/* 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. = PolynomialRing(QQ) g = P([-5, -9, -1, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([14, 14, w - 1]) primes_array = [ [2, 2, -w - 1],\ [5, 5, w],\ [7, 7, w^2 - 2*w - 8],\ [7, 7, -2*w^2 + 3*w + 16],\ [7, 7, -w^2 + 2*w + 6],\ [11, 11, w^2 - 2*w - 2],\ [19, 19, -w + 2],\ [23, 23, -w^2 + 3*w + 1],\ [25, 5, w^2 - w - 9],\ [27, 3, 3],\ [29, 29, -w^2 - w + 1],\ [29, 29, w^2 - 2*w - 4],\ [29, 29, -w^2 + w + 11],\ [43, 43, 2*w^2 - 3*w - 18],\ [47, 47, -2*w + 7],\ [53, 53, w^2 - w - 3],\ [61, 61, w^2 + 2*w + 2],\ [67, 67, -2*w^2 + 5*w + 8],\ [79, 79, w^2 - 3*w - 9],\ [97, 97, -w^2 - 2*w + 2],\ [109, 109, -2*w^2 + 4*w + 11],\ [109, 109, -3*w^2 + 4*w + 26],\ [109, 109, -3*w^2 + 4*w + 28],\ [113, 113, -4*w^2 + 8*w + 27],\ [121, 11, 5*w^2 - 8*w - 38],\ [127, 127, -3*w^2 + 4*w + 24],\ [131, 131, -4*w^2 + 5*w + 38],\ [139, 139, 2*w - 3],\ [149, 149, -w^2 + 3*w - 1],\ [151, 151, w^2 - 4*w - 6],\ [157, 157, 2*w^2 - 3*w - 12],\ [167, 167, -3*w^2 + 6*w + 22],\ [173, 173, -3*w^2 + 8*w + 8],\ [181, 181, w^2 - 2*w - 12],\ [181, 181, w^2 - 12],\ [181, 181, -4*w - 1],\ [191, 191, 2*w^2 - 6*w - 9],\ [193, 193, -8*w^2 + 12*w + 67],\ [197, 197, 2*w^2 - 3*w - 4],\ [199, 199, 2*w^2 - 2*w - 11],\ [211, 211, 4*w^2 - 6*w - 29],\ [223, 223, 3*w^2 - 5*w - 27],\ [227, 227, -4*w^2 + 7*w + 28],\ [229, 229, 3*w^2 - 8*w - 14],\ [229, 229, w^2 - 4*w - 8],\ [229, 229, -2*w - 7],\ [233, 233, -5*w^2 + 7*w + 43],\ [241, 241, 2*w^2 - 4*w - 7],\ [251, 251, w^2 + 2],\ [251, 251, -2*w^2 + 2*w + 1],\ [251, 251, 2*w^2 - 5*w - 2],\ [263, 263, 2*w^2 - 5*w - 14],\ [269, 269, -w^2 + 5*w - 1],\ [271, 271, -2*w^2 - w + 2],\ [277, 277, -3*w^2 + 7*w + 7],\ [281, 281, -5*w^2 + 10*w + 34],\ [293, 293, -2*w^2 + 5*w + 16],\ [311, 311, w^2 + w + 3],\ [313, 313, 2*w^2 + w - 4],\ [317, 317, -7*w^2 + 13*w + 49],\ [331, 331, w^2 - 5*w - 7],\ [337, 337, -w^2 + 2*w - 2],\ [337, 337, 7*w^2 - 12*w - 52],\ [337, 337, -2*w^2 + 3],\ [347, 347, -10*w^2 + 17*w + 76],\ [349, 349, 2*w^2 - 2*w - 9],\ [353, 353, 2*w^2 - 3*w - 8],\ [359, 359, -4*w^2 + 5*w + 36],\ [361, 19, w^2 + w - 7],\ [367, 367, -4*w^2 + 8*w + 29],\ [373, 373, 2*w^2 - 6*w - 11],\ [373, 373, 8*w^2 - 14*w - 59],\ [379, 379, -2*w^2 + w + 4],\ [383, 383, -3*w^2 + 2*w + 34],\ [383, 383, 5*w^2 - 7*w - 41],\ [383, 383, 3*w^2 - 7*w - 19],\ [397, 397, -7*w^2 + 13*w + 51],\ [431, 431, -8*w^2 + 14*w + 63],\ [433, 433, -2*w^2 + 9*w + 8],\ [443, 443, -6*w^2 + 10*w + 51],\ [443, 443, 3*w - 4],\ [443, 443, w^2 - 7*w - 3],\ [449, 449, -7*w^2 + 10*w + 62],\ [461, 461, -6*w^2 + 11*w + 46],\ [461, 461, 2*w^2 - 4*w - 1],\ [461, 461, w^2 - 5*w - 9],\ [463, 463, 7*w + 6],\ [463, 463, -5*w^2 + 8*w + 36],\ [463, 463, 3*w^2 - 2*w - 16],\ [467, 467, 3*w^2 - 7*w - 9],\ [467, 467, 4*w - 1],\ [467, 467, 6*w^2 - 11*w - 48],\ [479, 479, 2*w^2 - 4*w - 21],\ [479, 479, w^2 + 7*w + 7],\ [479, 479, w^2 - 2*w - 14],\ [487, 487, w^2 - 5*w - 11],\ [487, 487, 3*w^2 - 3*w - 23],\ [487, 487, w^2 + 3*w - 3],\ [491, 491, w^2 - 8*w - 4],\ [499, 499, -6*w^2 + 14*w + 31],\ [509, 509, -w - 8],\ [523, 523, 2*w^2 + 1],\ [529, 23, 2*w^2 - 5*w - 22],\ [541, 541, -w^2 + 5*w - 7],\ [547, 547, 2*w^2 - 6*w - 13],\ [563, 563, 2*w^2 - 9*w - 4],\ [569, 569, 4*w^2 - 7*w - 26],\ [569, 569, -3*w^2 + 5*w + 3],\ [569, 569, 4*w^2 - 6*w - 27],\ [577, 577, 3*w^2 - 7*w - 21],\ [577, 577, -8*w^2 + 12*w + 69],\ [577, 577, -2*w^2 + w + 26],\ [587, 587, w^2 - 6*w - 8],\ [593, 593, -2*w^2 + w + 16],\ [593, 593, 3*w^2 - 5*w - 17],\ [593, 593, 2*w^2 - 8*w - 9],\ [601, 601, -6*w^2 + 8*w + 53],\ [607, 607, -2*w - 9],\ [613, 613, w^2 + 2*w - 6],\ [617, 617, 5*w^2 - 11*w - 31],\ [619, 619, -4*w^2 + 6*w + 37],\ [641, 641, 3*w^2 - 6*w - 14],\ [647, 647, -2*w^2 + 3*w + 22],\ [653, 653, 2*w^2 - 7*w - 22],\ [653, 653, 2*w^2 + w + 2],\ [653, 653, -3*w^2 + 5*w + 29],\ [659, 659, 4*w^2 - 11*w - 18],\ [659, 659, 3*w^2 - 6*w - 8],\ [659, 659, 4*w^2 - 8*w - 23],\ [661, 661, w^2 + w - 11],\ [677, 677, -w^2 - 4],\ [701, 701, -2*w^2 - 5*w + 2],\ [727, 727, -2*w^2 + w + 18],\ [733, 733, -9*w^2 + 13*w + 79],\ [733, 733, -7*w^2 + 10*w + 56],\ [733, 733, 2*w^2 - 13],\ [739, 739, 6*w^2 - 10*w - 43],\ [739, 739, 4*w^2 - 4*w - 29],\ [739, 739, -8*w^2 + 15*w + 58],\ [751, 751, -11*w^2 + 19*w + 87],\ [757, 757, 3*w^2 - 10*w - 12],\ [761, 761, 4*w - 3],\ [769, 769, 3*w^2 - w - 13],\ [811, 811, 5*w^2 - 9*w - 33],\ [823, 823, 2*w^2 - 7*w - 12],\ [853, 853, -7*w - 8],\ [853, 853, -3*w^2 + 4*w + 2],\ [853, 853, 2*w^2 + 5*w + 6],\ [857, 857, 10*w^2 - 15*w - 82],\ [857, 857, 3*w^2 - 8*w - 18],\ [857, 857, -4*w^2 + 3*w + 44],\ [859, 859, -w^2 + 4*w - 6],\ [877, 877, -4*w^2 + 13*w + 14],\ [881, 881, 2*w^2 - 9*w - 2],\ [887, 887, 3*w - 14],\ [907, 907, -11*w^2 + 21*w + 77],\ [919, 919, 12*w^2 - 21*w - 92],\ [929, 929, w^2 - 18],\ [937, 937, 9*w + 4],\ [941, 941, -2*w^2 + 2*w - 1],\ [947, 947, 4*w^2 - 9*w - 18],\ [953, 953, -7*w^2 + 14*w + 48],\ [977, 977, w^2 - 2*w - 16],\ [983, 983, 4*w^2 - 7*w - 24],\ [991, 991, -3*w^2 - 2*w + 4]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^5 - 7*x^4 + 6*x^3 + 39*x^2 - 64*x + 18 K. = NumberField(heckePol) hecke_eigenvalues_array = [-1, e, -2*e^4 + 6*e^3 + 13*e^2 - 28*e + 8, -2*e^4 + 7*e^3 + 11*e^2 - 36*e + 14, 1, -e^2 + e + 6, -e^4 + 3*e^3 + 7*e^2 - 15*e + 2, e^3 - 3*e^2 - 6*e + 12, -e^4 + 3*e^3 + 6*e^2 - 12*e + 8, e^3 - 3*e^2 - 6*e + 10, 4*e^4 - 12*e^3 - 26*e^2 + 55*e - 12, -e, e^4 - 4*e^3 - 5*e^2 + 24*e - 12, -e^4 + 4*e^3 + 5*e^2 - 24*e + 14, -3*e^4 + 8*e^3 + 22*e^2 - 34*e, 5*e^4 - 17*e^3 - 29*e^2 + 89*e - 30, 5*e^4 - 17*e^3 - 29*e^2 + 86*e - 28, -2*e^4 + 4*e^3 + 18*e^2 - 15*e - 4, 2*e^4 - 8*e^3 - 8*e^2 + 40*e - 22, e^4 - 2*e^3 - 10*e^2 + 8*e + 8, 8*e^4 - 27*e^3 - 46*e^2 + 137*e - 52, 3*e^4 - 12*e^3 - 13*e^2 + 63*e - 22, -2*e^4 + 9*e^3 + 7*e^2 - 54*e + 32, 5*e^4 - 13*e^3 - 38*e^2 + 54*e, -e^4 + 15*e^2 + 6*e - 22, 3*e^4 - 8*e^3 - 21*e^2 + 32*e + 2, -2*e^3 + 4*e^2 + 20*e - 24, 3*e^3 - 6*e^2 - 25*e + 20, 5*e^4 - 16*e^3 - 32*e^2 + 80*e - 18, 11*e^4 - 36*e^3 - 64*e^2 + 176*e - 70, -2*e^4 + 5*e^3 + 15*e^2 - 15*e - 10, 8*e^4 - 25*e^3 - 50*e^2 + 118*e - 36, -3*e, 4*e^4 - 10*e^3 - 30*e^2 + 36*e + 2, -4*e^4 + 11*e^3 + 28*e^2 - 49*e + 8, 11*e^4 - 34*e^3 - 70*e^2 + 163*e - 40, -4*e^4 + 13*e^3 + 24*e^2 - 62*e + 24, 9*e^4 - 31*e^3 - 49*e^2 + 156*e - 70, -7*e^4 + 26*e^3 + 36*e^2 - 145*e + 60, e^4 - 12*e^2 - 16*e + 20, e^4 - e^3 - 12*e^2 + 3*e + 20, 2*e^4 - 10*e^3 - 2*e^2 + 52*e - 46, 6*e^4 - 19*e^3 - 36*e^2 + 89*e - 24, -2*e^4 + 9*e^3 + 7*e^2 - 46*e + 20, -7*e^4 + 24*e^3 + 39*e^2 - 120*e + 44, -7*e^4 + 25*e^3 + 38*e^2 - 126*e + 38, 2*e^4 - 5*e^3 - 15*e^2 + 22*e - 6, -9*e^4 + 31*e^3 + 50*e^2 - 154*e + 56, e^4 - 14*e^2 - 8*e + 12, e^4 - 4*e^3 - 6*e^2 + 24*e, -4*e^4 + 9*e^3 + 35*e^2 - 38*e - 18, -10*e^4 + 31*e^3 + 65*e^2 - 154*e + 36, -e^4 + 7*e^3 - 5*e^2 - 39*e + 42, -11*e^4 + 35*e^3 + 69*e^2 - 172*e + 38, 9*e^4 - 32*e^3 - 47*e^2 + 163*e - 58, -4*e^4 + 11*e^3 + 30*e^2 - 50*e, 6*e^4 - 18*e^3 - 36*e^2 + 76*e - 42, e^4 - 4*e^3 - 2*e^2 + 24*e - 24, -12*e^4 + 36*e^3 + 81*e^2 - 172*e + 26, 5*e^4 - 20*e^3 - 23*e^2 + 107*e - 42, 14*e^4 - 47*e^3 - 80*e^2 + 234*e - 76, e^4 - 7*e^3 + 4*e^2 + 42*e - 40, -2*e^4 + 8*e^3 + 8*e^2 - 40*e + 32, -2*e^4 + 7*e^3 + 10*e^2 - 32*e - 4, 3*e^4 - 12*e^3 - 12*e^2 + 67*e - 36, 2*e^2 - e - 28, 4*e^2 - 6*e - 24, -2*e^4 + 10*e^3 + 3*e^2 - 54*e + 54, 14*e^4 - 44*e^3 - 90*e^2 + 220*e - 52, -e^4 + 4*e^3 + 5*e^2 - 32*e + 20, 14*e^4 - 51*e^3 - 70*e^2 + 266*e - 118, -5*e^4 + 15*e^3 + 33*e^2 - 73*e + 38, 6*e^4 - 16*e^3 - 47*e^2 + 75*e + 2, -19*e^4 + 62*e^3 + 114*e^2 - 314*e + 120, 5*e^4 - 22*e^3 - 14*e^2 + 120*e - 72, -12*e^4 + 40*e^3 + 69*e^2 - 200*e + 72, 9*e^4 - 31*e^3 - 48*e^2 + 154*e - 70, -12*e^4 + 36*e^3 + 77*e^2 - 160*e + 42, -17*e^4 + 60*e^3 + 93*e^2 - 320*e + 134, -2*e^4 + 10*e^3 + 3*e^2 - 52*e + 42, -4*e^4 + 14*e^3 + 20*e^2 - 75*e + 48, 15*e^4 - 49*e^3 - 91*e^2 + 243*e - 66, -2*e^4 + 13*e^3 - 4*e^2 - 78*e + 66, 12*e^4 - 38*e^3 - 77*e^2 + 188*e - 36, -17*e^4 + 54*e^3 + 108*e^2 - 268*e + 54, 8*e^4 - 30*e^3 - 39*e^2 + 164*e - 72, 7*e^4 - 23*e^3 - 44*e^2 + 122*e - 10, 10*e^4 - 37*e^3 - 47*e^2 + 184*e - 94, 18*e^4 - 64*e^3 - 96*e^2 + 332*e - 136, 10*e^4 - 31*e^3 - 63*e^2 + 147*e - 30, -10*e^4 + 33*e^3 + 57*e^2 - 167*e + 90, -5*e^4 + 16*e^3 + 33*e^2 - 80*e + 6, -8*e^4 + 24*e^3 + 56*e^2 - 120*e + 6, 15*e^4 - 43*e^3 - 101*e^2 + 186*e - 36, 5*e^4 - 13*e^3 - 36*e^2 + 46*e - 12, -6*e^4 + 19*e^3 + 40*e^2 - 98*e + 32, -10*e^4 + 30*e^3 + 63*e^2 - 144*e + 62, 10*e^4 - 33*e^3 - 60*e^2 + 164*e - 40, 5*e^4 - 14*e^3 - 32*e^2 + 55*e - 12, -13*e^4 + 40*e^3 + 81*e^2 - 193*e + 86, 18*e^4 - 60*e^3 - 105*e^2 + 304*e - 108, e^4 - 4*e^3 - 8*e^2 + 24*e + 8, -5*e^3 + 9*e^2 + 38*e - 34, 18*e^4 - 60*e^3 - 106*e^2 + 304*e - 106, -11*e^4 + 38*e^3 + 64*e^2 - 196*e + 44, -17*e^4 + 54*e^3 + 105*e^2 - 259*e + 78, -5*e^4 + 23*e^3 + 14*e^2 - 128*e + 72, 5*e^4 - 17*e^3 - 35*e^2 + 94*e + 6, 3*e^4 - 4*e^3 - 31*e^2 + 8*e + 30, -2*e^4 + 12*e^3 - e^2 - 64*e + 44, -7*e^4 + 18*e^3 + 50*e^2 - 68*e + 8, 2*e^4 - 7*e^3 - 7*e^2 + 34*e - 58, -8*e^4 + 32*e^3 + 34*e^2 - 168*e + 84, -16*e^4 + 48*e^3 + 106*e^2 - 226*e + 36, -2*e^4 + 10*e^3 + e^2 - 60*e + 42, -7*e^4 + 21*e^3 + 44*e^2 - 94*e + 48, -9*e^4 + 28*e^3 + 58*e^2 - 126*e + 8, -22*e^4 + 70*e^3 + 139*e^2 - 348*e + 86, -10*e^4 + 34*e^3 + 59*e^2 - 179*e + 62, -13*e^4 + 38*e^3 + 86*e^2 - 180*e + 66, -10*e^4 + 35*e^3 + 59*e^2 - 194*e + 62, -6*e^4 + 20*e^3 + 37*e^2 - 92*e - 6, 13*e^4 - 44*e^3 - 73*e^2 + 224*e - 102, 15*e^4 - 56*e^3 - 70*e^2 + 288*e - 138, 13*e^4 - 40*e^3 - 82*e^2 + 192*e - 54, e^4 - 3*e^3 - 6*e^2 + 2*e + 6, 2*e^4 - 4*e^3 - 17*e^2 + 8*e + 30, 17*e^4 - 54*e^3 - 106*e^2 + 269*e - 96, 4*e^4 - 11*e^3 - 25*e^2 + 45*e - 18, -12*e^4 + 40*e^3 + 69*e^2 - 211*e + 98, 7*e^4 - 22*e^3 - 39*e^2 + 100*e - 60, 4*e^4 - 16*e^3 - 15*e^2 + 85*e - 30, 25*e^4 - 78*e^3 - 153*e^2 + 362*e - 118, -13*e^4 + 44*e^3 + 70*e^2 - 216*e + 98, 2*e^4 - 5*e^3 - 17*e^2 + 29*e + 14, -7*e^4 + 26*e^3 + 36*e^2 - 139*e + 44, -8*e^4 + 27*e^3 + 46*e^2 - 135*e + 68, 10*e^4 - 27*e^3 - 69*e^2 + 113*e - 34, 2*e^4 - 15*e^3 + 7*e^2 + 90*e - 58, -3*e^4 + 13*e^3 + 13*e^2 - 70*e + 14, 21*e^4 - 72*e^3 - 120*e^2 + 376*e - 130, -e^4 + e^3 + 11*e^2 + 8*e - 6, 16*e^4 - 49*e^3 - 104*e^2 + 228*e - 40, -11*e^4 + 35*e^3 + 67*e^2 - 161*e + 50, 3*e^4 - 13*e^3 - 10*e^2 + 78*e - 64, -e^3 + 2*e^2 + 14*e + 14, 6*e^4 - 20*e^3 - 35*e^2 + 96*e - 28, -7*e^4 + 23*e^3 + 44*e^2 - 122*e + 14, -3*e^4 + 6*e^3 + 28*e^2 - 24*e - 12, -6*e^3 + 15*e^2 + 44*e - 48, 16*e^4 - 48*e^3 - 111*e^2 + 242*e - 30, 31*e^4 - 100*e^3 - 190*e^2 + 504*e - 172, 7*e^4 - 30*e^3 - 26*e^2 + 180*e - 118, 10*e^4 - 30*e^3 - 63*e^2 + 130*e - 30, 14*e^4 - 39*e^3 - 102*e^2 + 186*e - 6, -9*e^4 + 35*e^3 + 38*e^2 - 194*e + 104, e^4 - 7*e^3 + 4*e^2 + 42*e - 34, 9*e^4 - 33*e^3 - 47*e^2 + 174*e - 84, 12*e^4 - 44*e^3 - 61*e^2 + 228*e - 106, -6*e^4 + 19*e^3 + 31*e^2 - 90*e + 72, -16*e^4 + 47*e^3 + 111*e^2 - 229*e + 30, 14*e^4 - 43*e^3 - 87*e^2 + 210*e - 114, -11*e^4 + 36*e^3 + 69*e^2 - 192*e + 48, -e^4 - e^3 + 17*e^2 + 30*e - 66, -9*e^4 + 31*e^3 + 53*e^2 - 166*e + 38] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([2, 2, -w - 1])] = 1 AL_eigenvalues[ZF.ideal([7, 7, -w^2 + 2*w + 6])] = -1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]