/*
  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, -2, 7, 2, -7, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {169,13,7*w^5 - w^4 - 51*w^3 - 29*w^2 + 32*w + 12}>;

primesArray := [
[29, 29, -9*w^5 + 3*w^4 + 64*w^3 + 26*w^2 - 40*w - 10],
[29, 29, w^5 - 7*w^3 - 5*w^2 + 2*w + 2],
[29, 29, w^4 - w^3 - 6*w^2 + 2],
[29, 29, 5*w^5 - w^4 - 36*w^3 - 19*w^2 + 21*w + 9],
[29, 29, -w^5 + w^4 + 7*w^3 - 2*w^2 - 6*w + 1],
[29, 29, 2*w^5 - 15*w^3 - 10*w^2 + 11*w + 5],
[41, 41, 5*w^5 - w^4 - 36*w^3 - 18*w^2 + 21*w + 5],
[41, 41, -5*w^5 + 2*w^4 + 36*w^3 + 11*w^2 - 25*w - 2],
[41, 41, 6*w^5 - w^4 - 44*w^3 - 23*w^2 + 30*w + 8],
[41, 41, 13*w^5 - 4*w^4 - 93*w^3 - 39*w^2 + 59*w + 16],
[41, 41, -4*w^5 + 30*w^3 + 19*w^2 - 19*w - 8],
[41, 41, w^5 - 7*w^3 - 6*w^2 + 2*w + 3],
[49, 7, -5*w^5 + w^4 + 36*w^3 + 19*w^2 - 22*w - 6],
[64, 2, -2],
[71, 71, -8*w^5 + w^4 + 58*w^3 + 34*w^2 - 34*w - 16],
[71, 71, -6*w^5 + 2*w^4 + 42*w^3 + 18*w^2 - 23*w - 6],
[71, 71, -8*w^5 + 2*w^4 + 58*w^3 + 27*w^2 - 38*w - 10],
[71, 71, 4*w^5 - 30*w^3 - 19*w^2 + 20*w + 8],
[71, 71, -10*w^5 + 3*w^4 + 72*w^3 + 30*w^2 - 48*w - 10],
[71, 71, -8*w^5 + 2*w^4 + 58*w^3 + 26*w^2 - 37*w - 8],
[125, 5, 4*w^5 - w^4 - 29*w^3 - 13*w^2 + 19*w + 2],
[139, 139, -w^5 - w^4 + 8*w^3 + 12*w^2 - 3*w - 5],
[139, 139, -11*w^5 + 4*w^4 + 78*w^3 + 29*w^2 - 49*w - 11],
[139, 139, -12*w^5 + 3*w^4 + 86*w^3 + 41*w^2 - 52*w - 18],
[139, 139, -4*w^5 + w^4 + 28*w^3 + 15*w^2 - 14*w - 7],
[139, 139, 8*w^5 - 2*w^4 - 58*w^3 - 27*w^2 + 39*w + 10],
[139, 139, 7*w^5 - 2*w^4 - 51*w^3 - 21*w^2 + 35*w + 7],
[169, 13, -8*w^5 + 2*w^4 + 57*w^3 + 28*w^2 - 34*w - 13],
[169, 13, w^5 - w^4 - 6*w^3 + w^2 + 2*w - 2],
[169, 13, -7*w^5 + w^4 + 51*w^3 + 29*w^2 - 32*w - 12],
[181, 181, -5*w^5 + 36*w^3 + 26*w^2 - 18*w - 12],
[181, 181, -6*w^5 + 44*w^3 + 30*w^2 - 23*w - 13],
[181, 181, 9*w^5 - 4*w^4 - 64*w^3 - 19*w^2 + 44*w + 7],
[181, 181, 2*w^4 - w^3 - 14*w^2 - 2*w + 8],
[181, 181, -12*w^5 + 4*w^4 + 85*w^3 + 35*w^2 - 53*w - 16],
[181, 181, -11*w^5 + 3*w^4 + 78*w^3 + 37*w^2 - 45*w - 18],
[211, 211, -10*w^5 + 3*w^4 + 71*w^3 + 32*w^2 - 43*w - 15],
[211, 211, -7*w^5 + 3*w^4 + 49*w^3 + 16*w^2 - 31*w - 5],
[211, 211, 8*w^5 - w^4 - 58*w^3 - 34*w^2 + 35*w + 15],
[211, 211, -w^4 + w^3 + 7*w^2 - 2*w - 4],
[211, 211, 6*w^5 - 2*w^4 - 42*w^3 - 17*w^2 + 23*w + 6],
[211, 211, w^3 - w^2 - 5*w],
[239, 239, w^3 - w^2 - 5*w + 1],
[239, 239, w^4 - w^3 - 7*w^2 + 2*w + 5],
[239, 239, 8*w^5 - w^4 - 58*w^3 - 34*w^2 + 35*w + 14],
[239, 239, 7*w^5 - 3*w^4 - 49*w^3 - 16*w^2 + 31*w + 6],
[239, 239, 10*w^5 - 3*w^4 - 71*w^3 - 32*w^2 + 43*w + 14],
[239, 239, -11*w^5 + 2*w^4 + 80*w^3 + 42*w^2 - 50*w - 18],
[251, 251, 6*w^5 - w^4 - 44*w^3 - 23*w^2 + 31*w + 8],
[251, 251, 11*w^5 - 2*w^4 - 79*w^3 - 43*w^2 + 47*w + 18],
[251, 251, 10*w^5 - w^4 - 73*w^3 - 44*w^2 + 45*w + 18],
[251, 251, -11*w^5 + 4*w^4 + 78*w^3 + 29*w^2 - 48*w - 13],
[251, 251, 6*w^5 - 2*w^4 - 42*w^3 - 17*w^2 + 23*w + 5],
[251, 251, -16*w^5 + 4*w^4 + 115*w^3 + 54*w^2 - 70*w - 22],
[281, 281, 8*w^5 - 3*w^4 - 57*w^3 - 20*w^2 + 36*w + 7],
[281, 281, 3*w^5 - w^4 - 22*w^3 - 8*w^2 + 18*w + 3],
[281, 281, 2*w^5 - 2*w^4 - 13*w^3 + 2*w^2 + 9*w],
[281, 281, w^5 + w^4 - 8*w^3 - 11*w^2 + 3*w + 4],
[281, 281, w^5 - w^4 - 7*w^3 + 2*w^2 + 8*w - 2],
[281, 281, -13*w^5 + 4*w^4 + 93*w^3 + 39*w^2 - 58*w - 15],
[349, 349, 7*w^5 - 2*w^4 - 51*w^3 - 21*w^2 + 36*w + 5],
[349, 349, 7*w^5 - w^4 - 51*w^3 - 28*w^2 + 32*w + 8],
[349, 349, 3*w^5 - w^4 - 21*w^3 - 8*w^2 + 12*w],
[349, 349, 10*w^5 - 2*w^4 - 72*w^3 - 38*w^2 + 45*w + 17],
[349, 349, 7*w^5 - 2*w^4 - 50*w^3 - 23*w^2 + 33*w + 12],
[349, 349, 17*w^5 - 5*w^4 - 122*w^3 - 53*w^2 + 79*w + 21],
[379, 379, 2*w^5 - 14*w^3 - 11*w^2 + 7*w + 4],
[379, 379, -10*w^5 + 3*w^4 + 71*w^3 + 32*w^2 - 42*w - 15],
[379, 379, -15*w^5 + 5*w^4 + 106*w^3 + 44*w^2 - 63*w - 17],
[379, 379, 2*w^5 - 3*w^4 - 12*w^3 + 9*w^2 + 8*w - 4],
[379, 379, 4*w^5 - 29*w^3 - 20*w^2 + 14*w + 10],
[379, 379, 13*w^5 - 2*w^4 - 94*w^3 - 53*w^2 + 55*w + 24],
[419, 419, 3*w^5 - 2*w^4 - 20*w^3 - 4*w^2 + 12*w + 4],
[419, 419, 3*w^5 - 22*w^3 - 16*w^2 + 15*w + 8],
[419, 419, -11*w^5 + 4*w^4 + 77*w^3 + 31*w^2 - 45*w - 14],
[419, 419, -19*w^5 + 5*w^4 + 137*w^3 + 62*w^2 - 87*w - 24],
[419, 419, -16*w^5 + 5*w^4 + 114*w^3 + 48*w^2 - 72*w - 17],
[419, 419, 17*w^5 - 5*w^4 - 121*w^3 - 54*w^2 + 73*w + 23],
[421, 421, 4*w^5 - w^4 - 29*w^3 - 13*w^2 + 20*w + 1],
[421, 421, 4*w^5 - 2*w^4 - 29*w^3 - 6*w^2 + 22*w + 1],
[421, 421, -8*w^5 + 3*w^4 + 57*w^3 + 20*w^2 - 37*w - 10],
[421, 421, -6*w^5 + 45*w^3 + 30*w^2 - 31*w - 14],
[421, 421, 5*w^5 + w^4 - 38*w^3 - 31*w^2 + 22*w + 14],
[421, 421, -6*w^5 + 3*w^4 + 41*w^3 + 12*w^2 - 24*w - 5],
[449, 449, -9*w^5 + w^4 + 66*w^3 + 38*w^2 - 43*w - 16],
[449, 449, 2*w^5 + w^4 - 16*w^3 - 17*w^2 + 11*w + 9],
[449, 449, 10*w^5 - w^4 - 74*w^3 - 43*w^2 + 49*w + 18],
[449, 449, 17*w^5 - 6*w^4 - 121*w^3 - 46*w^2 + 78*w + 16],
[449, 449, -11*w^5 + 2*w^4 + 79*w^3 + 44*w^2 - 47*w - 21],
[449, 449, 6*w^5 - w^4 - 44*w^3 - 22*w^2 + 28*w + 5],
[461, 461, -6*w^5 + 3*w^4 + 42*w^3 + 10*w^2 - 28*w - 4],
[461, 461, -15*w^5 + 5*w^4 + 107*w^3 + 43*w^2 - 70*w - 16],
[461, 461, -10*w^5 + 2*w^4 + 73*w^3 + 36*w^2 - 46*w - 13],
[461, 461, -9*w^5 + 2*w^4 + 66*w^3 + 31*w^2 - 46*w - 10],
[461, 461, 12*w^5 - 4*w^4 - 85*w^3 - 34*w^2 + 50*w + 13],
[461, 461, -18*w^5 + 4*w^4 + 130*w^3 + 64*w^2 - 81*w - 28],
[491, 491, 17*w^5 - 5*w^4 - 121*w^3 - 54*w^2 + 73*w + 21],
[491, 491, -10*w^5 + 4*w^4 + 70*w^3 + 26*w^2 - 43*w - 11],
[491, 491, -6*w^5 + 3*w^4 + 41*w^3 + 12*w^2 - 23*w - 3],
[491, 491, 2*w^5 - 16*w^3 - 9*w^2 + 16*w + 3],
[491, 491, 20*w^5 - 5*w^4 - 144*w^3 - 68*w^2 + 90*w + 29],
[491, 491, -6*w^5 + w^4 + 44*w^3 + 24*w^2 - 31*w - 13],
[601, 601, -10*w^5 + 2*w^4 + 73*w^3 + 36*w^2 - 47*w - 12],
[601, 601, -7*w^5 + 2*w^4 + 49*w^3 + 24*w^2 - 26*w - 10],
[601, 601, -15*w^5 + 3*w^4 + 108*w^3 + 56*w^2 - 64*w - 24],
[601, 601, -3*w^5 + 3*w^4 + 20*w^3 - 5*w^2 - 15*w + 3],
[601, 601, 9*w^5 - 2*w^4 - 66*w^3 - 31*w^2 + 45*w + 11],
[601, 601, 14*w^5 - 2*w^4 - 102*w^3 - 57*w^2 + 63*w + 25],
[631, 631, 5*w^5 - 2*w^4 - 35*w^3 - 13*w^2 + 20*w + 9],
[631, 631, -10*w^5 + 4*w^4 + 71*w^3 + 24*w^2 - 47*w - 6],
[631, 631, -5*w^5 + w^4 + 36*w^3 + 18*w^2 - 20*w - 9],
[631, 631, -21*w^5 + 5*w^4 + 151*w^3 + 73*w^2 - 93*w - 31],
[631, 631, 8*w^5 - 2*w^4 - 57*w^3 - 27*w^2 + 34*w + 8],
[631, 631, -14*w^5 + 3*w^4 + 102*w^3 + 50*w^2 - 68*w - 19],
[659, 659, 14*w^5 - 3*w^4 - 102*w^3 - 49*w^2 + 67*w + 19],
[659, 659, 7*w^5 - 2*w^4 - 50*w^3 - 22*w^2 + 30*w + 5],
[659, 659, -6*w^5 + 2*w^4 + 43*w^3 + 17*w^2 - 29*w - 10],
[659, 659, -18*w^5 + 6*w^4 + 128*w^3 + 52*w^2 - 80*w - 21],
[659, 659, 14*w^5 - 2*w^4 - 102*w^3 - 57*w^2 + 64*w + 24],
[659, 659, -16*w^5 + 5*w^4 + 114*w^3 + 48*w^2 - 72*w - 18],
[701, 701, -7*w^5 + 4*w^4 + 49*w^3 + 9*w^2 - 34*w - 4],
[701, 701, 15*w^5 - 3*w^4 - 108*w^3 - 57*w^2 + 65*w + 26],
[701, 701, -12*w^5 + 2*w^4 + 87*w^3 + 48*w^2 - 55*w - 22],
[701, 701, 7*w^5 - 3*w^4 - 49*w^3 - 16*w^2 + 28*w + 5],
[701, 701, -8*w^5 + 2*w^4 + 57*w^3 + 27*w^2 - 32*w - 12],
[701, 701, -9*w^5 + 3*w^4 + 64*w^3 + 26*w^2 - 38*w - 12],
[729, 3, -3],
[769, 769, -12*w^5 + w^4 + 88*w^3 + 53*w^2 - 54*w - 21],
[769, 769, 14*w^5 - 2*w^4 - 102*w^3 - 57*w^2 + 62*w + 25],
[769, 769, -3*w^5 + 23*w^3 + 13*w^2 - 17*w - 1],
[769, 769, 9*w^5 - 2*w^4 - 65*w^3 - 32*w^2 + 43*w + 12],
[769, 769, -13*w^5 + 4*w^4 + 94*w^3 + 38*w^2 - 63*w - 14],
[769, 769, 16*w^5 - 4*w^4 - 116*w^3 - 53*w^2 + 75*w + 21],
[811, 811, 20*w^5 - 7*w^4 - 142*w^3 - 55*w^2 + 89*w + 21],
[811, 811, 11*w^5 - 2*w^4 - 80*w^3 - 42*w^2 + 53*w + 17],
[811, 811, -w^5 + 7*w^3 + 4*w^2 - w + 1],
[811, 811, 9*w^5 - 2*w^4 - 64*w^3 - 34*w^2 + 35*w + 16],
[811, 811, 5*w^5 - 2*w^4 - 35*w^3 - 13*w^2 + 20*w + 3],
[811, 811, -7*w^5 + 52*w^3 + 34*w^2 - 31*w - 15],
[839, 839, -14*w^5 + 5*w^4 + 99*w^3 + 38*w^2 - 60*w - 16],
[839, 839, -20*w^5 + 6*w^4 + 143*w^3 + 62*w^2 - 90*w - 25],
[839, 839, 7*w^5 - w^4 - 50*w^3 - 29*w^2 + 26*w + 12],
[839, 839, -19*w^5 + 5*w^4 + 137*w^3 + 62*w^2 - 88*w - 22],
[839, 839, -14*w^5 + 3*w^4 + 101*w^3 + 50*w^2 - 62*w - 21],
[839, 839, -w^5 - w^4 + 9*w^3 + 11*w^2 - 10*w - 6],
[881, 881, -14*w^5 + 3*w^4 + 102*w^3 + 49*w^2 - 67*w - 18],
[881, 881, 12*w^5 - 5*w^4 - 84*w^3 - 29*w^2 + 52*w + 13],
[881, 881, -3*w^5 + 2*w^4 + 21*w^3 + 2*w^2 - 15*w - 3],
[881, 881, -3*w^5 + 21*w^3 + 16*w^2 - 8*w - 8],
[881, 881, -17*w^5 + 4*w^4 + 123*w^3 + 58*w^2 - 79*w - 22],
[881, 881, 19*w^5 - 5*w^4 - 136*w^3 - 64*w^2 + 83*w + 26],
[911, 911, -6*w^5 + 2*w^4 + 44*w^3 + 15*w^2 - 32*w - 2],
[911, 911, -15*w^5 + 4*w^4 + 107*w^3 + 51*w^2 - 66*w - 23],
[911, 911, 13*w^5 - 4*w^4 - 94*w^3 - 37*w^2 + 63*w + 13],
[911, 911, -15*w^5 + 3*w^4 + 109*w^3 + 55*w^2 - 71*w - 20],
[911, 911, -7*w^5 + 3*w^4 + 48*w^3 + 17*w^2 - 25*w - 6],
[911, 911, 12*w^5 - 4*w^4 - 86*w^3 - 34*w^2 + 55*w + 13]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^6 - 9*x^5 - 48*x^4 + 304*x^3 + 816*x^2 - 2112*x - 4800;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, 31/4176*e^5 - 295/4176*e^4 - 275/1044*e^3 + 213/116*e^2 + 499/174*e - 634/87, 31/4176*e^5 - 295/4176*e^4 - 275/1044*e^3 + 213/116*e^2 + 499/174*e - 634/87, -e + 2, e, -e + 2, -1/1044*e^5 + 5/2088*e^4 + 259/2088*e^3 - 49/116*e^2 - 331/174*e + 514/87, -43/4176*e^5 + 499/4176*e^4 + 185/1044*e^3 - 389/116*e^2 - 50/87*e + 1306/87, 1/174*e^5 - 17/174*e^4 + 5/29*e^3 + 88/29*e^2 - 162/29*e - 622/29, -43/4176*e^5 + 499/4176*e^4 + 185/1044*e^3 - 389/116*e^2 - 50/87*e + 1306/87, -1/1044*e^5 + 5/2088*e^4 + 259/2088*e^3 - 49/116*e^2 - 331/174*e + 514/87, 1/174*e^5 - 17/174*e^4 + 5/29*e^3 + 88/29*e^2 - 162/29*e - 622/29, 7/174*e^5 - 151/348*e^4 - 121/116*e^3 + 384/29*e^2 + 84/29*e - 1918/29, 61/4176*e^5 - 805/4176*e^4 - 25/522*e^3 + 341/58*e^2 - 445/87*e - 2575/87, 97/4176*e^5 - 1069/4176*e^4 - 119/261*e^3 + 833/116*e^2 - 130/87*e - 3460/87, 37/1392*e^5 - 397/1392*e^4 - 115/174*e^3 + 495/58*e^2 + 12/29*e - 1144/29, -23/2088*e^5 + 47/522*e^4 + 1021/2088*e^3 - 259/116*e^2 - 805/174*e + 604/87, -23/2088*e^5 + 47/522*e^4 + 1021/2088*e^3 - 259/116*e^2 - 805/174*e + 604/87, 97/4176*e^5 - 1069/4176*e^4 - 119/261*e^3 + 833/116*e^2 - 130/87*e - 3460/87, 37/1392*e^5 - 397/1392*e^4 - 115/174*e^3 + 495/58*e^2 + 12/29*e - 1144/29, -47/2088*e^5 + 509/2088*e^4 + 629/1044*e^3 - 248/29*e^2 + 178/87*e + 5728/87, 49/2088*e^5 - 601/2088*e^4 - 70/261*e^3 + 253/29*e^2 - 491/87*e - 3980/87, -35/4176*e^5 + 479/4176*e^4 - 37/522*e^3 - 349/116*e^2 + 1171/174*e + 1496/87, 17/1392*e^5 - 173/1392*e^4 - 37/87*e^3 + 487/116*e^2 + 40/29*e - 720/29, 17/1392*e^5 - 173/1392*e^4 - 37/87*e^3 + 487/116*e^2 + 40/29*e - 720/29, 49/2088*e^5 - 601/2088*e^4 - 70/261*e^3 + 253/29*e^2 - 491/87*e - 3980/87, -35/4176*e^5 + 479/4176*e^4 - 37/522*e^3 - 349/116*e^2 + 1171/174*e + 1496/87, -181/4176*e^5 + 2149/4176*e^4 + 803/1044*e^3 - 1949/116*e^2 + 613/87*e + 9730/87, -3/116*e^5 + 31/87*e^4 + 49/348*e^3 - 367/29*e^2 + 236/29*e + 2770/29, -1, 163/4176*e^5 - 1843/4176*e^4 - 469/522*e^3 + 414/29*e^2 + 56/87*e - 7330/87, -11/2088*e^5 + 71/2088*e^4 + 53/261*e^3 + 3/58*e^2 - 62/87*e - 914/87, -139/2088*e^5 + 1609/2088*e^4 + 1279/1044*e^3 - 1333/58*e^2 + 221/87*e + 11276/87, -139/2088*e^5 + 1609/2088*e^4 + 1279/1044*e^3 - 1333/58*e^2 + 221/87*e + 11276/87, -11/2088*e^5 + 71/2088*e^4 + 53/261*e^3 + 3/58*e^2 - 62/87*e - 914/87, 163/4176*e^5 - 1843/4176*e^4 - 469/522*e^3 + 414/29*e^2 + 56/87*e - 7330/87, -85/1044*e^5 + 238/261*e^4 + 1829/1044*e^3 - 802/29*e^2 + 70/87*e + 13240/87, -17/2088*e^5 + 173/2088*e^4 + 61/522*e^3 - 28/29*e^2 + 7/87*e - 938/87, -85/1044*e^5 + 238/261*e^4 + 1829/1044*e^3 - 802/29*e^2 + 70/87*e + 13240/87, -11/1044*e^5 + 79/522*e^4 - 11/1044*e^3 - 142/29*e^2 + 398/87*e + 2000/87, -11/1044*e^5 + 79/522*e^4 - 11/1044*e^3 - 142/29*e^2 + 398/87*e + 2000/87, -17/2088*e^5 + 173/2088*e^4 + 61/522*e^3 - 28/29*e^2 + 7/87*e - 938/87, 11/2088*e^5 - 71/2088*e^4 - 53/261*e^3 - 3/58*e^2 + 149/87*e + 44/87, 25/696*e^5 - 35/87*e^4 - 671/696*e^3 + 1591/116*e^2 + 295/58*e - 2278/29, -16/261*e^5 + 769/1044*e^4 + 1067/1044*e^3 - 697/29*e^2 + 718/87*e + 12712/87, 11/2088*e^5 - 71/2088*e^4 - 53/261*e^3 - 3/58*e^2 + 149/87*e + 44/87, -16/261*e^5 + 769/1044*e^4 + 1067/1044*e^3 - 697/29*e^2 + 718/87*e + 12712/87, 25/696*e^5 - 35/87*e^4 - 671/696*e^3 + 1591/116*e^2 + 295/58*e - 2278/29, -11/464*e^5 + 445/1392*e^4 + 7/87*e^3 - 1249/116*e^2 + 597/58*e + 1848/29, -91/2088*e^5 + 967/2088*e^4 + 521/522*e^3 - 358/29*e^2 + 17/87*e + 4508/87, -11/464*e^5 + 445/1392*e^4 + 7/87*e^3 - 1249/116*e^2 + 597/58*e + 1848/29, -91/2088*e^5 + 967/2088*e^4 + 521/522*e^3 - 358/29*e^2 + 17/87*e + 4508/87, 1/348*e^5 - 5/696*e^4 - 259/696*e^3 + 31/116*e^2 + 563/58*e + 240/29, 1/348*e^5 - 5/696*e^4 - 259/696*e^3 + 31/116*e^2 + 563/58*e + 240/29, 113/4176*e^5 - 1457/4176*e^4 - 149/522*e^3 + 337/29*e^2 - 512/87*e - 6038/87, -17/464*e^5 + 635/1392*e^4 + 125/348*e^3 - 1577/116*e^2 + 485/58*e + 2392/29, -41/696*e^5 + 155/232*e^4 + 100/87*e^3 - 554/29*e^2 - 36/29*e + 2678/29, -41/696*e^5 + 155/232*e^4 + 100/87*e^3 - 554/29*e^2 - 36/29*e + 2678/29, -17/464*e^5 + 635/1392*e^4 + 125/348*e^3 - 1577/116*e^2 + 485/58*e + 2392/29, 113/4176*e^5 - 1457/4176*e^4 - 149/522*e^3 + 337/29*e^2 - 512/87*e - 6038/87, -247/4176*e^5 + 2923/4176*e^4 + 251/261*e^3 - 2511/116*e^2 + 775/87*e + 11686/87, -2/87*e^5 + 13/58*e^4 + 56/87*e^3 - 269/58*e^2 - 135/29*e - 238/29, -17/2088*e^5 + 65/522*e^4 + 331/2088*e^3 - 721/116*e^2 + 101/174*e + 4456/87, -2/87*e^5 + 13/58*e^4 + 56/87*e^3 - 269/58*e^2 - 135/29*e - 238/29, -17/2088*e^5 + 65/522*e^4 + 331/2088*e^3 - 721/116*e^2 + 101/174*e + 4456/87, -247/4176*e^5 + 2923/4176*e^4 + 251/261*e^3 - 2511/116*e^2 + 775/87*e + 11686/87, -13/1044*e^5 + 163/1044*e^4 + 62/261*e^3 - 333/58*e^2 - 107/87*e + 4072/87, 41/1044*e^5 - 247/522*e^4 - 481/1044*e^3 + 787/58*e^2 - 566/87*e - 5588/87, -13/1044*e^5 + 163/1044*e^4 + 62/261*e^3 - 333/58*e^2 - 107/87*e + 4072/87, 41/1044*e^5 - 247/522*e^4 - 481/1044*e^3 + 787/58*e^2 - 566/87*e - 5588/87, 1/16*e^5 - 31/48*e^4 - 11/6*e^3 + 75/4*e^2 + 23/2*e - 86, 1/16*e^5 - 31/48*e^4 - 11/6*e^3 + 75/4*e^2 + 23/2*e - 86, -313/4176*e^5 + 3349/4176*e^4 + 1901/1044*e^3 - 1319/58*e^2 - 542/87*e + 10336/87, -223/4176*e^5 + 2515/4176*e^4 + 296/261*e^3 - 2159/116*e^2 + 115/87*e + 10168/87, -223/4176*e^5 + 2515/4176*e^4 + 296/261*e^3 - 2159/116*e^2 + 115/87*e + 10168/87, 61/2088*e^5 - 631/2088*e^4 - 1057/1044*e^3 + 653/58*e^2 + 328/87*e - 7064/87, 61/2088*e^5 - 631/2088*e^4 - 1057/1044*e^3 + 653/58*e^2 + 328/87*e - 7064/87, -313/4176*e^5 + 3349/4176*e^4 + 1901/1044*e^3 - 1319/58*e^2 - 542/87*e + 10336/87, 103/4176*e^5 - 1519/4176*e^4 + 265/1044*e^3 + 1269/116*e^2 - 2591/174*e - 4840/87, 293/4176*e^5 - 3125/4176*e^4 - 1993/1044*e^3 + 2625/116*e^2 + 995/174*e - 9158/87, 293/4176*e^5 - 3125/4176*e^4 - 1993/1044*e^3 + 2625/116*e^2 + 995/174*e - 9158/87, 103/4176*e^5 - 1519/4176*e^4 + 265/1044*e^3 + 1269/116*e^2 - 2591/174*e - 4840/87, 3/116*e^5 - 31/87*e^4 - 49/348*e^3 + 367/29*e^2 - 207/29*e - 2886/29, 3/116*e^5 - 31/87*e^4 - 49/348*e^3 + 367/29*e^2 - 207/29*e - 2886/29, -35/696*e^5 + 167/348*e^4 + 433/232*e^3 - 1659/116*e^2 - 935/58*e + 2064/29, -13/174*e^5 + 163/174*e^4 + 161/174*e^3 - 1737/58*e^2 + 337/29*e + 5418/29, -13/174*e^5 + 163/174*e^4 + 161/174*e^3 - 1737/58*e^2 + 337/29*e + 5418/29, 85/2088*e^5 - 389/1044*e^4 - 3221/2088*e^3 + 1227/116*e^2 + 1757/174*e - 3140/87, -35/696*e^5 + 167/348*e^4 + 433/232*e^3 - 1659/116*e^2 - 935/58*e + 2064/29, 85/2088*e^5 - 389/1044*e^4 - 3221/2088*e^3 + 1227/116*e^2 + 1757/174*e - 3140/87, 61/4176*e^5 - 805/4176*e^4 + 211/1044*e^3 + 479/116*e^2 - 2021/174*e - 2488/87, -77/1044*e^5 + 1777/2088*e^4 + 2543/2088*e^3 - 2845/116*e^2 + 1657/174*e + 11912/87, -10/261*e^5 + 112/261*e^4 + 415/522*e^3 - 719/58*e^2 - 182/87*e + 5074/87, 61/4176*e^5 - 805/4176*e^4 + 211/1044*e^3 + 479/116*e^2 - 2021/174*e - 2488/87, -10/261*e^5 + 112/261*e^4 + 415/522*e^3 - 719/58*e^2 - 182/87*e + 5074/87, -77/1044*e^5 + 1777/2088*e^4 + 2543/2088*e^3 - 2845/116*e^2 + 1657/174*e + 11912/87, 89/4176*e^5 - 1397/4176*e^4 + 109/522*e^3 + 1373/116*e^2 - 2749/174*e - 6086/87, -163/2088*e^5 + 2017/2088*e^4 + 919/1044*e^3 - 1685/58*e^2 + 1106/87*e + 14660/87, -163/2088*e^5 + 2017/2088*e^4 + 919/1044*e^3 - 1685/58*e^2 + 1106/87*e + 14660/87, 253/2088*e^5 - 691/522*e^4 - 5663/2088*e^3 + 4357/116*e^2 + 851/174*e - 16040/87, 89/4176*e^5 - 1397/4176*e^4 + 109/522*e^3 + 1373/116*e^2 - 2749/174*e - 6086/87, 253/2088*e^5 - 691/522*e^4 - 5663/2088*e^3 + 4357/116*e^2 + 851/174*e - 16040/87, -139/4176*e^5 + 1783/4176*e^4 + 211/522*e^3 - 1739/116*e^2 + 763/87*e + 8770/87, 4/87*e^5 - 185/348*e^4 - 101/116*e^3 + 501/29*e^2 - 165/29*e - 3410/29, 4/87*e^5 - 185/348*e^4 - 101/116*e^3 + 501/29*e^2 - 165/29*e - 3410/29, -199/2088*e^5 + 2107/2088*e^4 + 682/261*e^3 - 1807/58*e^2 - 568/87*e + 13646/87, -139/4176*e^5 + 1783/4176*e^4 + 211/522*e^3 - 1739/116*e^2 + 763/87*e + 8770/87, -199/2088*e^5 + 2107/2088*e^4 + 682/261*e^3 - 1807/58*e^2 - 568/87*e + 13646/87, 37/4176*e^5 - 745/4176*e^4 + 233/522*e^3 + 765/116*e^2 - 1699/87*e - 3580/87, -41/2088*e^5 + 407/2088*e^4 + 719/1044*e^3 - 379/58*e^2 - 65/87*e + 3664/87, -23/522*e^5 + 275/522*e^4 + 325/522*e^3 - 895/58*e^2 + 565/87*e + 6940/87, -23/522*e^5 + 275/522*e^4 + 325/522*e^3 - 895/58*e^2 + 565/87*e + 6940/87, 37/4176*e^5 - 745/4176*e^4 + 233/522*e^3 + 765/116*e^2 - 1699/87*e - 3580/87, -41/2088*e^5 + 407/2088*e^4 + 719/1044*e^3 - 379/58*e^2 - 65/87*e + 3664/87, 73/2088*e^5 - 661/2088*e^4 - 328/261*e^3 + 423/58*e^2 + 973/87*e - 578/87, 13/2088*e^5 - 163/2088*e^4 - 385/1044*e^3 + 134/29*e^2 + 706/87*e - 3428/87, -31/522*e^5 + 677/1044*e^4 + 1243/1044*e^3 - 997/58*e^2 + 92/87*e + 5942/87, 73/2088*e^5 - 661/2088*e^4 - 328/261*e^3 + 423/58*e^2 + 973/87*e - 578/87, -31/522*e^5 + 677/1044*e^4 + 1243/1044*e^3 - 997/58*e^2 + 92/87*e + 5942/87, 13/2088*e^5 - 163/2088*e^4 - 385/1044*e^3 + 134/29*e^2 + 706/87*e - 3428/87, 91/1392*e^5 - 967/1392*e^4 - 152/87*e^3 + 2351/116*e^2 + 354/29*e - 2950/29, -11/348*e^5 + 25/87*e^4 + 151/116*e^3 - 475/58*e^2 - 414/29*e + 1188/29, -11/348*e^5 + 25/87*e^4 + 151/116*e^3 - 475/58*e^2 - 414/29*e + 1188/29, 79/1044*e^5 - 937/1044*e^4 - 787/522*e^3 + 888/29*e^2 - 715/87*e - 17638/87, 91/1392*e^5 - 967/1392*e^4 - 152/87*e^3 + 2351/116*e^2 + 354/29*e - 2950/29, 79/1044*e^5 - 937/1044*e^4 - 787/522*e^3 + 888/29*e^2 - 715/87*e - 17638/87, 149/4176*e^5 - 1373/4176*e^4 - 1159/1044*e^3 + 803/116*e^2 + 586/87*e + 1690/87, -113/4176*e^5 + 1457/4176*e^4 + 149/522*e^3 - 337/29*e^2 + 860/87*e + 4298/87, 79/1044*e^5 - 937/1044*e^4 - 263/261*e^3 + 1515/58*e^2 - 1498/87*e - 12766/87, -113/4176*e^5 + 1457/4176*e^4 + 149/522*e^3 - 337/29*e^2 + 860/87*e + 4298/87, 79/1044*e^5 - 937/1044*e^4 - 263/261*e^3 + 1515/58*e^2 - 1498/87*e - 12766/87, -149/4176*e^5 + 1373/4176*e^4 + 1681/1044*e^3 - 1383/116*e^2 - 2477/174*e + 6662/87, -149/4176*e^5 + 1373/4176*e^4 + 1681/1044*e^3 - 1383/116*e^2 - 2477/174*e + 6662/87, 187/2088*e^5 - 995/1044*e^4 - 4859/2088*e^3 + 3233/116*e^2 + 1499/174*e - 13346/87, 5/116*e^5 - 113/174*e^4 + 131/348*e^3 + 631/29*e^2 - 693/29*e - 4288/29, -119/2088*e^5 + 1385/2088*e^4 + 941/1044*e^3 - 1175/58*e^2 + 1267/87*e + 9442/87, 5/116*e^5 - 113/174*e^4 + 131/348*e^3 + 631/29*e^2 - 693/29*e - 4288/29, -119/2088*e^5 + 1385/2088*e^4 + 941/1044*e^3 - 1175/58*e^2 + 1267/87*e + 9442/87, 187/2088*e^5 - 995/1044*e^4 - 4859/2088*e^3 + 3233/116*e^2 + 1499/174*e - 13346/87, -397/4176*e^5 + 4777/4176*e^4 + 383/261*e^3 - 4189/116*e^2 + 724/87*e + 19216/87, 89/1044*e^5 - 2011/2088*e^4 - 3737/2088*e^3 + 3375/116*e^2 + 53/174*e - 13382/87, 89/4176*e^5 - 1049/4176*e^4 - 217/1044*e^3 + 735/116*e^2 - 461/87*e - 344/87, 89/1044*e^5 - 2011/2088*e^4 - 3737/2088*e^3 + 3375/116*e^2 + 53/174*e - 13382/87, 89/4176*e^5 - 1049/4176*e^4 - 217/1044*e^3 + 735/116*e^2 - 461/87*e - 344/87, -397/4176*e^5 + 4777/4176*e^4 + 383/261*e^3 - 4189/116*e^2 + 724/87*e + 19216/87, -55/2088*e^5 + 529/2088*e^4 + 1147/1044*e^3 - 268/29*e^2 - 658/87*e + 7088/87, -17/2088*e^5 - 1/2088*e^4 + 1079/1044*e^3 + 31/58*e^2 - 1385/87*e - 2330/87, -55/2088*e^5 + 529/2088*e^4 + 1147/1044*e^3 - 268/29*e^2 - 658/87*e + 7088/87, -17/2088*e^5 - 1/2088*e^4 + 1079/1044*e^3 + 31/58*e^2 - 1385/87*e - 2330/87, -337/4176*e^5 + 3757/4176*e^4 + 991/522*e^3 - 3251/116*e^2 + 118/87*e + 13246/87, -337/4176*e^5 + 3757/4176*e^4 + 991/522*e^3 - 3251/116*e^2 + 118/87*e + 13246/87, 523/4176*e^5 - 6223/4176*e^4 - 560/261*e^3 + 5515/116*e^2 - 883/87*e - 24532/87, -35/522*e^5 + 1655/2088*e^4 + 2731/2088*e^3 - 3053/116*e^2 + 929/174*e + 13882/87, 523/4176*e^5 - 6223/4176*e^4 - 560/261*e^3 + 5515/116*e^2 - 883/87*e - 24532/87, 95/696*e^5 - 1151/696*e^4 - 173/87*e^3 + 1481/29*e^2 - 411/29*e - 8610/29, 95/696*e^5 - 1151/696*e^4 - 173/87*e^3 + 1481/29*e^2 - 411/29*e - 8610/29, -35/522*e^5 + 1655/2088*e^4 + 2731/2088*e^3 - 3053/116*e^2 + 929/174*e + 13882/87];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {169,13,7*w^5 - w^4 - 51*w^3 - 29*w^2 + 32*w + 12}>] := 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;