/*
  This code can be loaded, or copied and paste using cpaste, into Sage.
  It will load the data associated to the HMF, including
  the field, level, and Hecke and Atkin-Lehner eigenvalue data.
*/

P.<x> = PolynomialRing(QQ)
g = P([-27, -1, 1])
F.<w> = NumberField(g)
ZF = F.ring_of_integers()

NN = ZF.ideal([7,7,-w - 4])

primes_array = [
[3, 3, -w + 6],\
[3, 3, w + 5],\
[4, 2, 2],\
[5, 5, -3*w + 17],\
[5, 5, -3*w - 14],\
[7, 7, w - 5],\
[7, 7, w + 4],\
[29, 29, -w - 7],\
[29, 29, -w + 8],\
[31, 31, -5*w + 28],\
[31, 31, -5*w - 23],\
[43, 43, 6*w + 29],\
[43, 43, -6*w + 35],\
[61, 61, 3*w - 19],\
[61, 61, -3*w - 16],\
[71, 71, -7*w - 34],\
[71, 71, 7*w - 41],\
[73, 73, 2*w - 7],\
[73, 73, -2*w - 5],\
[83, 83, -w - 10],\
[83, 83, w - 11],\
[89, 89, 3*w - 14],\
[89, 89, 3*w + 11],\
[97, 97, 3*w + 17],\
[97, 97, 3*w - 20],\
[109, 109, 2*w - 1],\
[113, 113, -3*w - 10],\
[113, 113, 3*w - 13],\
[121, 11, -11],\
[131, 131, 5*w - 31],\
[131, 131, -5*w - 26],\
[137, 137, -9*w - 41],\
[137, 137, 9*w - 50],\
[157, 157, -14*w + 79],\
[157, 157, 14*w + 65],\
[169, 13, -13],\
[173, 173, -3*w - 7],\
[173, 173, 3*w - 10],\
[191, 191, -10*w - 49],\
[191, 191, 10*w - 59],\
[193, 193, 22*w + 103],\
[193, 193, 22*w - 125],\
[197, 197, 6*w + 25],\
[197, 197, -6*w + 31],\
[211, 211, -4*w - 13],\
[211, 211, 4*w - 17],\
[223, 223, 8*w + 35],\
[223, 223, -8*w + 43],\
[227, 227, 9*w - 49],\
[227, 227, 9*w + 40],\
[233, 233, 3*w - 5],\
[233, 233, -3*w - 2],\
[239, 239, -3*w - 1],\
[239, 239, 3*w - 4],\
[263, 263, 33*w - 188],\
[263, 263, 33*w + 155],\
[281, 281, 8*w - 49],\
[281, 281, -8*w - 41],\
[289, 17, -17],\
[293, 293, 4*w - 29],\
[293, 293, 4*w + 25],\
[307, 307, 3*w - 25],\
[307, 307, -3*w - 22],\
[311, 311, -5*w - 29],\
[311, 311, 5*w - 34],\
[331, 331, 19*w + 88],\
[331, 331, -19*w + 107],\
[347, 347, 23*w - 133],\
[347, 347, 36*w - 205],\
[349, 349, 15*w - 88],\
[349, 349, -15*w - 73],\
[353, 353, -w - 19],\
[353, 353, w - 20],\
[361, 19, -19],\
[373, 373, 23*w - 130],\
[373, 373, 23*w + 107],\
[401, 401, -9*w + 47],\
[401, 401, 9*w + 38],\
[409, 409, 5*w - 19],\
[409, 409, -5*w - 14],\
[421, 421, -10*w + 53],\
[421, 421, 10*w + 43],\
[431, 431, -15*w - 68],\
[431, 431, 15*w - 83],\
[433, 433, 43*w - 245],\
[433, 433, 43*w + 202],\
[439, 439, 35*w - 199],\
[439, 439, 35*w + 164],\
[443, 443, 12*w + 53],\
[443, 443, 12*w - 65],\
[457, 457, 3*w - 28],\
[457, 457, -3*w - 25],\
[461, 461, 21*w - 118],\
[461, 461, 21*w + 97],\
[463, 463, -6*w - 35],\
[463, 463, 6*w - 41],\
[467, 467, 2*w - 25],\
[467, 467, -2*w - 23],\
[479, 479, -w - 22],\
[479, 479, w - 23],\
[499, 499, -5*w - 11],\
[499, 499, 5*w - 16],\
[509, 509, -5*w - 32],\
[509, 509, 5*w - 37],\
[523, 523, -7*w - 25],\
[523, 523, 7*w - 32],\
[529, 23, -23],\
[541, 541, 11*w + 47],\
[541, 541, -11*w + 58],\
[557, 557, 7*w - 47],\
[557, 557, -7*w - 40],\
[571, 571, -5*w - 8],\
[571, 571, 5*w - 13],\
[593, 593, -16*w - 79],\
[593, 593, 16*w - 95],\
[619, 619, 6*w - 43],\
[619, 619, 6*w + 37],\
[647, 647, -9*w + 44],\
[647, 647, -9*w - 35],\
[653, 653, -4*w - 31],\
[653, 653, 4*w - 35],\
[659, 659, 45*w + 211],\
[659, 659, 45*w - 256],\
[661, 661, 5*w - 7],\
[661, 661, -5*w - 2],\
[683, 683, 27*w + 125],\
[683, 683, -27*w + 152],\
[727, 727, -27*w - 130],\
[727, 727, 27*w - 157],\
[743, 743, 14*w - 85],\
[743, 743, -14*w - 71],\
[751, 751, 40*w + 187],\
[751, 751, 40*w - 227],\
[797, 797, -13*w - 67],\
[797, 797, 13*w - 80],\
[809, 809, -30*w + 169],\
[809, 809, 30*w + 139],\
[811, 811, 3*w - 34],\
[811, 811, -3*w - 31],\
[823, 823, -13*w + 68],\
[823, 823, 13*w + 55],\
[827, 827, 7*w - 50],\
[827, 827, 7*w + 43],\
[829, 829, -7*w - 19],\
[829, 829, 7*w - 26],\
[857, 857, -8*w - 47],\
[857, 857, 8*w - 55],\
[863, 863, -11*w + 70],\
[863, 863, -11*w - 59],\
[877, 877, -3*w - 32],\
[877, 877, 3*w - 35],\
[881, 881, -6*w - 7],\
[881, 881, 6*w - 13],\
[887, 887, -21*w - 95],\
[887, 887, 21*w - 116],\
[907, 907, 19*w - 104],\
[907, 907, 19*w + 85],\
[947, 947, 9*w - 40],\
[947, 947, -9*w - 31],\
[953, 953, 29*w + 140],\
[953, 953, -29*w + 169],\
[977, 977, 6*w - 5],\
[977, 977, 6*w - 1],\
[997, 997, -26*w - 119],\
[997, 997, 26*w - 145]]
primes = [ZF.ideal(I) for I in primes_array]

heckePol = x^6 - 9*x^4 + 2*x^3 + 9*x^2 + x - 1
K.<e> = NumberField(heckePol)

hecke_eigenvalues_array = [e, e^5 - 9*e^3 + 2*e^2 + 8*e, -2*e^5 + e^4 + 17*e^3 - 13*e^2 - 8*e + 2, 3*e^5 - 2*e^4 - 26*e^3 + 23*e^2 + 14*e - 6, -e^5 + e^4 + 9*e^3 - 10*e^2 - 6*e + 2, -3*e^5 + e^4 + 26*e^3 - 15*e^2 - 18*e + 3, 1, -3*e^5 + 26*e^3 - 6*e^2 - 17*e - 3, 5*e^5 - 2*e^4 - 43*e^3 + 27*e^2 + 26*e - 1, -5*e^5 + 4*e^4 + 43*e^3 - 45*e^2 - 19*e + 10, e^5 - e^4 - 8*e^3 + 11*e^2 - 2*e - 6, 7*e^5 - 3*e^4 - 61*e^3 + 39*e^2 + 39*e - 2, 8*e^5 - 5*e^4 - 70*e^3 + 59*e^2 + 42*e - 16, 7*e^5 - 2*e^4 - 61*e^3 + 32*e^2 + 41*e - 5, -e^5 - e^4 + 10*e^3 + 7*e^2 - 18*e - 7, -11*e^5 + 6*e^4 + 94*e^3 - 74*e^2 - 45*e + 19, 7*e^5 - 3*e^4 - 60*e^3 + 40*e^2 + 31*e - 5, 3*e^5 - 2*e^4 - 24*e^3 + 26*e^2 - e - 12, 7*e^5 - 2*e^4 - 61*e^3 + 31*e^2 + 44*e - 8, 2*e^5 - 2*e^4 - 15*e^3 + 20*e^2 - 10*e - 5, -5*e^5 + 2*e^4 + 45*e^3 - 27*e^2 - 41*e + 6, -e^5 + 9*e^3 - 9*e - 7, -9*e^5 + 6*e^4 + 79*e^3 - 68*e^2 - 50*e + 10, 3*e^5 + 2*e^4 - 28*e^3 - 10*e^2 + 39*e + 5, 7*e^5 - 4*e^4 - 61*e^3 + 47*e^2 + 38*e - 16, -10*e^5 + 6*e^4 + 86*e^3 - 73*e^2 - 44*e + 22, -e^5 + 7*e^3 - e^2 + 4*e - 13, 4*e^5 - 6*e^4 - 34*e^3 + 58*e^2 + 9*e - 22, -4*e^4 + 32*e^2 - 12*e - 18, 4*e^5 - 5*e^4 - 35*e^3 + 50*e^2 + 10*e - 21, -6*e^5 + 4*e^4 + 51*e^3 - 46*e^2 - 24*e + 9, -10*e^5 + 8*e^4 + 86*e^3 - 91*e^2 - 39*e + 27, -4*e^5 + 5*e^4 + 35*e^3 - 53*e^2 - 18*e + 25, e^5 - 4*e^4 - 10*e^3 + 34*e^2 + 6*e - 23, 3*e^5 - 3*e^4 - 29*e^3 + 30*e^2 + 29*e - 8, 5*e^4 - 3*e^3 - 42*e^2 + 34*e + 16, -3*e^5 + 4*e^4 + 23*e^3 - 42*e^2 + 20*e + 18, -e^5 + 5*e^4 + 11*e^3 - 43*e^2 - 8*e + 22, -e^5 - 4*e^4 + 9*e^3 + 31*e^2 - 19*e - 17, -e^5 - 2*e^4 + 7*e^3 + 16*e^2 + 6*e - 14, 12*e^5 - 4*e^4 - 103*e^3 + 57*e^2 + 61*e + 6, 17*e^5 - 8*e^4 - 148*e^3 + 103*e^2 + 88*e - 22, -21*e^5 + 13*e^4 + 182*e^3 - 153*e^2 - 103*e + 36, 11*e^5 - 2*e^4 - 97*e^3 + 40*e^2 + 80*e - 5, -4*e^5 + e^4 + 34*e^3 - 20*e^2 - 17*e + 13, 2*e^5 - 19*e^3 + 6*e^2 + 25*e - 8, 2*e^4 - 2*e^3 - 17*e^2 + 16*e + 3, e^5 - 9*e^3 + 2*e^2 + 11*e - 7, -13*e^5 + 7*e^4 + 113*e^3 - 89*e^2 - 68*e + 18, 3*e^5 - 6*e^4 - 28*e^3 + 58*e^2 + 14*e - 36, -11*e^5 + 5*e^4 + 97*e^3 - 66*e^2 - 71*e + 16, 14*e^5 - 6*e^4 - 122*e^3 + 78*e^2 + 74*e - 10, 9*e^5 - 5*e^4 - 78*e^3 + 64*e^2 + 48*e - 11, 11*e^5 - 4*e^4 - 96*e^3 + 55*e^2 + 68*e - 5, -6*e^5 + 4*e^4 + 50*e^3 - 51*e^2 - 17*e + 32, e^5 - 13*e^3 + e^2 + 37*e - 3, 17*e^5 - 8*e^4 - 149*e^3 + 101*e^2 + 104*e - 7, 16*e^5 - 11*e^4 - 138*e^3 + 130*e^2 + 74*e - 44, 4*e^5 - 4*e^4 - 34*e^3 + 46*e^2 + 11*e - 27, 13*e^5 - 10*e^4 - 115*e^3 + 112*e^2 + 77*e - 28, -23*e^5 + 7*e^4 + 199*e^3 - 107*e^2 - 117*e + 9, 27*e^5 - 10*e^4 - 233*e^3 + 143*e^2 + 146*e - 26, -23*e^5 + 10*e^4 + 200*e^3 - 128*e^2 - 127*e + 14, 9*e^5 - 4*e^4 - 78*e^3 + 53*e^2 + 45*e - 19, -11*e^5 + 5*e^4 + 95*e^3 - 66*e^2 - 58*e, 11*e^5 - 96*e^3 + 29*e^2 + 77*e - 8, -9*e^5 + 10*e^4 + 79*e^3 - 102*e^2 - 45*e + 29, -19*e^5 + 12*e^4 + 161*e^3 - 144*e^2 - 74*e + 41, 2*e^5 - 3*e^4 - 15*e^3 + 29*e^2 - 11*e - 26, 10*e^5 - 7*e^4 - 86*e^3 + 83*e^2 + 44*e - 40, -4*e^5 + 37*e^3 - 11*e^2 - 45*e + 1, 21*e^5 - 11*e^4 - 183*e^3 + 138*e^2 + 109*e - 33, -8*e^5 + 4*e^4 + 66*e^3 - 54*e^2 - 18*e + 27, 9*e^5 - 6*e^4 - 76*e^3 + 74*e^2 + 30*e - 37, 32*e^5 - 14*e^4 - 276*e^3 + 184*e^2 + 162*e - 35, e^4 - 3*e^3 - 11*e^2 + 18*e + 11, -13*e^5 + 9*e^4 + 112*e^3 - 103*e^2 - 54*e + 22, 18*e^5 - 13*e^4 - 155*e^3 + 151*e^2 + 70*e - 61, -4*e^5 - 5*e^4 + 35*e^3 + 29*e^2 - 38*e - 9, -5*e^5 + 2*e^4 + 43*e^3 - 25*e^2 - 18*e - 4, -12*e^5 + 7*e^4 + 104*e^3 - 86*e^2 - 55*e + 28, 19*e^5 - 9*e^4 - 160*e^3 + 120*e^2 + 70*e - 21, -33*e^5 + 16*e^4 + 284*e^3 - 206*e^2 - 153*e + 48, -e^5 + 5*e^4 + 10*e^3 - 43*e^2 - 4*e + 19, -9*e^5 + 6*e^4 + 76*e^3 - 73*e^2 - 34*e + 45, 9*e^5 - 8*e^4 - 73*e^3 + 86*e^2 - e - 33, -22*e^5 + 12*e^4 + 193*e^3 - 142*e^2 - 124*e + 19, -17*e^5 + 13*e^4 + 146*e^3 - 143*e^2 - 57*e + 35, e^5 - 6*e^4 - 10*e^3 + 51*e^2 + 3*e - 15, -e^5 - e^4 + 12*e^3 + 8*e^2 - 42*e - 8, -6*e^5 - 4*e^4 + 55*e^3 + 19*e^2 - 67*e - 10, 13*e^5 - 9*e^4 - 112*e^3 + 107*e^2 + 61*e - 26, -6*e^5 + e^4 + 52*e^3 - 18*e^2 - 44*e - 20, 5*e^5 + 3*e^4 - 45*e^3 - 10*e^2 + 55*e + 7, 5*e^4 + 2*e^3 - 43*e^2 - 2*e + 21, 4*e^5 - 9*e^4 - 33*e^3 + 88*e^2 - 5*e - 40, -32*e^5 + 12*e^4 + 279*e^3 - 169*e^2 - 177*e + 20, 18*e^5 - 10*e^4 - 151*e^3 + 122*e^2 + 55*e - 28, -18*e^5 + 11*e^4 + 156*e^3 - 127*e^2 - 89*e + 27, -9*e^5 - e^4 + 77*e^3 - 10*e^2 - 57*e - 5, 12*e^5 - 2*e^4 - 103*e^3 + 37*e^2 + 64*e + 20, -11*e^5 + 3*e^4 + 92*e^3 - 53*e^2 - 41*e + 10, 19*e^5 - 3*e^4 - 164*e^3 + 65*e^2 + 112*e - 5, 7*e^5 - 10*e^4 - 59*e^3 + 102*e^2 + 15*e - 42, 9*e^5 + e^4 - 79*e^3 + 15*e^2 + 71*e + 11, -12*e^5 + 8*e^4 + 107*e^3 - 91*e^2 - 71*e + 7, -6*e^5 + 8*e^4 + 55*e^3 - 76*e^2 - 37*e + 12, -28*e^5 + 14*e^4 + 240*e^3 - 178*e^2 - 122*e + 48, -31*e^5 + 14*e^4 + 273*e^3 - 179*e^2 - 192*e + 24, -17*e^5 + 16*e^4 + 143*e^3 - 174*e^2 - 38*e + 66, 5*e^5 - 6*e^4 - 41*e^3 + 64*e^2 + 5*e - 32, -34*e^5 + 19*e^4 + 295*e^3 - 231*e^2 - 178*e + 41, -4*e^5 - 3*e^4 + 38*e^3 + 18*e^2 - 46*e - 14, -11*e^5 - 5*e^4 + 95*e^3 + 17*e^2 - 87*e - 21, 20*e^5 - 18*e^4 - 174*e^3 + 195*e^2 + 89*e - 59, -30*e^5 + 7*e^4 + 260*e^3 - 125*e^2 - 184*e + 22, -2*e^5 + 3*e^4 + 17*e^3 - 26*e^2 - 11*e - 13, 10*e^5 - 2*e^4 - 90*e^3 + 35*e^2 + 84*e + 16, 3*e^4 - e^3 - 27*e^2 + 3*e + 26, 11*e^5 - 5*e^4 - 94*e^3 + 67*e^2 + 47*e - 5, 3*e^5 + 2*e^4 - 25*e^3 - 17*e^2 + 11*e + 31, e^5 - 3*e^4 - 7*e^3 + 31*e^2 - 15*e - 31, -13*e^5 + 11*e^4 + 109*e^3 - 123*e^2 - 24*e + 61, 6*e^5 - 9*e^4 - 49*e^3 + 92*e^2 + e - 40, 4*e^5 - 32*e^3 + 5*e^2 - e + 15, -22*e^5 + 12*e^4 + 190*e^3 - 146*e^2 - 105*e + 47, -7*e^5 + 63*e^3 - 17*e^2 - 71*e + 11, -14*e^5 + 12*e^4 + 119*e^3 - 139*e^2 - 40*e + 65, 20*e^5 - 11*e^4 - 172*e^3 + 136*e^2 + 104*e - 39, -9*e^5 + 3*e^4 + 75*e^3 - 52*e^2 - 26*e + 29, -16*e^5 + 6*e^4 + 140*e^3 - 87*e^2 - 91*e + 36, -4*e^5 + 4*e^4 + 30*e^3 - 48*e^2 + 6*e + 27, -13*e^5 + 6*e^4 + 109*e^3 - 80*e^2 - 43*e + 12, -12*e^5 + 6*e^4 + 108*e^3 - 77*e^2 - 88*e + 1, 10*e^5 - 3*e^4 - 86*e^3 + 50*e^2 + 57*e - 21, -17*e^5 + 11*e^4 + 154*e^3 - 127*e^2 - 125*e + 35, -11*e^5 + 13*e^4 + 94*e^3 - 133*e^2 - 27*e + 47, 16*e^5 - 3*e^4 - 138*e^3 + 58*e^2 + 93*e + 17, -4*e^5 + 4*e^4 + 31*e^3 - 45*e^2 + 12*e + 10, -e^5 - e^4 + 6*e^3 + 11*e^2 + 12*e - 18, -8*e^5 + 71*e^3 - 21*e^2 - 65*e + 6, 21*e^5 - 14*e^4 - 182*e^3 + 160*e^2 + 100*e - 36, e^4 + e^3 - 9*e^2 - 13*e - 4, -32*e^5 + 8*e^4 + 274*e^3 - 139*e^2 - 161*e + 16, -2*e^5 + 4*e^4 + 19*e^3 - 41*e^2 - 23*e + 6, 21*e^5 - 23*e^4 - 185*e^3 + 238*e^2 + 87*e - 91, 39*e^5 - 29*e^4 - 334*e^3 + 334*e^2 + 152*e - 100, 15*e^5 - 10*e^4 - 132*e^3 + 118*e^2 + 81*e - 48, -18*e^5 + 12*e^4 + 154*e^3 - 133*e^2 - 66*e + 26, 4*e^5 - 8*e^4 - 35*e^3 + 76*e^2 + 12*e - 35, 11*e^5 - 8*e^4 - 98*e^3 + 94*e^2 + 63*e - 37, 3*e^5 - 3*e^4 - 31*e^3 + 28*e^2 + 34*e - 4, -9*e^5 - 5*e^4 + 77*e^3 + 23*e^2 - 53*e - 42, e^5 - 4*e^4 - 10*e^3 + 38*e^2 + 9*e - 7, -33*e^5 + 17*e^4 + 284*e^3 - 215*e^2 - 162*e + 35, -15*e^5 + 15*e^4 + 125*e^3 - 157*e^2 - 13*e + 46, 3*e^5 - e^4 - 22*e^3 + 13*e^2 - 20*e + 20, -14*e^5 + 10*e^4 + 125*e^3 - 112*e^2 - 93*e + 48, 8*e^5 + 9*e^4 - 67*e^3 - 58*e^2 + 61*e + 39, 4*e^5 - 9*e^4 - 28*e^3 + 89*e^2 - 35*e - 43, 15*e^5 + 2*e^4 - 132*e^3 + 13*e^2 + 98*e + 29, -25*e^5 + 19*e^4 + 220*e^3 - 212*e^2 - 129*e + 52, 9*e^5 - 74*e^3 + 25*e^2 + 36*e - 18, 30*e^5 - 17*e^4 - 256*e^3 + 215*e^2 + 125*e - 82, -5*e^5 + 7*e^4 + 44*e^3 - 68*e^2 - 22*e + 4]
hecke_eigenvalues = {}
for i in range(len(hecke_eigenvalues_array)):
    hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i]

AL_eigenvalues = {}
AL_eigenvalues[ZF.ideal([7,7,-w - 4])] = -1

# EXAMPLE:
# pp = ZF.ideal(2).factor()[0][0]
# hecke_eigenvalues[pp]