/* 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([-114, 0, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([3, 3, w]) primes_array = [ [2, 2, -3*w - 32],\ [3, 3, w],\ [5, 5, w + 2],\ [5, 5, w + 3],\ [7, 7, -w + 11],\ [7, 7, w + 11],\ [11, 11, w + 2],\ [11, 11, w + 9],\ [13, 13, w + 6],\ [13, 13, w + 7],\ [19, 19, w],\ [37, 37, w + 15],\ [37, 37, w + 22],\ [41, 41, 37*w + 395],\ [41, 41, 5*w + 53],\ [67, 67, w + 28],\ [67, 67, w + 39],\ [71, 71, 40*w + 427],\ [71, 71, 8*w + 85],\ [73, 73, -2*w + 23],\ [73, 73, 2*w + 23],\ [83, 83, w + 23],\ [83, 83, w + 60],\ [89, 89, -w - 5],\ [89, 89, w - 5],\ [101, 101, w + 35],\ [101, 101, w + 66],\ [109, 109, w + 21],\ [109, 109, w + 88],\ [113, 113, -w - 1],\ [113, 113, w - 1],\ [131, 131, w + 30],\ [131, 131, w + 101],\ [149, 149, w + 73],\ [149, 149, w + 76],\ [167, 167, 2*w - 17],\ [167, 167, -2*w - 17],\ [181, 181, w + 64],\ [181, 181, w + 117],\ [197, 197, w + 36],\ [197, 197, w + 161],\ [199, 199, -3*w - 35],\ [199, 199, 3*w - 35],\ [211, 211, w + 89],\ [211, 211, w + 122],\ [251, 251, w + 37],\ [251, 251, w + 214],\ [257, 257, 7*w - 73],\ [257, 257, -7*w - 73],\ [271, 271, -15*w - 161],\ [271, 271, 81*w + 865],\ [281, 281, 55*w + 587],\ [281, 281, 23*w + 245],\ [289, 17, -17],\ [307, 307, w + 81],\ [307, 307, w + 226],\ [313, 313, 78*w + 833],\ [313, 313, -18*w - 193],\ [331, 331, w + 86],\ [331, 331, w + 245],\ [347, 347, w + 43],\ [347, 347, w + 304],\ [367, 367, 11*w - 119],\ [367, 367, 11*w + 119],\ [373, 373, w + 62],\ [373, 373, w + 311],\ [379, 379, w + 71],\ [379, 379, w + 308],\ [383, 383, 6*w + 61],\ [383, 383, 6*w - 61],\ [389, 389, w + 170],\ [389, 389, w + 219],\ [401, 401, -3*w - 25],\ [401, 401, 3*w - 25],\ [419, 419, w + 47],\ [419, 419, w + 372],\ [421, 421, w + 172],\ [421, 421, w + 249],\ [431, 431, 2*w - 5],\ [431, 431, -2*w - 5],\ [443, 443, w + 185],\ [443, 443, w + 258],\ [449, 449, 5*w - 49],\ [449, 449, 5*w + 49],\ [457, 457, -14*w - 151],\ [457, 457, 14*w - 151],\ [461, 461, w + 110],\ [461, 461, w + 351],\ [463, 463, -33*w - 353],\ [463, 463, 63*w + 673],\ [467, 467, w + 54],\ [467, 467, w + 413],\ [491, 491, w + 144],\ [491, 491, w + 347],\ [521, 521, 67*w + 715],\ [521, 521, 35*w + 373],\ [523, 523, w + 239],\ [523, 523, w + 284],\ [529, 23, -23],\ [547, 547, w + 85],\ [547, 547, w + 462],\ [557, 557, w + 79],\ [557, 557, w + 478],\ [569, 569, 11*w - 115],\ [569, 569, -11*w - 115],\ [577, 577, 4*w - 49],\ [577, 577, -4*w - 49],\ [587, 587, w + 124],\ [587, 587, w + 463],\ [599, 599, 4*w - 35],\ [599, 599, -4*w - 35],\ [631, 631, -5*w - 59],\ [631, 631, 5*w - 59],\ [641, 641, 5*w - 47],\ [641, 641, -5*w - 47],\ [653, 653, w + 296],\ [653, 653, w + 357],\ [661, 661, w + 82],\ [661, 661, w + 579],\ [701, 701, w + 157],\ [701, 701, w + 544],\ [727, 727, -w - 29],\ [727, 727, w - 29],\ [743, 743, 76*w + 811],\ [743, 743, 44*w + 469],\ [769, 769, 2*w - 35],\ [769, 769, -2*w - 35],\ [787, 787, w + 129],\ [787, 787, w + 658],\ [811, 811, w + 134],\ [811, 811, w + 677],\ [821, 821, w + 186],\ [821, 821, w + 635],\ [823, 823, 3*w - 43],\ [823, 823, -3*w - 43],\ [829, 829, w + 51],\ [829, 829, w + 778],\ [839, 839, 126*w + 1345],\ [839, 839, 30*w + 319],\ [841, 29, -29],\ [857, 857, -3*w - 13],\ [857, 857, 3*w - 13],\ [863, 863, 4*w - 31],\ [863, 863, -4*w - 31],\ [877, 877, w + 248],\ [877, 877, w + 629],\ [887, 887, -22*w + 233],\ [887, 887, 22*w + 233],\ [907, 907, w + 387],\ [907, 907, w + 520],\ [911, 911, 82*w + 875],\ [911, 911, 50*w + 533],\ [919, 919, -15*w + 163],\ [919, 919, -15*w - 163],\ [937, 937, -6*w - 71],\ [937, 937, 6*w - 71],\ [947, 947, w + 190],\ [947, 947, w + 757],\ [953, 953, 9*w + 91],\ [953, 953, 9*w - 91],\ [961, 31, -31],\ [967, 967, -9*w - 101],\ [967, 967, 9*w - 101],\ [977, 977, -3*w - 7],\ [977, 977, 3*w - 7],\ [983, 983, -4*w - 29],\ [983, 983, 4*w - 29]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^6 - 23*x^4 + 152*x^2 - 236 K. = NumberField(heckePol) hecke_eigenvalues_array = [0, 1, e, -e, -1/4*e^4 + 13/4*e^2 - 13/2, -1/4*e^4 + 13/4*e^2 - 13/2, -1/8*e^5 + 17/8*e^3 - 29/4*e, 1/8*e^5 - 17/8*e^3 + 29/4*e, 3/8*e^4 - 43/8*e^2 + 47/4, 3/8*e^4 - 43/8*e^2 + 47/4, -1/4*e^4 + 9/4*e^2 + 3/2, -1/8*e^4 + 25/8*e^2 - 53/4, -1/8*e^4 + 25/8*e^2 - 53/4, 1/8*e^5 - 9/8*e^3 - 3/4*e, -1/8*e^5 + 9/8*e^3 + 3/4*e, 3/4*e^4 - 43/4*e^2 + 47/2, 3/4*e^4 - 43/4*e^2 + 47/2, 1/4*e^5 - 17/4*e^3 + 33/2*e, -1/4*e^5 + 17/4*e^3 - 33/2*e, 1/2*e^4 - 11/2*e^2 + 3, 1/2*e^4 - 11/2*e^2 + 3, -1/8*e^5 + 9/8*e^3 + 11/4*e, 1/8*e^5 - 9/8*e^3 - 11/4*e, 1/4*e^5 - 17/4*e^3 + 33/2*e, -1/4*e^5 + 17/4*e^3 - 33/2*e, -1/4*e^5 + 13/4*e^3 - 11/2*e, 1/4*e^5 - 13/4*e^3 + 11/2*e, 5/8*e^4 - 61/8*e^2 + 25/4, 5/8*e^4 - 61/8*e^2 + 25/4, -1/8*e^5 + 9/8*e^3 + 19/4*e, 1/8*e^5 - 9/8*e^3 - 19/4*e, -1/8*e^5 + 17/8*e^3 - 37/4*e, 1/8*e^5 - 17/8*e^3 + 37/4*e, -3*e, 3*e, -1/4*e^5 + 17/4*e^3 - 25/2*e, 1/4*e^5 - 17/4*e^3 + 25/2*e, 11/8*e^4 - 147/8*e^2 + 183/4, 11/8*e^4 - 147/8*e^2 + 183/4, -e^3 + 7*e, e^3 - 7*e, -e^2 + 4, -e^2 + 4, -3/2*e^4 + 43/2*e^2 - 59, -3/2*e^4 + 43/2*e^2 - 59, -3/8*e^5 + 35/8*e^3 - 7/4*e, 3/8*e^5 - 35/8*e^3 + 7/4*e, 1/4*e^5 - 17/4*e^3 + 41/2*e, -1/4*e^5 + 17/4*e^3 - 41/2*e, -3/2*e^4 + 39/2*e^2 - 37, -3/2*e^4 + 39/2*e^2 - 37, 5/8*e^5 - 77/8*e^3 + 113/4*e, -5/8*e^5 + 77/8*e^3 - 113/4*e, e^2 - 36, -1/4*e^4 + 9/4*e^2 + 3/2, -1/4*e^4 + 9/4*e^2 + 3/2, 3/2*e^4 - 47/2*e^2 + 63, 3/2*e^4 - 47/2*e^2 + 63, 5/4*e^4 - 61/4*e^2 + 25/2, 5/4*e^4 - 61/4*e^2 + 25/2, -1/8*e^5 + 1/8*e^3 + 35/4*e, 1/8*e^5 - 1/8*e^3 - 35/4*e, 2*e^2 - 10, 2*e^2 - 10, -3/8*e^4 + 43/8*e^2 - 95/4, -3/8*e^4 + 43/8*e^2 - 95/4, -3/2*e^4 + 35/2*e^2 - 35, -3/2*e^4 + 35/2*e^2 - 35, -e^5 + 15*e^3 - 48*e, e^5 - 15*e^3 + 48*e, 1/4*e^5 - 17/4*e^3 + 27/2*e, -1/4*e^5 + 17/4*e^3 - 27/2*e, -1/8*e^5 + 9/8*e^3 + 51/4*e, 1/8*e^5 - 9/8*e^3 - 51/4*e, 7/8*e^5 - 95/8*e^3 + 99/4*e, -7/8*e^5 + 95/8*e^3 - 99/4*e, 5/8*e^4 - 93/8*e^2 + 121/4, 5/8*e^4 - 93/8*e^2 + 121/4, 1/4*e^5 - 25/4*e^3 + 61/2*e, -1/4*e^5 + 25/4*e^3 - 61/2*e, 7/8*e^5 - 87/8*e^3 + 59/4*e, -7/8*e^5 + 87/8*e^3 - 59/4*e, -3/4*e^5 + 35/4*e^3 - 19/2*e, 3/4*e^5 - 35/4*e^3 + 19/2*e, -5*e^2 + 24, -5*e^2 + 24, 1/4*e^5 - 25/4*e^3 + 75/2*e, -1/4*e^5 + 25/4*e^3 - 75/2*e, 1/2*e^4 - 15/2*e^2 + 37, 1/2*e^4 - 15/2*e^2 + 37, 1/8*e^5 - 1/8*e^3 - 67/4*e, -1/8*e^5 + 1/8*e^3 + 67/4*e, -3/8*e^5 + 43/8*e^3 - 55/4*e, 3/8*e^5 - 43/8*e^3 + 55/4*e, 1/4*e^5 - 17/4*e^3 + 9/2*e, -1/4*e^5 + 17/4*e^3 - 9/2*e, 3/2*e^4 - 35/2*e^2 + 11, 3/2*e^4 - 35/2*e^2 + 11, -2*e^4 + 28*e^2 - 96, -5/2*e^4 + 61/2*e^2 - 37, -5/2*e^4 + 61/2*e^2 - 37, -1/4*e^5 + 9/4*e^3 + 21/2*e, 1/4*e^5 - 9/4*e^3 - 21/2*e, -1/4*e^5 + 17/4*e^3 - 33/2*e, 1/4*e^5 - 17/4*e^3 + 33/2*e, 1/2*e^4 - 3/2*e^2 - 15, 1/2*e^4 - 3/2*e^2 - 15, -1/8*e^5 + 1/8*e^3 + 43/4*e, 1/8*e^5 - 1/8*e^3 - 43/4*e, -2*e^3 + 14*e, 2*e^3 - 14*e, 1/4*e^4 - 29/4*e^2 + 109/2, 1/4*e^4 - 29/4*e^2 + 109/2, -1/4*e^5 + 17/4*e^3 - 49/2*e, 1/4*e^5 - 17/4*e^3 + 49/2*e, 1/4*e^5 - 1/4*e^3 - 49/2*e, -1/4*e^5 + 1/4*e^3 + 49/2*e, -13/8*e^4 + 229/8*e^2 - 401/4, -13/8*e^4 + 229/8*e^2 - 401/4, -e^3 + 15*e, e^3 - 15*e, 7*e^2 - 56, 7*e^2 - 56, 4*e^3 - 44*e, -4*e^3 + 44*e, 3/2*e^4 - 53/2*e^2 + 81, 3/2*e^4 - 53/2*e^2 + 81, -7/4*e^4 + 111/4*e^2 - 163/2, -7/4*e^4 + 111/4*e^2 - 163/2, -3/2*e^4 + 35/2*e^2 - 47, -3/2*e^4 + 35/2*e^2 - 47, 1/2*e^5 - 21/2*e^3 + 52*e, -1/2*e^5 + 21/2*e^3 - 52*e, -13/4*e^4 + 161/4*e^2 - 89/2, -13/4*e^4 + 161/4*e^2 - 89/2, -5/8*e^4 + 61/8*e^2 - 121/4, -5/8*e^4 + 61/8*e^2 - 121/4, e^5 - 17*e^3 + 58*e, -e^5 + 17*e^3 - 58*e, -2*e^4 + 30*e^2 - 114, 3/4*e^5 - 35/4*e^3 + 11/2*e, -3/4*e^5 + 35/4*e^3 - 11/2*e, 1/4*e^5 - 9/4*e^3 + 5/2*e, -1/4*e^5 + 9/4*e^3 - 5/2*e, -1/8*e^4 - 7/8*e^2 + 75/4, -1/8*e^4 - 7/8*e^2 + 75/4, -1/4*e^5 + 33/4*e^3 - 105/2*e, 1/4*e^5 - 33/4*e^3 + 105/2*e, 1/2*e^4 - 9/2*e^2 - 23, 1/2*e^4 - 9/2*e^2 - 23, -e^5 + 15*e^3 - 44*e, e^5 - 15*e^3 + 44*e, -1/4*e^4 - 15/4*e^2 + 63/2, -1/4*e^4 - 15/4*e^2 + 63/2, 3*e^4 - 42*e^2 + 100, 3*e^4 - 42*e^2 + 100, 5/8*e^5 - 61/8*e^3 + 49/4*e, -5/8*e^5 + 61/8*e^3 - 49/4*e, -1/8*e^5 + 9/8*e^3 + 51/4*e, 1/8*e^5 - 9/8*e^3 - 51/4*e, 4*e^4 - 62*e^2 + 180, -3/4*e^4 + 83/4*e^2 - 195/2, -3/4*e^4 + 83/4*e^2 - 195/2, -5/8*e^5 + 77/8*e^3 - 97/4*e, 5/8*e^5 - 77/8*e^3 + 97/4*e, -3/4*e^5 + 43/4*e^3 - 63/2*e, 3/4*e^5 - 43/4*e^3 + 63/2*e] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([3, 3, w])] = -1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]