/* 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([-1, 0, 8, 4, -6, -1, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([59, 59, w^4 - w^3 - 5*w^2 + 3*w + 4]) primes_array = [ [5, 5, w - 1],\ [11, 11, w^3 - 3*w],\ [19, 19, -w^5 + w^4 + 5*w^3 - 4*w^2 - 5*w],\ [25, 5, w^5 - 2*w^4 - 6*w^3 + 8*w^2 + 8*w],\ [29, 29, -w^4 + 4*w^2 + 1],\ [31, 31, w^5 - 2*w^4 - 5*w^3 + 9*w^2 + 3*w - 4],\ [31, 31, -w^2 + 2],\ [59, 59, -w^5 + w^4 + 5*w^3 - 3*w^2 - 4*w - 2],\ [59, 59, w^4 - w^3 - 5*w^2 + 3*w + 4],\ [59, 59, -w^4 - w^3 + 4*w^2 + 5*w],\ [61, 61, w^5 - w^4 - 4*w^3 + 5*w^2 - 4],\ [64, 2, 2],\ [71, 71, w^5 - w^4 - 6*w^3 + 4*w^2 + 6*w],\ [71, 71, -w^5 + 2*w^4 + 4*w^3 - 9*w^2 + w + 4],\ [71, 71, 2*w^5 - 3*w^4 - 9*w^3 + 13*w^2 + 4*w - 4],\ [71, 71, w^3 - 5*w],\ [79, 79, w^5 - w^4 - 4*w^3 + 4*w^2 + w - 2],\ [89, 89, w^3 - w^2 - 4*w + 1],\ [101, 101, -2*w^5 + 3*w^4 + 11*w^3 - 13*w^2 - 12*w + 2],\ [101, 101, 2*w^5 - 2*w^4 - 13*w^3 + 7*w^2 + 20*w + 4],\ [101, 101, w^5 - w^4 - 5*w^3 + 5*w^2 + 2*w - 3],\ [101, 101, w^5 - 2*w^4 - 6*w^3 + 9*w^2 + 8*w - 1],\ [109, 109, -w^5 + w^4 + 7*w^3 - 4*w^2 - 11*w - 1],\ [121, 11, -w^5 + w^4 + 4*w^3 - 5*w^2 + w + 4],\ [131, 131, w^4 - 4*w^2 + w - 1],\ [131, 131, -2*w^5 + 2*w^4 + 10*w^3 - 9*w^2 - 7*w + 2],\ [139, 139, w^4 + w^3 - 3*w^2 - 4*w - 4],\ [149, 149, -w^5 + 3*w^4 + 6*w^3 - 13*w^2 - 8*w + 2],\ [149, 149, 2*w^5 - 11*w^3 - 2*w^2 + 11*w + 6],\ [151, 151, 3*w^5 - 5*w^4 - 16*w^3 + 21*w^2 + 16*w - 6],\ [169, 13, w^5 - 3*w^4 - 5*w^3 + 13*w^2 + 6*w - 3],\ [179, 179, w^5 - 2*w^4 - 7*w^3 + 9*w^2 + 12*w - 1],\ [179, 179, 2*w^5 - 4*w^4 - 9*w^3 + 18*w^2 + 4*w - 7],\ [179, 179, -3*w^5 + 3*w^4 + 15*w^3 - 13*w^2 - 12*w + 3],\ [179, 179, -w^5 + 3*w^4 + 5*w^3 - 14*w^2 - 5*w + 6],\ [181, 181, w^3 + w^2 - 4*w - 6],\ [191, 191, -w^5 + w^4 + 4*w^3 - 5*w^2 + w + 3],\ [191, 191, -w^5 + w^4 + 6*w^3 - 4*w^2 - 7*w + 3],\ [191, 191, -w^5 + 2*w^4 + 4*w^3 - 9*w^2 - w + 3],\ [191, 191, -w^5 + w^4 + 7*w^3 - 4*w^2 - 12*w - 2],\ [199, 199, w^5 - w^4 - 6*w^3 + 4*w^2 + 9*w - 1],\ [199, 199, w^4 + w^3 - 5*w^2 - 3*w + 3],\ [199, 199, -w^5 + w^4 + 3*w^3 - 4*w^2 + 3*w + 2],\ [199, 199, w^2 + w - 3],\ [211, 211, 3*w^5 - 4*w^4 - 16*w^3 + 17*w^2 + 16*w - 3],\ [239, 239, -3*w^5 + 3*w^4 + 16*w^3 - 12*w^2 - 16*w + 3],\ [239, 239, 4*w^5 - 6*w^4 - 21*w^3 + 25*w^2 + 19*w - 5],\ [239, 239, -w^5 + w^4 + 7*w^3 - 5*w^2 - 10*w + 2],\ [241, 241, w^5 - 2*w^4 - 5*w^3 + 8*w^2 + 4*w + 2],\ [251, 251, 3*w^5 - 4*w^4 - 16*w^3 + 18*w^2 + 15*w - 5],\ [251, 251, w^5 - 2*w^4 - 5*w^3 + 9*w^2 + 6*w - 2],\ [251, 251, -3*w^5 + 4*w^4 + 16*w^3 - 17*w^2 - 15*w + 2],\ [251, 251, -w^5 + 2*w^4 + 4*w^3 - 9*w^2 - w + 4],\ [269, 269, w^5 - w^4 - 6*w^3 + 6*w^2 + 7*w - 5],\ [271, 271, 3*w^5 - 3*w^4 - 15*w^3 + 12*w^2 + 11*w - 1],\ [271, 271, -w^5 + 2*w^4 + 5*w^3 - 8*w^2 - 2*w],\ [281, 281, w^2 - w - 4],\ [281, 281, -3*w^5 + 4*w^4 + 16*w^3 - 18*w^2 - 14*w + 4],\ [281, 281, w^5 - w^4 - 6*w^3 + 4*w^2 + 9*w],\ [281, 281, -3*w^5 + 3*w^4 + 14*w^3 - 13*w^2 - 8*w + 4],\ [289, 17, 2*w^5 - 2*w^4 - 11*w^3 + 8*w^2 + 10*w - 1],\ [311, 311, -w^5 - w^4 + 6*w^3 + 6*w^2 - 8*w - 5],\ [311, 311, -2*w^5 + 3*w^4 + 10*w^3 - 13*w^2 - 8*w + 7],\ [331, 331, 2*w^5 - 3*w^4 - 11*w^3 + 12*w^2 + 12*w - 4],\ [331, 331, -3*w^5 + 4*w^4 + 15*w^3 - 16*w^2 - 12*w + 3],\ [331, 331, -w^5 + w^4 + 4*w^3 - 3*w^2 + w - 1],\ [331, 331, -2*w^5 + 5*w^4 + 10*w^3 - 21*w^2 - 8*w + 5],\ [359, 359, 2*w^4 - 2*w^3 - 9*w^2 + 7*w + 3],\ [361, 19, -w^4 + 2*w^3 + 4*w^2 - 6*w - 2],\ [379, 379, 3*w^5 - 2*w^4 - 15*w^3 + 8*w^2 + 11*w - 1],\ [379, 379, 2*w^5 - 2*w^4 - 11*w^3 + 8*w^2 + 13*w - 3],\ [389, 389, w^5 + w^4 - 5*w^3 - 5*w^2 + 3*w + 4],\ [389, 389, 3*w^5 - 5*w^4 - 16*w^3 + 21*w^2 + 16*w - 5],\ [389, 389, -w^5 + 4*w^4 + 5*w^3 - 17*w^2 - 4*w + 4],\ [389, 389, 3*w^5 - 4*w^4 - 15*w^3 + 18*w^2 + 10*w - 6],\ [401, 401, w^5 - 2*w^4 - 6*w^3 + 7*w^2 + 8*w + 3],\ [401, 401, -2*w^5 + 4*w^4 + 11*w^3 - 17*w^2 - 13*w + 3],\ [409, 409, w^3 + w^2 - 3*w - 5],\ [419, 419, -w^5 + w^4 + 4*w^3 - 6*w^2 + 4],\ [419, 419, -w^4 + 2*w^3 + 5*w^2 - 8*w - 6],\ [421, 421, w^5 - w^4 - 5*w^3 + 5*w^2 + 3*w - 6],\ [431, 431, 2*w^3 + w^2 - 9*w - 3],\ [439, 439, -w^5 + 6*w^3 + w^2 - 6*w - 4],\ [439, 439, w^5 - 2*w^4 - 4*w^3 + 9*w^2 - 7],\ [439, 439, 3*w^5 - 5*w^4 - 16*w^3 + 21*w^2 + 14*w - 4],\ [439, 439, -2*w^5 + 3*w^4 + 11*w^3 - 11*w^2 - 13*w - 1],\ [439, 439, w^4 - w^3 - 3*w^2 + 3*w - 3],\ [439, 439, -3*w^5 + 4*w^4 + 16*w^3 - 16*w^2 - 17*w + 2],\ [449, 449, -w^3 - 2*w^2 + 5*w + 6],\ [461, 461, w^5 - w^4 - 6*w^3 + 3*w^2 + 10*w + 1],\ [479, 479, 3*w^5 - 5*w^4 - 16*w^3 + 21*w^2 + 15*w - 2],\ [491, 491, w^5 - w^4 - 6*w^3 + 3*w^2 + 10*w + 2],\ [499, 499, w^5 - 5*w^3 - 2*w^2 + 4*w + 4],\ [509, 509, -w^5 + 2*w^4 + 5*w^3 - 10*w^2 - 4*w + 4],\ [509, 509, -w^5 + w^4 + 6*w^3 - 6*w^2 - 7*w + 4],\ [521, 521, -3*w^5 + 5*w^4 + 16*w^3 - 22*w^2 - 14*w + 5],\ [541, 541, -w^4 + w^3 + 5*w^2 - 2*w - 5],\ [541, 541, 2*w^5 - 3*w^4 - 9*w^3 + 14*w^2 + 4*w - 5],\ [541, 541, -2*w^5 + 3*w^4 + 9*w^3 - 13*w^2 - 2*w + 4],\ [569, 569, 2*w^5 - w^4 - 11*w^3 + 5*w^2 + 11*w - 3],\ [571, 571, w^5 - w^4 - 6*w^3 + 2*w^2 + 8*w + 4],\ [571, 571, 3*w^5 - 5*w^4 - 15*w^3 + 22*w^2 + 10*w - 7],\ [599, 599, -w^4 + w^3 + 3*w^2 - 4*w],\ [599, 599, -w^4 + 6*w^2 + w - 4],\ [601, 601, -2*w^5 + 3*w^4 + 10*w^3 - 12*w^2 - 7*w + 4],\ [619, 619, -3*w^5 + 4*w^4 + 15*w^3 - 16*w^2 - 11*w + 2],\ [631, 631, w^4 + 2*w^3 - 5*w^2 - 8*w + 2],\ [631, 631, -w^5 + 4*w^4 + 5*w^3 - 18*w^2 - 4*w + 6],\ [631, 631, w^4 + 2*w^3 - 4*w^2 - 8*w + 1],\ [641, 641, -w^5 - w^4 + 5*w^3 + 5*w^2 - 5*w - 2],\ [641, 641, -w^4 + 6*w^2 + w - 5],\ [641, 641, w^5 - 2*w^4 - 4*w^3 + 9*w^2 + 2*w - 5],\ [659, 659, -w^4 - 2*w^3 + 5*w^2 + 8*w - 1],\ [661, 661, 2*w^5 - 2*w^4 - 9*w^3 + 8*w^2 + 3*w + 1],\ [661, 661, 2*w^5 - 3*w^4 - 11*w^3 + 12*w^2 + 10*w - 1],\ [661, 661, w^5 - 3*w^4 - 6*w^3 + 12*w^2 + 9*w],\ [661, 661, -2*w^5 + 3*w^4 + 12*w^3 - 13*w^2 - 14*w + 1],\ [691, 691, w^5 - 6*w^3 - w^2 + 9*w + 1],\ [691, 691, 3*w^5 - 5*w^4 - 15*w^3 + 21*w^2 + 13*w - 5],\ [691, 691, w^5 - 2*w^4 - 6*w^3 + 8*w^2 + 6*w + 2],\ [691, 691, 2*w^5 - 3*w^4 - 12*w^3 + 12*w^2 + 16*w - 2],\ [701, 701, -w^5 - w^4 + 6*w^3 + 4*w^2 - 8*w - 2],\ [709, 709, -3*w^5 + 3*w^4 + 14*w^3 - 12*w^2 - 8*w + 3],\ [709, 709, w^5 + w^4 - 6*w^3 - 4*w^2 + 8*w + 1],\ [709, 709, 2*w^5 - 4*w^4 - 10*w^3 + 16*w^2 + 9*w - 2],\ [709, 709, 2*w^5 - 2*w^4 - 12*w^3 + 9*w^2 + 14*w - 3],\ [719, 719, w^4 - w^3 - 6*w^2 + 3*w + 4],\ [729, 3, -3],\ [739, 739, -w^5 + w^4 + 6*w^3 - 3*w^2 - 9*w - 5],\ [751, 751, -2*w^5 + 3*w^4 + 11*w^3 - 13*w^2 - 12*w + 1],\ [751, 751, -w^5 + w^4 + 4*w^3 - 4*w^2 - 2*w - 1],\ [761, 761, -2*w^5 + 2*w^4 + 12*w^3 - 9*w^2 - 15*w],\ [769, 769, -3*w^5 + 5*w^4 + 17*w^3 - 21*w^2 - 19*w + 3],\ [769, 769, -2*w^5 + 2*w^4 + 9*w^3 - 7*w^2 - 5*w - 1],\ [769, 769, -2*w^5 + 4*w^4 + 10*w^3 - 18*w^2 - 9*w + 7],\ [769, 769, 2*w^5 - w^4 - 12*w^3 + 3*w^2 + 14*w + 2],\ [811, 811, -3*w^5 + 2*w^4 + 16*w^3 - 9*w^2 - 15*w + 2],\ [821, 821, 2*w^3 - w^2 - 8*w + 1],\ [829, 829, w^5 - w^4 - 7*w^3 + 3*w^2 + 11*w + 1],\ [829, 829, 2*w^5 - 2*w^4 - 10*w^3 + 9*w^2 + 8*w - 6],\ [839, 839, -w^4 - 2*w^3 + 5*w^2 + 6*w - 1],\ [841, 29, -2*w^5 + 2*w^4 + 12*w^3 - 9*w^2 - 16*w + 2],\ [859, 859, -w^5 + 2*w^4 + 4*w^3 - 10*w^2 - w + 4],\ [859, 859, 2*w^5 - 3*w^4 - 10*w^3 + 11*w^2 + 7*w + 1],\ [881, 881, -2*w^5 + 3*w^4 + 9*w^3 - 14*w^2 - 4*w + 4],\ [881, 881, 2*w^5 - 2*w^4 - 10*w^3 + 10*w^2 + 6*w - 5],\ [911, 911, w^5 - w^4 - 4*w^3 + 5*w^2 - 2*w - 5],\ [911, 911, w^3 - 4*w - 4],\ [929, 929, -w^5 + 3*w^4 + 6*w^3 - 12*w^2 - 9*w + 1],\ [941, 941, 2*w^4 - 10*w^2 + w + 6],\ [941, 941, 3*w^5 - 3*w^4 - 15*w^3 + 14*w^2 + 10*w - 5],\ [941, 941, -w^5 + 5*w^3 + w^2 - 2*w - 4],\ [961, 31, -w^4 - w^3 + 6*w^2 + 4*w - 4],\ [961, 31, 2*w^5 - 3*w^4 - 12*w^3 + 12*w^2 + 15*w - 3],\ [971, 971, -2*w^5 + 3*w^4 + 12*w^3 - 13*w^2 - 14*w],\ [991, 991, w^5 - 3*w^4 - 4*w^3 + 14*w^2 + w - 6]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x K = QQ e = 1 hecke_eigenvalues_array = [0, -2, 6, -4, -2, -4, 2, 12, 1, -12, 8, 7, 0, -8, 2, 8, -6, 2, -10, -18, 10, -12, -14, -4, -12, -4, -8, 6, -14, -10, 6, 4, -20, 0, -4, -10, 14, -8, 8, -8, -16, 4, 4, 10, -22, 0, -12, 18, -2, -22, -6, -24, 28, -6, -20, 18, -18, -12, 18, 30, -10, -28, 0, 20, -24, 0, -28, 30, 12, 20, 20, 24, -22, -38, 0, -2, 0, -22, 24, 2, 10, -36, -26, 36, 8, 10, 4, 26, -6, -30, -10, 20, 8, -42, -40, 30, 10, 34, 6, 10, -16, 30, 24, -32, -2, 28, -4, 32, 44, 12, -16, -46, 48, -22, -14, 32, 40, 6, 18, 12, 44, -30, -50, -26, 38, 0, 8, -2, -12, -36, -44, -2, -30, -26, 26, -44, 16, -30, -6, -2, 56, -26, -16, -28, -26, -42, -2, -40, 46, -18, -16, -46, 24, -46, 12, -32] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([59, 59, w^4 - w^3 - 5*w^2 + 3*w + 4])] = -1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]