/*
  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![-67, 0, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {3, 3, -w + 8}>;

primesArray := [
[2, 2, -27*w + 221],
[3, 3, -w + 8],
[3, 3, -w - 8],
[7, 7, -11*w + 90],
[7, 7, -11*w - 90],
[11, 11, 6*w - 49],
[11, 11, 6*w + 49],
[17, 17, 4*w + 33],
[17, 17, -4*w + 33],
[25, 5, -5],
[29, 29, -70*w + 573],
[29, 29, 151*w - 1236],
[31, 31, -w - 6],
[31, 31, w - 6],
[37, 37, -21*w - 172],
[37, 37, -21*w + 172],
[43, 43, 2*w - 15],
[43, 43, 2*w + 15],
[67, 67, -w],
[73, 73, -3*w - 26],
[73, 73, -3*w + 26],
[79, 79, 92*w - 753],
[79, 79, 313*w - 2562],
[89, 89, -5*w - 42],
[89, 89, -5*w + 42],
[139, 139, 10*w + 81],
[139, 139, 10*w - 81],
[149, 149, 19*w - 156],
[149, 149, -19*w - 156],
[157, 157, -102*w + 835],
[157, 157, 561*w - 4592],
[169, 13, -13],
[173, 173, 2*w - 21],
[173, 173, -2*w - 21],
[179, 179, 87*w - 712],
[179, 179, 87*w + 712],
[181, 181, 3*w - 28],
[181, 181, 3*w + 28],
[191, 191, 15*w - 122],
[191, 191, 15*w + 122],
[193, 193, 36*w + 295],
[193, 193, -36*w + 295],
[239, 239, 12*w + 97],
[239, 239, 12*w - 97],
[241, 241, -183*w + 1498],
[241, 241, 480*w - 3929],
[251, 251, 45*w + 368],
[251, 251, 45*w - 368],
[257, 257, -w - 18],
[257, 257, w - 18],
[269, 269, 41*w + 336],
[269, 269, -41*w + 336],
[271, 271, 40*w - 327],
[271, 271, -40*w - 327],
[277, 277, 426*w - 3487],
[277, 277, -237*w + 1940],
[293, 293, -998*w + 8169],
[293, 293, 107*w - 876],
[311, 311, 804*w - 6581],
[311, 311, 141*w - 1154],
[317, 317, -7*w + 60],
[317, 317, -7*w - 60],
[331, 331, 254*w - 2079],
[331, 331, 475*w - 3888],
[347, 347, -3*w - 16],
[347, 347, 3*w - 16],
[349, 349, -9*w - 76],
[349, 349, -9*w + 76],
[361, 19, -19],
[367, 367, 7*w + 54],
[367, 367, 7*w - 54],
[379, 379, 5*w + 36],
[379, 379, 5*w - 36],
[383, 383, 168*w - 1375],
[383, 383, 831*w - 6802],
[389, 389, 34*w - 279],
[389, 389, -34*w - 279],
[397, 397, -6*w - 53],
[397, 397, -6*w + 53],
[421, 421, -3*w - 32],
[421, 421, 3*w - 32],
[443, 443, -27*w - 220],
[443, 443, -27*w + 220],
[449, 449, -4*w - 39],
[449, 449, 4*w - 39],
[457, 457, 51*w + 418],
[457, 457, -51*w + 418],
[461, 461, 2*w - 27],
[461, 461, -2*w - 27],
[463, 463, 136*w + 1113],
[463, 463, 136*w - 1113],
[487, 487, 32*w + 261],
[487, 487, -32*w + 261],
[499, 499, -62*w - 507],
[499, 499, 62*w - 507],
[503, 503, -3*w - 10],
[503, 503, 3*w - 10],
[509, 509, -w - 24],
[509, 509, w - 24],
[529, 23, -23],
[547, 547, 47*w + 384],
[547, 547, -47*w + 384],
[557, 557, 14*w - 117],
[557, 557, 14*w + 117],
[563, 563, 6*w + 43],
[563, 563, 6*w - 43],
[569, 569, -56*w - 459],
[569, 569, 56*w - 459],
[587, 587, -3*w - 4],
[587, 587, 3*w - 4],
[599, 599, 3*w - 2],
[599, 599, -3*w - 2],
[601, 601, 117*w - 958],
[601, 601, 117*w + 958],
[613, 613, 6*w - 55],
[613, 613, -6*w - 55],
[617, 617, -269*w + 2202],
[617, 617, 836*w - 6843],
[631, 631, -4*w - 21],
[631, 631, 4*w - 21],
[647, 647, 21*w - 170],
[647, 647, 21*w + 170],
[683, 683, 18*w + 145],
[683, 683, 18*w - 145],
[709, 709, 30*w - 247],
[709, 709, 30*w + 247],
[727, 727, 416*w - 3405],
[727, 727, 637*w - 5214],
[739, 739, 158*w - 1293],
[739, 739, 158*w + 1293],
[761, 761, -100*w - 819],
[761, 761, -100*w + 819],
[773, 773, 122*w + 999],
[773, 773, 122*w - 999],
[787, 787, -1322*w + 10821],
[787, 787, -217*w + 1776],
[797, 797, 674*w - 5517],
[797, 797, -431*w + 3528],
[811, 811, 14*w - 111],
[811, 811, 14*w + 111],
[821, 821, 2*w - 33],
[821, 821, -2*w - 33],
[829, 829, -66*w - 541],
[829, 829, 66*w - 541],
[853, 853, -171*w - 1400],
[853, 853, 171*w - 1400],
[877, 877, 54*w - 443],
[877, 877, -54*w - 443],
[881, 881, 1300*w - 10641],
[881, 881, -247*w + 2022],
[883, 883, 31*w + 252],
[883, 883, 31*w - 252],
[919, 919, 13*w + 102],
[919, 919, 13*w - 102],
[953, 953, 4*w - 45],
[953, 953, -4*w - 45],
[977, 977, -71*w - 582],
[977, 977, 71*w - 582],
[983, 983, 36*w - 293],
[983, 983, 36*w + 293],
[991, 991, -4*w - 9],
[991, 991, 4*w - 9],
[997, 997, 3*w - 40],
[997, 997, -3*w - 40]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^7 - 2*x^6 - 6*x^5 + 12*x^4 + 9*x^3 - 18*x^2 - 4*x + 7;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, -1, -e^6 + e^5 + 6*e^4 - 4*e^3 - 8*e^2 + e + 2, -2*e^6 + e^5 + 13*e^4 - 4*e^3 - 23*e^2 + e + 12, 2*e^6 - 15*e^4 - e^3 + 32*e^2 + 3*e - 17, e^6 - 2*e^5 - 5*e^4 + 9*e^3 + 5*e^2 - 5*e, -e^6 + 3*e^5 + 5*e^4 - 15*e^3 - 4*e^2 + 12*e - 1, -e^6 + e^5 + 7*e^4 - 4*e^3 - 15*e^2 + 2*e + 11, -4*e^6 + 4*e^5 + 24*e^4 - 20*e^3 - 34*e^2 + 17*e + 10, -e^6 + 7*e^4 + 2*e^3 - 13*e^2 - 5*e + 2, -5*e^6 + 5*e^5 + 29*e^4 - 22*e^3 - 35*e^2 + 12*e + 3, 2*e^5 - e^4 - 13*e^3 + 5*e^2 + 16*e - 5, -4*e^6 + 3*e^5 + 26*e^4 - 16*e^3 - 44*e^2 + 19*e + 18, -e^5 + 6*e^3 - 7*e + 5, 3*e^6 - 3*e^5 - 20*e^4 + 14*e^3 + 35*e^2 - 8*e - 16, -2*e^6 + 2*e^5 + 10*e^4 - 7*e^3 - 4*e^2 - 3*e - 7, -e^6 - e^5 + 10*e^4 + 3*e^3 - 26*e^2 + 3*e + 18, 3*e^6 - 3*e^5 - 19*e^4 + 16*e^3 + 33*e^2 - 17*e - 15, -5*e^6 + 6*e^5 + 27*e^4 - 29*e^3 - 32*e^2 + 22*e + 14, 3*e^6 - 3*e^5 - 17*e^4 + 12*e^3 + 18*e^2 - 6*e + 4, e^6 - 6*e^4 + 8*e^2 + 3*e - 3, 4*e^6 - 5*e^5 - 25*e^4 + 24*e^3 + 41*e^2 - 20*e - 18, -3*e^6 + 3*e^5 + 13*e^4 - 12*e^3 - 4*e^2 + 4*e - 4, -8*e^6 + 10*e^5 + 47*e^4 - 49*e^3 - 63*e^2 + 39*e + 20, e^6 + 2*e^5 - 6*e^4 - 10*e^3 + 8*e^2 + 6*e - 2, 4*e^5 + e^4 - 19*e^3 - 10*e^2 + 13*e + 17, -2*e^6 - 3*e^5 + 17*e^4 + 16*e^3 - 41*e^2 - 19*e + 33, e^6 - 2*e^5 - 3*e^4 + 7*e^3 - 8*e^2 + 3*e + 13, -e^6 + 2*e^5 + e^4 - 15*e^3 + 17*e^2 + 25*e - 20, -3*e^6 - 4*e^5 + 24*e^4 + 26*e^3 - 49*e^2 - 38*e + 20, 3*e^5 - 2*e^4 - 16*e^3 + 7*e^2 + 16*e + 1, 6*e^6 - 7*e^5 - 41*e^4 + 35*e^3 + 78*e^2 - 30*e - 45, -6*e^5 + 5*e^4 + 31*e^3 - 26*e^2 - 24*e + 16, e^6 + 3*e^5 - 12*e^4 - 21*e^3 + 36*e^2 + 35*e - 23, e^6 + 3*e^5 - 15*e^4 - 18*e^3 + 45*e^2 + 22*e - 19, 3*e^6 - 8*e^5 - 11*e^4 + 43*e^3 - 6*e^2 - 40*e + 13, -e^6 + 6*e^5 + e^4 - 32*e^3 + 12*e^2 + 28*e - 4, e^6 - 5*e^5 - e^4 + 21*e^3 - 9*e^2 - 8*e + 5, -2*e^6 + 4*e^5 + 18*e^4 - 24*e^3 - 49*e^2 + 21*e + 40, -5*e^6 + 8*e^5 + 25*e^4 - 40*e^3 - 10*e^2 + 33*e - 20, -2*e^6 - 4*e^5 + 25*e^4 + 20*e^3 - 77*e^2 - 17*e + 52, 5*e^6 - e^5 - 36*e^4 + 6*e^3 + 72*e^2 - 12*e - 38, 5*e^6 - 10*e^5 - 27*e^4 + 49*e^3 + 30*e^2 - 40*e - 4, -12*e^6 + 19*e^5 + 63*e^4 - 97*e^3 - 60*e^2 + 85*e + 12, 5*e^6 + 6*e^5 - 42*e^4 - 38*e^3 + 92*e^2 + 49*e - 33, 8*e^6 - 17*e^5 - 43*e^4 + 87*e^3 + 46*e^2 - 80*e - 9, 19*e^6 - 22*e^5 - 108*e^4 + 105*e^3 + 139*e^2 - 81*e - 39, 2*e^6 - 4*e^5 - 6*e^4 + 20*e^3 - 14*e^2 - 16*e + 16, 3*e^5 + 5*e^4 - 18*e^3 - 27*e^2 + 14*e + 28, 9*e^6 - 15*e^5 - 50*e^4 + 68*e^3 + 59*e^2 - 41*e - 14, -8*e^6 + 5*e^5 + 57*e^4 - 27*e^3 - 111*e^2 + 28*e + 48, 7*e^6 - 51*e^4 - 5*e^3 + 99*e^2 + 6*e - 37, e^6 + e^5 - 13*e^4 - 4*e^3 + 44*e^2 + 6*e - 34, -12*e^6 + 20*e^5 + 63*e^4 - 101*e^3 - 67*e^2 + 84*e + 19, -4*e^6 + 9*e^5 + 19*e^4 - 50*e^3 - 13*e^2 + 60*e + 2, 3*e^6 - 8*e^5 - 10*e^4 + 37*e^3 - 13*e^2 - 25*e + 20, 10*e^6 + 2*e^5 - 78*e^4 - 17*e^3 + 169*e^2 + 25*e - 83, -7*e^6 + 10*e^5 + 42*e^4 - 47*e^3 - 74*e^2 + 31*e + 53, -8*e^6 + 8*e^5 + 51*e^4 - 41*e^3 - 82*e^2 + 40*e + 29, -2*e^6 - 10*e^5 + 29*e^4 + 50*e^3 - 105*e^2 - 32*e + 88, -12*e^6 + 7*e^5 + 75*e^4 - 32*e^3 - 108*e^2 + 23*e + 20, 7*e^6 - 11*e^5 - 43*e^4 + 56*e^3 + 71*e^2 - 52*e - 38, -2*e^6 + e^5 + 14*e^4 - 34*e^2 - 10*e + 23, 2*e^6 - 10*e^4 + 7*e^2 - 4*e + 10, -10*e^6 + 14*e^5 + 61*e^4 - 74*e^3 - 89*e^2 + 63*e + 27, 12*e^6 - 3*e^5 - 81*e^4 + 3*e^3 + 151*e^2 + 23*e - 74, -2*e^6 + 2*e^5 + 14*e^4 + e^3 - 31*e^2 - 23*e + 16, 9*e^6 - 5*e^5 - 58*e^4 + 17*e^3 + 104*e^2 + 3*e - 52, 16*e^6 - 19*e^5 - 90*e^4 + 94*e^3 + 108*e^2 - 73*e - 25, -2*e^6 - e^5 + 8*e^4 + 10*e^3 - 4*e^2 - 11*e + 14, 6*e^6 - 4*e^5 - 28*e^4 + 14*e^3 + 13*e^2 + 27, 5*e^6 - 3*e^5 - 28*e^4 + 13*e^3 + 39*e^2 - 9*e - 8, -4*e^6 + 10*e^5 + 13*e^4 - 53*e^3 + 11*e^2 + 64*e - 5, -8*e^6 + 6*e^5 + 56*e^4 - 29*e^3 - 104*e^2 + 29*e + 39, 10*e^5 - 16*e^4 - 42*e^3 + 80*e^2 + 17*e - 66, -e^6 - e^5 + 15*e^4 + 4*e^3 - 60*e^2 + 13*e + 55, -17*e^6 + 19*e^5 + 101*e^4 - 92*e^3 - 139*e^2 + 69*e + 51, -10*e^6 + 13*e^5 + 57*e^4 - 62*e^3 - 79*e^2 + 58*e + 35, 2*e^6 + 6*e^5 - 31*e^4 - 34*e^3 + 113*e^2 + 34*e - 76, -6*e^6 - 2*e^5 + 52*e^4 + 8*e^3 - 121*e^2 + 4*e + 70, 12*e^6 - 19*e^5 - 65*e^4 + 90*e^3 + 72*e^2 - 69*e - 4, 3*e^6 - 9*e^5 - 8*e^4 + 45*e^3 - 23*e^2 - 36*e + 36, 5*e^6 - 6*e^5 - 22*e^4 + 29*e^3 - 4*e^2 - 27*e + 36, 7*e^6 - 7*e^5 - 37*e^4 + 41*e^3 + 30*e^2 - 48*e + 4, 9*e^6 - 3*e^5 - 54*e^4 + 7*e^3 + 75*e^2 + 11*e - 25, -10*e^6 + 5*e^5 + 61*e^4 - 16*e^3 - 89*e^2 - 11*e + 17, 13*e^6 - 22*e^5 - 62*e^4 + 105*e^3 + 34*e^2 - 76*e + 24, 9*e^6 - 4*e^5 - 56*e^4 + 7*e^3 + 88*e^2 + 28*e - 36, 14*e^6 - 16*e^5 - 81*e^4 + 74*e^3 + 92*e^2 - 46*e - 4, -7*e^6 + 3*e^5 + 48*e^4 - 10*e^3 - 90*e^2 - 8*e + 45, 12*e^6 - 4*e^5 - 87*e^4 + 13*e^3 + 175*e^2 - 2*e - 90, -4*e^6 + 15*e^5 + 13*e^4 - 80*e^3 + 15*e^2 + 71*e - 16, -10*e^6 + 14*e^5 + 60*e^4 - 73*e^3 - 88*e^2 + 67*e + 26, 11*e^6 - 16*e^5 - 65*e^4 + 81*e^3 + 80*e^2 - 63*e - 7, 13*e^6 - 5*e^5 - 90*e^4 + 19*e^3 + 174*e^2 + 3*e - 88, 17*e^6 - 16*e^5 - 100*e^4 + 68*e^3 + 144*e^2 - 34*e - 58, -5*e^6 - 7*e^5 + 32*e^4 + 48*e^3 - 53*e^2 - 74*e + 23, 15*e^6 - 16*e^5 - 101*e^4 + 80*e^3 + 173*e^2 - 63*e - 67, -2*e^6 + 12*e^5 + 10*e^4 - 60*e^3 - 22*e^2 + 51*e + 29, 2*e^6 + 4*e^5 - 23*e^4 - 19*e^3 + 69*e^2 + 17*e - 46, 14*e^6 - 10*e^5 - 85*e^4 + 50*e^3 + 109*e^2 - 37*e - 6, 6*e^6 - 3*e^5 - 54*e^4 + 13*e^3 + 146*e^2 - 17*e - 97, -9*e^6 + 13*e^5 + 56*e^4 - 64*e^3 - 98*e^2 + 50*e + 63, -22*e^6 + 2*e^5 + 153*e^4 + 3*e^3 - 289*e^2 - 33*e + 137, 14*e^6 - 6*e^5 - 96*e^4 + 30*e^3 + 175*e^2 - 36*e - 71, 11*e^6 - 7*e^5 - 76*e^4 + 33*e^3 + 141*e^2 - 16*e - 69, -e^6 + 6*e^5 + 10*e^4 - 26*e^3 - 30*e^2 + 8*e + 13, 11*e^6 - 8*e^5 - 71*e^4 + 41*e^3 + 117*e^2 - 50*e - 29, -10*e^6 + 2*e^5 + 74*e^4 - 7*e^3 - 153*e^2 + 13*e + 88, -e^6 - 12*e^5 + 28*e^4 + 62*e^3 - 112*e^2 - 60*e + 74, -11*e^5 + 61*e^3 + e^2 - 58*e - 18, 15*e^6 - 13*e^5 - 87*e^4 + 64*e^3 + 108*e^2 - 53*e - 16, -18*e^6 + 17*e^5 + 105*e^4 - 86*e^3 - 139*e^2 + 76*e + 37, -5*e^6 + 10*e^5 + 22*e^4 - 41*e^3 + 6*e^2 + 6*e - 32, 15*e^6 - 12*e^5 - 87*e^4 + 43*e^3 + 116*e^2 - e - 22, e^6 + 2*e^5 - 5*e^4 - 11*e^3 - e^2 + 5*e + 20, -4*e^6 + 2*e^5 + 32*e^4 - 12*e^3 - 72*e^2 + 14*e + 31, 4*e^6 - 6*e^5 - 22*e^4 + 25*e^3 + 43*e^2 - e - 51, -7*e^6 + e^5 + 45*e^4 - 7*e^3 - 60*e^2 + 8*e + 2, 8*e^6 - 10*e^5 - 42*e^4 + 48*e^3 + 56*e^2 - 38*e - 41, -e^6 + 4*e^5 - 19*e^3 + 18*e^2 + 13*e + 18, 3*e^6 - 7*e^5 - 4*e^4 + 38*e^3 - 36*e^2 - 44*e + 24, -5*e^6 + 12*e^5 + 30*e^4 - 66*e^3 - 52*e^2 + 83*e + 30, 3*e^6 - 4*e^5 - 9*e^4 + 12*e^3 - 22*e^2 + 8*e + 37, 2*e^6 + 11*e^5 - 20*e^4 - 69*e^3 + 53*e^2 + 73*e - 23, -2*e^6 - 20*e^5 + 40*e^4 + 107*e^3 - 144*e^2 - 113*e + 83, -14*e^6 + 15*e^5 + 85*e^4 - 74*e^3 - 123*e^2 + 58*e + 52, 16*e^6 - 22*e^5 - 81*e^4 + 105*e^3 + 68*e^2 - 78*e - 5, 8*e^6 - 5*e^5 - 57*e^4 + 20*e^3 + 117*e^2 - 11*e - 52, -e^6 - 6*e^4 + 13*e^3 + 54*e^2 - 36*e - 39, 21*e^6 - 20*e^5 - 131*e^4 + 93*e^3 + 207*e^2 - 67*e - 78, -21*e^6 + 18*e^5 + 117*e^4 - 68*e^3 - 134*e^2 + 13*e + 12, e^6 - 15*e^5 + 14*e^4 + 86*e^3 - 83*e^2 - 101*e + 52, 16*e^6 - 21*e^5 - 82*e^4 + 101*e^3 + 78*e^2 - 92*e - 11, -17*e^6 + 9*e^5 + 112*e^4 - 43*e^3 - 189*e^2 + 24*e + 95, 3*e^6 + 12*e^5 - 32*e^4 - 66*e^3 + 92*e^2 + 67*e - 43, 16*e^6 - 13*e^5 - 110*e^4 + 65*e^3 + 209*e^2 - 59*e - 108, -4*e^6 + 15*e^5 + 17*e^4 - 79*e^3 - 7*e^2 + 65*e - 9, -11*e^6 + e^5 + 80*e^4 + 11*e^3 - 147*e^2 - 48*e + 51, -7*e^6 + 5*e^5 + 33*e^4 - 29*e^3 - 7*e^2 + 38*e - 29, 3*e^6 - 2*e^5 - 7*e^4 + 4*e^3 - 25*e^2 + 21*e + 31, 3*e^6 + 3*e^5 - 28*e^4 - 19*e^3 + 71*e^2 + 31*e - 54, 3*e^6 + 3*e^5 - 29*e^4 - 15*e^3 + 93*e^2 + 11*e - 81, 12*e^6 + 5*e^5 - 101*e^4 - 34*e^3 + 240*e^2 + 36*e - 143, -e^6 + 8*e^5 + 2*e^4 - 35*e^3 + 17*e^2 + 12*e - 35, 5*e^6 - 9*e^5 - 32*e^4 + 40*e^3 + 55*e^2 - 13*e - 22, 9*e^6 - 25*e^5 - 33*e^4 + 124*e^3 - 26*e^2 - 95*e + 60, 10*e^6 + e^5 - 77*e^4 - 8*e^3 + 153*e^2 + 3*e - 65, -4*e^6 + 12*e^5 + 9*e^4 - 59*e^3 + 30*e^2 + 50*e - 31, e^6 + 9*e^5 - 7*e^4 - 45*e^3 + 8*e^2 + 23*e + 19, -2*e^6 + 7*e^5 + 11*e^4 - 32*e^3 - 19*e^2 + 8*e + 26, 3*e^6 + 3*e^5 - 13*e^4 - 26*e^3 + 3*e^2 + 50*e + 16, -4*e^6 + 11*e^5 + 16*e^4 - 47*e^3 + 10*e^2 + 4*e - 28, -26*e^6 + 9*e^5 + 181*e^4 - 38*e^3 - 340*e^2 + 18*e + 172, -e^6 + 10*e^5 - 13*e^4 - 53*e^3 + 68*e^2 + 48*e - 22, -9*e^6 - 7*e^5 + 66*e^4 + 34*e^3 - 123*e^2 - 22*e + 32, 7*e^6 - 18*e^5 - 23*e^4 + 87*e^3 - 19*e^2 - 62*e + 33, -12*e^6 + 7*e^5 + 74*e^4 - 36*e^3 - 101*e^2 + 32*e + 7, 15*e^6 - 11*e^5 - 106*e^4 + 47*e^3 + 205*e^2 - 14*e - 115, 2*e^6 + 22*e^5 - 36*e^4 - 128*e^3 + 128*e^2 + 141*e - 71, 23*e^6 - 32*e^5 - 127*e^4 + 148*e^3 + 150*e^2 - 105*e - 47, 2*e^6 + 17*e^5 - 34*e^4 - 98*e^3 + 109*e^2 + 108*e - 51, -18*e^6 + 19*e^5 + 105*e^4 - 85*e^3 - 131*e^2 + 45*e + 47, 7*e^6 - 9*e^5 - 21*e^4 + 41*e^3 - 43*e^2 - 37*e + 77];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {3, 3, -w + 8}>] := 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;