, -312783/830269*e^8 + 2838577/830269*e^7 - 483386/830269*e^6 - 43137181/830269*e^5 + 47401697/830269*e^4 + 209919312/830269*e^3 - 21441653/75479*e^2 - 321249684/830269*e + 250062399/830269, 13696/75479*e^8 - 83268/75479*e^7 - 268684/75479*e^6 + 1457697/75479*e^5 + 1969604/75479*e^4 - 6920877/75479*e^3 - 5822798/75479*e^2 + 4850895/75479*e + 1463074/75479, 135665/830269*e^8 - 935402/830269*e^7 - 2405966/830269*e^6 + 20534119/830269*e^5 + 3670350/830269*e^4 - 120912937/830269*e^3 + 5048966/75479*e^2 + 176034136/830269*e - 135036865/830269, 56743/830269*e^8 + 142479/830269*e^7 - 5563893/830269*e^6 + 9717562/830269*e^5 + 51371490/830269*e^4 - 89037426/830269*e^3 - 10829839/75479*e^2 + 131232675/830269*e - 19927205/830269, 243378/830269*e^8 - 2678916/830269*e^7 + 3245877/830269*e^6 + 42874960/830269*e^5 - 87695781/830269*e^4 - 234847044/830269*e^3 + 39646178/75479*e^2 + 467887496/830269*e - 433842491/830269, 193726/830269*e^8 - 2307474/830269*e^7 + 4484376/830269*e^6 + 29833218/830269*e^5 - 81664468/830269*e^4 - 136866328/830269*e^3 + 30484686/75479*e^2 + 265220133/830269*e - 282062991/830269, -48460/75479*e^8 + 388708/75479*e^7 + 306016/75479*e^6 - 6351861/75479*e^5 + 2546763/75479*e^4 + 32080882/75479*e^3 - 20230291/75479*e^2 - 46557262/75479*e + 30538455/75479, 47818/830269*e^8 + 346398/830269*e^7 - 6025108/830269*e^6 + 3289138/830269*e^5 + 68259280/830269*e^4 - 37588398/830269*e^3 - 20701503/75479*e^2 + 5980454/830269*e + 107531327/830269, -103456/830269*e^8 + 1051526/830269*e^7 - 1242834/830269*e^6 - 12946181/830269*e^5 + 25315047/830269*e^4 + 52458550/830269*e^3 - 8995684/75479*e^2 - 83924518/830269*e + 89387945/830269, 353843/830269*e^8 - 2940868/830269*e^7 - 1829845/830269*e^6 + 49995652/830269*e^5 - 28283766/830269*e^4 - 268024895/830269*e^3 + 18962262/75479*e^2 + 431951265/830269*e - 322450259/830269, 274422/830269*e^8 - 2342212/830269*e^7 - 1605037/830269*e^6 + 45473832/830269*e^5 - 35186443/830269*e^4 - 268450022/830269*e^3 + 25083546/75479*e^2 + 477268937/830269*e - 378493985/830269, -283572/830269*e^8 + 2300098/830269*e^7 + 2276431/830269*e^6 - 43608132/830269*e^5 + 22526567/830269*e^4 + 251962813/830269*e^3 - 19173985/75479*e^2 - 435194550/830269*e + 316501487/830269, 490976/830269*e^8 - 3566261/830269*e^7 - 5267440/830269*e^6 + 56279202/830269*e^5 + 14469292/830269*e^4 - 266965140/830269*e^3 + 568822/75479*e^2 + 314072190/830269*e - 130882616/830269, 112082/830269*e^8 - 685705/830269*e^7 - 2931140/830269*e^6 + 18850300/830269*e^5 + 10824300/830269*e^4 - 125821094/830269*e^3 + 3600536/75479*e^2 + 212554528/830269*e - 139524115/830269, -92505/830269*e^8 + 638612/830269*e^7 + 1422240/830269*e^6 - 11381648/830269*e^5 - 9331581/830269*e^4 + 64767434/830269*e^3 + 2377807/75479*e^2 - 101843410/830269*e + 3990062/830269, -65393/830269*e^8 + 362147/830269*e^7 + 1965588/830269*e^6 - 10784894/830269*e^5 - 8333114/830269*e^4 + 73609952/830269*e^3 - 2043298/75479*e^2 - 128334082/830269*e + 79302139/830269, -112826/830269*e^8 + 1100954/830269*e^7 - 1382844/830269*e^6 - 10885902/830269*e^5 + 23450978/830269*e^4 + 26870260/830269*e^3 - 6844031/75479*e^2 - 15845141/830269*e + 63928515/830269, -19001/75479*e^8 + 139527/75479*e^7 + 176365/75479*e^6 - 2030350/75479*e^5 - 593019/75479*e^4 + 8859496/75479*e^3 + 1284635/75479*e^2 - 8818202/75479*e + 2868937/75479, -63420/830269*e^8 - 104067/830269*e^7 + 4683475/830269*e^6 - 1196742/830269*e^5 - 59188937/830269*e^4 + 3075759/830269*e^3 + 21181819/75479*e^2 + 74754799/830269*e - 151735033/830269, 335380/830269*e^8 - 3236544/830269*e^7 + 2112076/830269*e^6 + 46274256/830269*e^5 - 67603161/830269*e^4 - 225217422/830269*e^3 + 28283936/75479*e^2 + 392145970/830269*e - 320173071/830269, -131707/830269*e^8 + 813154/830269*e^7 + 3010055/830269*e^6 - 19136185/830269*e^5 - 12542952/830269*e^4 + 118890226/830269*e^3 - 1884755/75479*e^2 - 196372556/830269*e + 90950060/830269, 10090/75479*e^8 - 122583/75479*e^7 + 229228/75479*e^6 + 1756178/75479*e^5 - 4559206/75479*e^4 - 8783504/75479*e^3 + 19532125/75479*e^2 + 17320444/75479*e - 15680657/75479, -430282/830269*e^8 + 3113887/830269*e^7 + 4855846/830269*e^6 - 50429723/830269*e^5 - 14502747/830269*e^4 + 249294249/830269*e^3 - 791024/75479*e^2 - 320663836/830269*e + 138060438/830269, -53942/830269*e^8 + 495973/830269*e^7 + 360963/830269*e^6 - 12153393/830269*e^5 + 13548779/830269*e^4 + 81166688/830269*e^3 - 9754421/75479*e^2 - 162181236/830269*e + 121083571/830269, 72607/830269*e^8 - 1507109/830269*e^7 + 5831720/830269*e^6 + 21591040/830269*e^5 - 97715149/830269*e^4 - 122101360/830269*e^3 + 39494182/75479*e^2 + 307631186/830269*e - 370495232/830269, -34733/75479*e^8 + 284993/75479*e^7 + 166762/75479*e^6 - 4540413/75479*e^5 + 2289411/75479*e^4 + 22898691/75479*e^3 - 15744879/75479*e^2 - 34539470/75479*e + 23426941/75479, 16377/75479*e^8 - 149674/75479*e^7 + 124122/75479*e^6 + 1467849/75479*e^5 - 2096673/75479*e^4 - 3376898/75479*e^3 + 4020385/75479*e^2 - 289360/75479*e + 346067/75479, 32650/75479*e^8 - 225882/75479*e^7 - 422149/75479*e^6 + 3540938/75479*e^5 + 2147982/75479*e^4 - 16052835/75479*e^3 - 4752529/75479*e^2 + 14809187/75479*e - 6900471/75479, 357988/830269*e^8 - 2891846/830269*e^7 - 1034224/830269*e^6 + 37404215/830269*e^5 - 13495665/830269*e^4 - 145783433/830269*e^3 + 3794084/75479*e^2 + 140012769/830269*e - 60334904/830269, -255671/830269*e^8 + 2155752/830269*e^7 + 1261412/830269*e^6 - 37523705/830269*e^5 + 22724220/830269*e^4 + 203470375/830269*e^3 - 14601318/75479*e^2 - 319567311/830269*e + 222177641/830269, -367934/830269*e^8 + 3075277/830269*e^7 + 2000312/830269*e^6 - 54594895/830269*e^5 + 35489008/830269*e^4 + 300703607/830269*e^3 - 23965544/75479*e^2 - 491215503/830269*e + 345444593/830269, 110989/830269*e^8 - 1415928/830269*e^7 + 2578203/830269*e^6 + 24258943/830269*e^5 - 62039198/830269*e^4 - 144253196/830269*e^3 + 28466853/75479*e^2 + 319749918/830269*e - 295397225/830269, -80242/830269*e^8 + 796864/830269*e^7 - 1384199/830269*e^6 - 4667469/830269*e^5 + 12843702/830269*e^4 - 11107209/830269*e^3 + 369953/75479*e^2 + 65563684/830269*e - 54257824/830269, -388606/830269*e^8 + 3087386/830269*e^7 + 3320062/830269*e^6 - 56736162/830269*e^5 + 22874634/830269*e^4 + 316968188/830269*e^3 - 21475580/75479*e^2 - 514421278/830269*e + 381503456/830269, -219228/830269*e^8 + 1687581/830269*e^7 + 1665068/830269*e^6 - 25626919/830269*e^5 + 339440/830269*e^4 + 122054117/830269*e^3 - 2260998/75479*e^2 - 158371012/830269*e + 92617650/830269, -68215/830269*e^8 + 796687/830269*e^7 - 1604115/830269*e^6 - 9202272/830269*e^5 + 26643294/830269*e^4 + 37622886/830269*e^3 - 8944900/75479*e^2 - 80373519/830269*e + 60538767/830269, -75052/830269*e^8 + 826196/830269*e^7 - 1716023/830269*e^6 - 6351823/830269*e^5 + 19733275/830269*e^4 + 3462997/830269*e^3 - 2458129/75479*e^2 + 30779822/830269*e - 42889559/830269, 301123/830269*e^8 - 2138196/830269*e^7 - 3498383/830269*e^6 + 33523412/830269*e^5 + 12500101/830269*e^4 - 154262293/830269*e^3 - 1031132/75479*e^2 + 161025697/830269*e - 70658999/830269, 298136/830269*e^8 - 2050134/830269*e^7 - 4355563/830269*e^6 + 35874670/830269*e^5 + 18833961/830269*e^4 - 182337895/830269*e^3 - 1769164/75479*e^2 + 220602447/830269*e - 63683108/830269, 96951/830269*e^8 - 1024300/830269*e^7 + 574104/830269*e^6 + 20152499/830269*e^5 - 31700199/830269*e^4 - 127561068/830269*e^3 + 16911560/75479*e^2 + 269580818/830269*e - 212684203/830269, 639059/830269*e^8 - 5502352/830269*e^7 - 1001652/830269*e^6 + 84013112/830269*e^5 - 65696958/830269*e^4 - 411656930/830269*e^3 + 32599722/75479*e^2 + 619680401/830269*e - 444028887/830269, 344055/830269*e^8 - 2883564/830269*e^7 - 1518877/830269*e^6 + 48357313/830269*e^5 - 30973715/830269*e^4 - 257684996/830269*e^3 + 19317803/75479*e^2 + 421578697/830269*e - 284142689/830269, 5740/75479*e^8 - 48565/75479*e^7 - 39331/75479*e^6 + 961577/75479*e^5 - 588623/75479*e^4 - 6190735/75479*e^3 + 5283695/75479*e^2 + 12893810/75479*e - 6976212/75479, 204258/830269*e^8 - 1949758/830269*e^7 + 1014971/830269*e^6 + 29319042/830269*e^5 - 41846663/830269*e^4 - 147057404/830269*e^3 + 18805193/75479*e^2 + 252409535/830269*e - 192468014/830269, -536768/830269*e^8 + 4671406/830269*e^7 + 212103/830269*e^6 - 68583576/830269*e^5 + 58093969/830269*e^4 + 325137462/830269*e^3 - 26445615/75479*e^2 - 479829137/830269*e + 336909901/830269, -352697/830269*e^8 + 3098041/830269*e^7 - 42181/830269*e^6 - 45778537/830269*e^5 + 42914659/830269*e^4 + 218163466/830269*e^3 - 20490906/75479*e^2 - 328296150/830269*e + 282319205/830269, 233109/830269*e^8 - 1885567/830269*e^7 - 1553351/830269*e^6 + 32487569/830269*e^5 - 14863940/830269*e^4 - 175685562/830269*e^3 + 11527002/75479*e^2 + 292274335/830269*e - 205808213/830269, 112829/830269*e^8 - 898409/830269*e^7 - 1278048/830269*e^6 + 19310479/830269*e^5 - 7713638/830269*e^4 - 122253507/830269*e^3 + 8797346/75479*e^2 + 220151836/830269*e - 166791313/830269, 188811/830269*e^8 - 2036099/830269*e^7 + 1679110/830269*e^6 + 37708933/830269*e^5 - 67385069/830269*e^4 - 229106533/830269*e^3 + 34840035/75479*e^2 + 477140680/830269*e - 434578526/830269, 94015/830269*e^8 - 814049/830269*e^7 + 146555/830269*e^6 + 9630994/830269*e^5 - 5251721/830269*e^4 - 36077776/830269*e^3 - 511736/75479*e^2 + 42913351/830269*e + 23324880/830269, -653121/830269*e^8 + 5103889/830269*e^7 + 4509578/830269*e^6 - 79131424/830269*e^5 + 17462570/830269*e^4 + 378581683/830269*e^3 - 15550841/75479*e^2 - 494316641/830269*e + 338376651/830269, 33405/830269*e^8 - 502298/830269*e^7 + 777383/830269*e^6 + 13006234/830269*e^5 - 30353655/830269*e^4 - 94547176/830269*e^3 + 17267618/75479*e^2 + 208120426/830269*e - 207150486/830269];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {4,2,3*w^5 - 4*w^4 - 23*w^3 + 5*w^2 + 36*w + 9}>] := 1;
ALEigenvalues[ideal<ZF | {11,11,-w^5 + w^4 + 8*w^3 + w^2 - 12*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;