/*
  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([41, 13, -12, -2, 1])
F.<w> = NumberField(g)
ZF = F.ring_of_integers()

NN = ZF.ideal([9,3,w^3 - 8*w - 8])

primes_array = [
[9, 3, -w^3 + 3*w^2 + 5*w - 15],\
[9, 3, -w^3 + 8*w + 8],\
[16, 2, 2],\
[19, 19, w + 1],\
[19, 19, -w^2 + 6],\
[19, 19, -w^2 + 2*w + 5],\
[19, 19, -w + 2],\
[25, 5, 2*w^2 - 2*w - 13],\
[29, 29, -w^2 + 9],\
[29, 29, -w^2 + 2*w + 6],\
[29, 29, w^2 - 7],\
[29, 29, -w^2 + 2*w + 8],\
[31, 31, -2*w^2 + w + 12],\
[31, 31, 2*w^2 - 3*w - 11],\
[41, 41, -w],\
[41, 41, -w + 1],\
[49, 7, w^3 + 2*w^2 - 10*w - 20],\
[49, 7, w^3 - 5*w^2 - 3*w + 27],\
[61, 61, 2*w^2 - 3*w - 14],\
[61, 61, 2*w^2 - w - 15],\
[71, 71, -w^3 + w^2 + 7*w - 5],\
[71, 71, w^3 - 2*w^2 - 6*w + 2],\
[79, 79, w^3 - 2*w^2 - 6*w + 4],\
[79, 79, 3*w^2 - 2*w - 18],\
[89, 89, 2*w^3 - 6*w^2 - 11*w + 33],\
[89, 89, -4*w^2 + 3*w + 25],\
[101, 101, w^3 + w^2 - 9*w - 12],\
[101, 101, w^3 - 4*w^2 - 4*w + 19],\
[109, 109, -w^3 + 3*w^2 + 6*w - 19],\
[109, 109, -2*w^2 + w + 18],\
[121, 11, -3*w^2 + 3*w + 19],\
[121, 11, 3*w^2 - 3*w - 20],\
[131, 131, 2*w^2 - 3*w - 15],\
[131, 131, 2*w^2 - w - 16],\
[139, 139, 3*w^2 - 4*w - 21],\
[139, 139, -w^3 - w^2 + 7*w + 13],\
[151, 151, 4*w^2 - 3*w - 27],\
[151, 151, 2*w^3 - w^2 - 13*w - 8],\
[151, 151, w^3 + 3*w^2 - 13*w - 27],\
[151, 151, 4*w^2 - 5*w - 26],\
[179, 179, w^3 - 3*w^2 - 5*w + 11],\
[179, 179, 3*w^3 - 11*w^2 - 14*w + 61],\
[179, 179, 2*w^3 - 2*w^2 - 12*w + 3],\
[179, 179, w^3 - 8*w - 4],\
[181, 181, -w - 4],\
[181, 181, w^3 - 5*w^2 - 3*w + 25],\
[181, 181, w^3 + 2*w^2 - 10*w - 18],\
[181, 181, w - 5],\
[229, 229, w^2 - 2*w - 11],\
[229, 229, w^2 - 12],\
[239, 239, 6*w^2 - 7*w - 42],\
[239, 239, 2*w^3 - 2*w^2 - 14*w - 3],\
[241, 241, w^3 - 4*w^2 - 4*w + 18],\
[241, 241, 3*w^2 - 2*w - 16],\
[269, 269, 3*w^2 - 4*w - 22],\
[269, 269, -w^3 + 6*w + 8],\
[281, 281, w^3 - w^2 - 9*w + 4],\
[281, 281, 2*w^3 - 9*w^2 - 8*w + 49],\
[311, 311, -2*w^3 - w^2 + 17*w + 19],\
[311, 311, 2*w^3 - 7*w^2 - 9*w + 33],\
[331, 331, w^3 + 2*w^2 - 9*w - 21],\
[331, 331, w^3 - 5*w^2 - 2*w + 27],\
[349, 349, 2*w^3 - w^2 - 15*w - 4],\
[349, 349, -2*w^3 + 5*w^2 + 11*w - 18],\
[359, 359, -w^3 + 6*w^2 - 31],\
[359, 359, -w^3 - 3*w^2 + 9*w + 26],\
[379, 379, 2*w^3 - 5*w^2 - 11*w + 20],\
[379, 379, 2*w^3 - 14*w - 11],\
[401, 401, w^3 - 3*w^2 - 4*w + 16],\
[401, 401, -2*w^3 + 4*w^2 + 11*w - 16],\
[401, 401, 2*w^3 - 2*w^2 - 13*w - 3],\
[401, 401, w^3 - 7*w - 10],\
[409, 409, w^3 - 4*w^2 - 4*w + 17],\
[409, 409, 2*w^3 - 6*w^2 - 9*w + 22],\
[409, 409, -2*w^3 + 15*w + 9],\
[409, 409, -w^3 + 3*w^2 + 6*w - 12],\
[419, 419, -w^3 + 5*w^2 + 3*w - 24],\
[419, 419, w^3 + 2*w^2 - 10*w - 17],\
[421, 421, w^3 + 3*w^2 - 12*w - 25],\
[421, 421, -w^3 - w^2 + 9*w + 7],\
[431, 431, -w^3 - 2*w^2 + 12*w + 20],\
[431, 431, w^3 + 3*w^2 - 10*w - 28],\
[431, 431, -w^3 + 6*w^2 + w - 34],\
[431, 431, -w^3 + 5*w^2 + 5*w - 29],\
[439, 439, w^3 - 7*w^2 - 2*w + 39],\
[439, 439, w^3 + 4*w^2 - 13*w - 31],\
[449, 449, -w^3 - 4*w^2 + 13*w + 34],\
[449, 449, 6*w^2 - 5*w - 36],\
[461, 461, -w^3 + 5*w^2 + w - 25],\
[461, 461, 2*w^3 - 15*w - 12],\
[461, 461, -2*w^3 + 6*w^2 + 9*w - 25],\
[461, 461, w^3 + 2*w^2 - 8*w - 20],\
[479, 479, -w^3 - 4*w^2 + 12*w + 31],\
[479, 479, 2*w^3 - 15*w - 16],\
[479, 479, -2*w^3 + 6*w^2 + 9*w - 29],\
[479, 479, w^3 - 7*w^2 - w + 38],\
[491, 491, w^2 - 3*w - 7],\
[491, 491, -2*w^3 + 3*w^2 + 11*w - 2],\
[491, 491, 4*w^2 - 2*w - 29],\
[491, 491, w^2 + w - 9],\
[499, 499, 5*w^2 - 6*w - 33],\
[499, 499, 5*w^2 - 4*w - 34],\
[509, 509, 2*w - 3],\
[509, 509, -2*w - 1],\
[541, 541, -w^3 - 4*w^2 + 12*w + 29],\
[541, 541, -w^3 + 7*w^2 + w - 36],\
[599, 599, -w^3 - w^2 + 8*w + 7],\
[599, 599, w^3 - 4*w^2 - 3*w + 13],\
[619, 619, w^2 - 3*w - 6],\
[619, 619, w^2 + w - 8],\
[641, 641, -2*w^3 + 16*w + 11],\
[641, 641, -2*w^3 + 6*w^2 + 10*w - 25],\
[659, 659, 2*w^3 + w^2 - 16*w - 16],\
[659, 659, 2*w^3 - 7*w^2 - 8*w + 29],\
[691, 691, -w^3 + 5*w^2 + 3*w - 23],\
[691, 691, -w^3 + 3*w^2 + 6*w - 11],\
[701, 701, w^3 - w^2 - 9*w + 3],\
[701, 701, 2*w^3 - w^2 - 14*w - 5],\
[709, 709, 2*w^3 - 3*w^2 - 11*w + 6],\
[719, 719, w^2 - 2*w - 12],\
[719, 719, -w^2 + 13],\
[739, 739, -2*w^3 + 8*w^2 + 7*w - 36],\
[739, 739, w^2 + 2*w - 13],\
[751, 751, -2*w^3 + 5*w^2 + 10*w - 20],\
[751, 751, 2*w^3 - w^2 - 14*w - 7],\
[761, 761, w^3 + 4*w^2 - 12*w - 32],\
[761, 761, -w^3 + 7*w^2 + w - 39],\
[769, 769, -2*w^3 - 4*w^2 + 23*w + 45],\
[769, 769, 2*w^3 - 5*w^2 - 9*w + 21],\
[769, 769, 2*w^3 + 2*w^2 - 18*w - 31],\
[769, 769, 2*w^3 - 10*w^2 - 9*w + 62],\
[811, 811, -w^3 + 10*w + 5],\
[811, 811, -5*w^2 + 8*w + 30],\
[811, 811, w^3 - 8*w^2 + 2*w + 43],\
[811, 811, 2*w^3 - 2*w^2 - 13*w + 2],\
[829, 829, -2*w^3 - 2*w^2 + 18*w + 25],\
[829, 829, -2*w^3 + 8*w^2 + 8*w - 39],\
[859, 859, -w^3 - 4*w^2 + 12*w + 33],\
[859, 859, -w^3 + 7*w^2 + w - 40],\
[919, 919, w^3 - 4*w^2 - 3*w + 24],\
[919, 919, -w^3 + 5*w^2 + w - 26],\
[929, 929, 5*w^2 - 7*w - 28],\
[929, 929, -w^3 + 2*w^2 + 6*w - 13],\
[929, 929, w^3 - w^2 - 7*w - 6],\
[929, 929, -2*w^3 + 2*w^2 + 14*w - 5],\
[941, 941, -2*w^3 + 3*w^2 + 11*w - 1],\
[941, 941, 2*w^3 - 3*w^2 - 11*w + 11],\
[961, 31, 5*w^2 - 5*w - 33],\
[971, 971, -w^3 - 3*w^2 + 11*w + 22],\
[971, 971, 7*w^2 - 5*w - 45],\
[971, 971, 7*w^2 - 9*w - 43],\
[971, 971, w^3 - 6*w^2 - 2*w + 29],\
[991, 991, w^3 - 3*w^2 - 3*w + 15],\
[991, 991, -w^3 + 5*w^2 + 2*w - 29]]
primes = [ZF.ideal(I) for I in primes_array]

heckePol = x^5 + 4*x^4 - x^3 - 16*x^2 - 14*x - 1
K.<e> = NumberField(heckePol)

hecke_eigenvalues_array = [e, 1, 9*e^4 + 23*e^3 - 42*e^2 - 84*e - 8, -5*e^4 - 12*e^3 + 25*e^2 + 43*e - 4, e^4 + 2*e^3 - 6*e^2 - 7*e + 4, 6*e^4 + 17*e^3 - 25*e^2 - 64*e - 12, 4*e^4 + 9*e^3 - 22*e^2 - 33*e + 5, 2*e^4 + 5*e^3 - 10*e^2 - 17*e + 1, -8*e^4 - 21*e^3 + 37*e^2 + 78*e + 3, -3*e^4 - 7*e^3 + 15*e^2 + 23*e - 5, 10*e^4 + 26*e^3 - 45*e^2 - 93*e - 13, -e^4 - 3*e^3 + 2*e^2 + 8*e + 8, -4*e^4 - 9*e^3 + 22*e^2 + 29*e - 7, -11*e^4 - 28*e^3 + 53*e^2 + 102*e - 1, -5*e^4 - 13*e^3 + 24*e^2 + 46*e - 4, -8*e^4 - 22*e^3 + 34*e^2 + 84*e + 17, -7*e^4 - 21*e^3 + 28*e^2 + 82*e + 19, 8*e^4 + 21*e^3 - 37*e^2 - 74*e - 3, e^3 + 4*e^2 - 2*e - 13, -6*e^4 - 16*e^3 + 25*e^2 + 57*e + 5, 5*e^4 + 12*e^3 - 24*e^2 - 45*e - 9, -9*e^4 - 25*e^3 + 38*e^2 + 90*e + 19, -e^4 - 2*e^3 + 8*e^2 + 10*e - 12, -10*e^4 - 23*e^3 + 51*e^2 + 79*e - 4, 21*e^4 + 55*e^3 - 98*e^2 - 199*e - 9, -24*e^4 - 61*e^3 + 115*e^2 + 226*e + 12, 8*e^4 + 19*e^3 - 40*e^2 - 65*e + 10, -18*e^4 - 48*e^3 + 80*e^2 + 177*e + 20, 2*e^4 + 4*e^3 - 13*e^2 - 16*e + 3, 20*e^4 + 56*e^3 - 85*e^2 - 205*e - 31, -25*e^4 - 64*e^3 + 117*e^2 + 229*e + 9, -4*e^4 - 13*e^3 + 13*e^2 + 53*e + 16, 3*e^4 + 8*e^3 - 14*e^2 - 30*e + 1, 4*e^4 + 8*e^3 - 24*e^2 - 25*e + 11, -28*e^4 - 71*e^3 + 135*e^2 + 260*e + 11, -6*e^4 - 16*e^3 + 28*e^2 + 60*e - 2, 14*e^4 + 37*e^3 - 64*e^2 - 140*e - 16, -34*e^4 - 90*e^3 + 154*e^2 + 323*e + 33, 6*e^4 + 15*e^3 - 29*e^2 - 53*e - 9, -20*e^4 - 51*e^3 + 97*e^2 + 187*e + 8, 40*e^4 + 105*e^3 - 182*e^2 - 382*e - 36, -34*e^4 - 86*e^3 + 160*e^2 + 310*e + 10, 16*e^4 + 42*e^3 - 72*e^2 - 145*e - 15, -22*e^4 - 59*e^3 + 96*e^2 + 214*e + 36, -4*e^4 - 12*e^3 + 14*e^2 + 43*e + 11, 17*e^4 + 43*e^3 - 82*e^2 - 157*e - 16, -31*e^4 - 81*e^3 + 141*e^2 + 296*e + 32, 27*e^4 + 69*e^3 - 129*e^2 - 252*e - 14, 2*e^4 + 6*e^3 - 7*e^2 - 25*e - 17, -21*e^4 - 53*e^3 + 102*e^2 + 195*e + 6, 6*e^4 + 20*e^3 - 22*e^2 - 86*e - 23, 36*e^4 + 93*e^3 - 171*e^2 - 337*e - 8, 36*e^4 + 94*e^3 - 169*e^2 - 342*e - 22, 45*e^4 + 117*e^3 - 207*e^2 - 433*e - 45, 41*e^4 + 108*e^3 - 186*e^2 - 398*e - 54, 42*e^4 + 109*e^3 - 196*e^2 - 403*e - 38, -39*e^4 - 102*e^3 + 181*e^2 + 380*e + 39, -15*e^4 - 39*e^3 + 74*e^2 + 146*e - 1, 4*e^4 + 8*e^3 - 25*e^2 - 28*e + 15, -22*e^4 - 59*e^3 + 95*e^2 + 214*e + 25, 12*e^4 + 35*e^3 - 46*e^2 - 133*e - 34, -39*e^4 - 102*e^3 + 185*e^2 + 373*e + 8, 19*e^4 + 53*e^3 - 82*e^2 - 201*e - 26, 13*e^4 + 37*e^3 - 57*e^2 - 144*e - 29, 20*e^4 + 50*e^3 - 96*e^2 - 175*e - 9, 27*e^4 + 67*e^3 - 135*e^2 - 242*e + 7, 26*e^4 + 72*e^3 - 111*e^2 - 275*e - 50, 37*e^4 + 98*e^3 - 171*e^2 - 366*e - 35, 40*e^4 + 105*e^3 - 181*e^2 - 387*e - 60, 7*e^4 + 18*e^3 - 36*e^2 - 74*e + 1, 8*e^4 + 20*e^3 - 34*e^2 - 68*e - 28, -33*e^4 - 90*e^3 + 144*e^2 + 335*e + 58, 29*e^4 + 72*e^3 - 140*e^2 - 257*e - 9, -12*e^4 - 31*e^3 + 54*e^2 + 118*e + 8, 54*e^4 + 143*e^3 - 240*e^2 - 516*e - 66, -27*e^4 - 70*e^3 + 129*e^2 + 254*e - 6, -16*e^4 - 42*e^3 + 75*e^2 + 158*e - 5, -6*e^4 - 13*e^3 + 30*e^2 + 29*e + 1, 17*e^4 + 48*e^3 - 75*e^2 - 180*e - 11, 45*e^4 + 118*e^3 - 210*e^2 - 441*e - 40, 14*e^4 + 30*e^3 - 73*e^2 - 96*e + 9, -54*e^4 - 133*e^3 + 264*e^2 + 480*e + 2, 10*e^4 + 26*e^3 - 48*e^2 - 104*e - 12, -44*e^4 - 115*e^3 + 207*e^2 + 429*e + 28, -34*e^4 - 89*e^3 + 151*e^2 + 322*e + 50, -3*e^4 - 16*e^3 - 2*e^2 + 65*e + 38, 62*e^4 + 162*e^3 - 283*e^2 - 592*e - 50, 8*e^4 + 17*e^3 - 47*e^2 - 55*e + 36, -4*e^4 - 9*e^3 + 19*e^2 + 37*e + 7, 16*e^4 + 39*e^3 - 79*e^2 - 139*e - 9, 17*e^4 + 46*e^3 - 77*e^2 - 180*e - 38, 34*e^4 + 90*e^3 - 149*e^2 - 333*e - 57, 39*e^4 + 101*e^3 - 182*e^2 - 371*e - 27, -68*e^4 - 179*e^3 + 310*e^2 + 648*e + 69, -30*e^4 - 77*e^3 + 144*e^2 + 276*e - 4, 18*e^4 + 40*e^3 - 92*e^2 - 132*e + 14, 49*e^4 + 129*e^3 - 222*e^2 - 465*e - 55, -31*e^4 - 83*e^3 + 133*e^2 + 293*e + 51, -61*e^4 - 152*e^3 + 292*e^2 + 545*e + 9, 36*e^4 + 92*e^3 - 166*e^2 - 319*e - 24, 41*e^4 + 109*e^3 - 186*e^2 - 417*e - 55, -33*e^4 - 87*e^3 + 156*e^2 + 333*e + 23, -36*e^4 - 92*e^3 + 171*e^2 + 344*e + 29, 3*e^4 + 10*e^3 - 27*e - 41, 20*e^4 + 56*e^3 - 88*e^2 - 207*e - 15, -6*e^4 - 19*e^3 + 30*e^2 + 85*e - 6, 29*e^4 + 76*e^3 - 132*e^2 - 281*e - 22, 27*e^4 + 69*e^3 - 120*e^2 - 237*e - 30, -10*e^4 - 23*e^3 + 50*e^2 + 75*e - 22, 14*e^4 + 41*e^3 - 55*e^2 - 147*e - 23, 44*e^4 + 113*e^3 - 204*e^2 - 417*e - 56, -51*e^4 - 137*e^3 + 224*e^2 + 502*e + 52, -82*e^4 - 219*e^3 + 371*e^2 + 812*e + 109, -68*e^4 - 175*e^3 + 312*e^2 + 630*e + 46, 8*e^4 + 19*e^3 - 37*e^2 - 68*e - 14, -15*e^4 - 43*e^3 + 65*e^2 + 158*e + 25, -70*e^4 - 184*e^3 + 322*e^2 + 674*e + 57, -3*e^4 - 13*e^3 + 6*e^2 + 54*e + 26, 34*e^4 + 87*e^3 - 169*e^2 - 325*e + 5, 52*e^4 + 141*e^3 - 234*e^2 - 522*e - 55, 27*e^4 + 70*e^3 - 132*e^2 - 252*e + 5, 93*e^4 + 244*e^3 - 423*e^2 - 902*e - 110, -8*e^4 - 19*e^3 + 32*e^2 + 67*e + 36, -13*e^4 - 43*e^3 + 47*e^2 + 168*e + 43, 9*e^4 + 24*e^3 - 46*e^2 - 100*e - 6, -29*e^4 - 77*e^3 + 132*e^2 + 290*e + 60, 59*e^4 + 154*e^3 - 269*e^2 - 561*e - 45, 9*e^4 + 27*e^3 - 39*e^2 - 99*e - 8, 76*e^4 + 195*e^3 - 355*e^2 - 702*e - 36, 79*e^4 + 204*e^3 - 364*e^2 - 749*e - 67, -17*e^4 - 41*e^3 + 87*e^2 + 140*e - 21, 51*e^4 + 131*e^3 - 238*e^2 - 473*e - 11, 70*e^4 + 186*e^3 - 311*e^2 - 675*e - 69, -5*e^4 - 10*e^3 + 35*e^2 + 42*e - 27, 5*e^3 + 13*e^2 - 37*e - 38, 89*e^4 + 229*e^3 - 411*e^2 - 830*e - 60, 52*e^4 + 139*e^3 - 233*e^2 - 507*e - 47, -17*e^4 - 45*e^3 + 71*e^2 + 166*e + 31, -48*e^4 - 125*e^3 + 226*e^2 + 469*e + 41, e^4 + 2*e^3 + 5*e^2 + e - 25, 39*e^4 + 108*e^3 - 166*e^2 - 408*e - 57, -7*e^4 - 20*e^3 + 25*e^2 + 67*e + 44, -26*e^4 - 60*e^3 + 132*e^2 + 215*e + 11, -13*e^4 - 23*e^3 + 73*e^2 + 56*e - 43, -10*e^4 - 23*e^3 + 51*e^2 + 79*e - 27, 26*e^4 + 67*e^3 - 128*e^2 - 249*e - 13, 18*e^4 + 47*e^3 - 86*e^2 - 181*e - 23, -44*e^4 - 106*e^3 + 221*e^2 + 376*e - 15, -17*e^4 - 39*e^3 + 87*e^2 + 140*e + 27, -15*e^4 - 43*e^3 + 63*e^2 + 151*e + 6, -3*e^4 - 6*e^3 + 23*e^2 + 28*e - 9, 16*e^4 + 37*e^3 - 87*e^2 - 126*e + 43, -72*e^4 - 179*e^3 + 348*e^2 + 643*e + 6, -13*e^4 - 34*e^3 + 60*e^2 + 103*e - 3]
hecke_eigenvalues = {}
for i in range(len(hecke_eigenvalues_array)):
    hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i]

AL_eigenvalues = {}
AL_eigenvalues[ZF.ideal([9,3,w^3 - 8*w - 8])] = -1

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