/* 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, -5, 2, 8, -4, -2, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([59, 59, 2*w^5 - 3*w^4 - 9*w^3 + 11*w^2 + 9*w - 4]) primes_array = [ [9, 3, -w^2 + 2],\ [19, 19, -w^3 + 4*w],\ [19, 19, w^3 - 3*w - 1],\ [29, 29, w^5 - w^4 - 6*w^3 + 4*w^2 + 8*w - 3],\ [29, 29, w^4 - w^3 - 4*w^2 + 2*w + 2],\ [31, 31, -w^4 + 4*w^2 + w - 3],\ [41, 41, 2*w^5 - 2*w^4 - 10*w^3 + 7*w^2 + 11*w - 4],\ [49, 7, -2*w^5 + 3*w^4 + 10*w^3 - 12*w^2 - 11*w + 6],\ [59, 59, w^5 - 2*w^4 - 5*w^3 + 8*w^2 + 7*w - 5],\ [59, 59, w^5 - w^4 - 4*w^3 + 3*w^2 + 3*w - 3],\ [59, 59, 2*w^5 - 3*w^4 - 9*w^3 + 11*w^2 + 9*w - 4],\ [61, 61, -w^5 + w^4 + 5*w^3 - 3*w^2 - 7*w + 1],\ [64, 2, -2],\ [71, 71, -2*w^5 + 2*w^4 + 10*w^3 - 8*w^2 - 11*w + 6],\ [79, 79, -3*w^5 + 4*w^4 + 13*w^3 - 14*w^2 - 9*w + 5],\ [79, 79, -w^4 - w^3 + 5*w^2 + 4*w - 3],\ [79, 79, -2*w^5 + 2*w^4 + 9*w^3 - 7*w^2 - 8*w + 4],\ [81, 3, 3*w^5 - 5*w^4 - 14*w^3 + 19*w^2 + 13*w - 8],\ [89, 89, 2*w^5 - 2*w^4 - 9*w^3 + 6*w^2 + 8*w - 1],\ [101, 101, -w^4 - w^3 + 5*w^2 + 3*w - 3],\ [101, 101, -w^5 + 2*w^4 + 6*w^3 - 8*w^2 - 9*w + 4],\ [109, 109, -2*w^5 + 4*w^4 + 8*w^3 - 14*w^2 - 6*w + 5],\ [109, 109, -3*w^5 + 4*w^4 + 13*w^3 - 14*w^2 - 10*w + 7],\ [125, 5, w^5 - w^4 - 5*w^3 + 4*w^2 + 5*w],\ [131, 131, -w^2 - w + 3],\ [131, 131, w^5 - 3*w^4 - 4*w^3 + 11*w^2 + 3*w - 4],\ [131, 131, w^5 - 2*w^4 - 5*w^3 + 9*w^2 + 6*w - 5],\ [131, 131, -2*w^5 + 4*w^4 + 8*w^3 - 15*w^2 - 6*w + 7],\ [139, 139, w^4 - 3*w^3 - 3*w^2 + 9*w],\ [149, 149, 2*w^5 - 3*w^4 - 9*w^3 + 12*w^2 + 7*w - 5],\ [149, 149, w^5 - 2*w^4 - 4*w^3 + 8*w^2 + 2*w - 2],\ [151, 151, 3*w^5 - 4*w^4 - 13*w^3 + 14*w^2 + 10*w - 5],\ [151, 151, -3*w^5 + 4*w^4 + 14*w^3 - 15*w^2 - 12*w + 6],\ [151, 151, 4*w^5 - 5*w^4 - 18*w^3 + 18*w^2 + 15*w - 9],\ [151, 151, 2*w^5 - 2*w^4 - 9*w^3 + 7*w^2 + 8*w - 5],\ [169, 13, -2*w^5 + w^4 + 11*w^3 - 3*w^2 - 13*w + 1],\ [179, 179, w^3 - w^2 - 2*w + 3],\ [181, 181, w^5 - 2*w^4 - 6*w^3 + 7*w^2 + 10*w - 3],\ [181, 181, -2*w^4 + w^3 + 8*w^2 - 3*w - 2],\ [191, 191, w^5 - w^4 - 6*w^3 + 5*w^2 + 9*w - 4],\ [191, 191, -w^3 + 2*w^2 + 3*w - 4],\ [191, 191, -2*w^5 + w^4 + 11*w^3 - 3*w^2 - 12*w],\ [191, 191, w^4 - 2*w^3 - 4*w^2 + 7*w],\ [199, 199, -3*w^5 + 4*w^4 + 15*w^3 - 14*w^2 - 16*w + 6],\ [199, 199, -2*w^5 + 3*w^4 + 10*w^3 - 12*w^2 - 12*w + 7],\ [211, 211, -2*w^5 + 4*w^4 + 9*w^3 - 15*w^2 - 8*w + 5],\ [211, 211, -3*w^5 + 5*w^4 + 13*w^3 - 18*w^2 - 10*w + 8],\ [229, 229, -3*w^5 + 4*w^4 + 13*w^3 - 14*w^2 - 10*w + 6],\ [229, 229, w^4 - 4*w^2 - 1],\ [229, 229, w^5 - w^4 - 3*w^3 + 3*w^2 - 3*w - 2],\ [239, 239, -2*w^5 + 3*w^4 + 9*w^3 - 11*w^2 - 8*w + 8],\ [241, 241, w^5 - 3*w^4 - 4*w^3 + 10*w^2 + 3*w - 2],\ [241, 241, w^5 - w^4 - 5*w^3 + 3*w^2 + 4*w + 1],\ [251, 251, -3*w^5 + 3*w^4 + 15*w^3 - 10*w^2 - 17*w + 3],\ [271, 271, -2*w^5 + 4*w^4 + 9*w^3 - 14*w^2 - 8*w + 2],\ [271, 271, -2*w^5 + 3*w^4 + 8*w^3 - 9*w^2 - 4*w],\ [271, 271, w^5 - 2*w^4 - 3*w^3 + 7*w^2 - 2*w - 4],\ [289, 17, 4*w^5 - 6*w^4 - 17*w^3 + 22*w^2 + 12*w - 9],\ [311, 311, -2*w^5 + 3*w^4 + 7*w^3 - 9*w^2 - 3*w + 2],\ [331, 331, 3*w^5 - 5*w^4 - 14*w^3 + 19*w^2 + 14*w - 12],\ [349, 349, -4*w^5 + 6*w^4 + 18*w^3 - 21*w^2 - 17*w + 7],\ [349, 349, -w^4 + w^3 + 5*w^2 - 3*w - 3],\ [359, 359, -2*w^5 + 4*w^4 + 9*w^3 - 15*w^2 - 10*w + 9],\ [361, 19, 2*w^5 - 3*w^4 - 9*w^3 + 11*w^2 + 6*w - 4],\ [361, 19, -3*w^5 + 5*w^4 + 13*w^3 - 18*w^2 - 10*w + 9],\ [379, 379, -4*w^5 + 5*w^4 + 18*w^3 - 17*w^2 - 14*w + 7],\ [379, 379, 2*w^5 - 4*w^4 - 10*w^3 + 15*w^2 + 11*w - 5],\ [389, 389, -5*w^5 + 7*w^4 + 22*w^3 - 25*w^2 - 17*w + 11],\ [389, 389, w^5 - 3*w^4 - 3*w^3 + 12*w^2 - 2*w - 5],\ [401, 401, -4*w^5 + 6*w^4 + 18*w^3 - 23*w^2 - 15*w + 10],\ [401, 401, -w^5 + 3*w^4 + 4*w^3 - 12*w^2 - w + 4],\ [401, 401, 4*w^5 - 5*w^4 - 20*w^3 + 19*w^2 + 21*w - 11],\ [401, 401, -3*w^5 + 3*w^4 + 16*w^3 - 12*w^2 - 18*w + 6],\ [409, 409, -2*w^4 + 2*w^3 + 7*w^2 - 6*w - 2],\ [409, 409, -w^5 + 3*w^4 + 5*w^3 - 13*w^2 - 5*w + 9],\ [419, 419, 2*w^5 - 2*w^4 - 10*w^3 + 7*w^2 + 10*w - 6],\ [419, 419, -w^5 + 7*w^3 - w^2 - 10*w + 4],\ [419, 419, w^5 - 2*w^4 - 5*w^3 + 10*w^2 + 5*w - 9],\ [421, 421, 3*w^5 - 3*w^4 - 16*w^3 + 12*w^2 + 18*w - 7],\ [431, 431, 4*w^5 - 7*w^4 - 17*w^3 + 25*w^2 + 13*w - 10],\ [431, 431, -2*w^5 + 3*w^4 + 7*w^3 - 10*w^2 - w + 5],\ [431, 431, 4*w^5 - 6*w^4 - 19*w^3 + 23*w^2 + 17*w - 10],\ [439, 439, -2*w^5 + 4*w^4 + 10*w^3 - 15*w^2 - 11*w + 4],\ [461, 461, w^5 - 2*w^4 - 3*w^3 + 6*w^2 + 2],\ [461, 461, w^5 - w^4 - 5*w^3 + 5*w^2 + 6*w - 7],\ [479, 479, -2*w^5 + w^4 + 9*w^3 - 2*w^2 - 8*w],\ [479, 479, 2*w^5 - 2*w^4 - 10*w^3 + 8*w^2 + 12*w - 7],\ [491, 491, -3*w^5 + 3*w^4 + 14*w^3 - 12*w^2 - 12*w + 7],\ [491, 491, -3*w^5 + 3*w^4 + 15*w^3 - 12*w^2 - 14*w + 8],\ [491, 491, -2*w^2 + w + 6],\ [499, 499, -3*w^5 + 3*w^4 + 14*w^3 - 10*w^2 - 13*w + 6],\ [509, 509, -2*w^5 + 2*w^4 + 8*w^3 - 6*w^2 - 5*w],\ [521, 521, 3*w^5 - 4*w^4 - 14*w^3 + 15*w^2 + 15*w - 9],\ [521, 521, -3*w^5 + 5*w^4 + 14*w^3 - 19*w^2 - 15*w + 9],\ [521, 521, -4*w^5 + 5*w^4 + 18*w^3 - 17*w^2 - 16*w + 6],\ [529, 23, w^5 - 3*w^4 - 5*w^3 + 11*w^2 + 7*w - 5],\ [541, 541, -3*w^5 + 4*w^4 + 14*w^3 - 15*w^2 - 14*w + 10],\ [569, 569, -w^5 - w^4 + 6*w^3 + 4*w^2 - 6*w - 1],\ [571, 571, -2*w^4 + w^3 + 9*w^2 - 3*w - 4],\ [599, 599, -3*w^5 + 3*w^4 + 14*w^3 - 11*w^2 - 11*w + 4],\ [601, 601, w^4 - 2*w^2 - w - 2],\ [619, 619, 2*w^5 - 3*w^4 - 9*w^3 + 11*w^2 + 10*w - 5],\ [619, 619, w^5 - 3*w^4 - 4*w^3 + 12*w^2 + 3*w - 3],\ [619, 619, -3*w^5 + 5*w^4 + 13*w^3 - 20*w^2 - 10*w + 10],\ [619, 619, w^5 - 2*w^4 - 5*w^3 + 10*w^2 + 5*w - 8],\ [619, 619, -4*w^5 + 6*w^4 + 19*w^3 - 23*w^2 - 17*w + 11],\ [619, 619, -4*w^5 + 6*w^4 + 17*w^3 - 22*w^2 - 13*w + 10],\ [631, 631, w^5 - 3*w^4 - 5*w^3 + 13*w^2 + 5*w - 6],\ [641, 641, -w^3 + w^2 + 3*w - 5],\ [641, 641, -w^5 + 3*w^4 + 2*w^3 - 10*w^2 + 3*w + 4],\ [659, 659, -w^5 + 3*w^4 + 5*w^3 - 12*w^2 - 5*w + 5],\ [661, 661, -2*w^5 + 3*w^4 + 8*w^3 - 12*w^2 - 3*w + 8],\ [661, 661, 2*w^5 - 3*w^4 - 8*w^3 + 10*w^2 + 6*w - 1],\ [691, 691, -w^3 - w^2 + 5*w + 1],\ [691, 691, -3*w^5 + 4*w^4 + 13*w^3 - 13*w^2 - 10*w + 6],\ [691, 691, 3*w^5 - 4*w^4 - 15*w^3 + 15*w^2 + 14*w - 6],\ [701, 701, 3*w^5 - 6*w^4 - 13*w^3 + 23*w^2 + 10*w - 10],\ [701, 701, -3*w^5 + 6*w^4 + 13*w^3 - 23*w^2 - 11*w + 9],\ [709, 709, 3*w^5 - 6*w^4 - 13*w^3 + 22*w^2 + 11*w - 10],\ [709, 709, -w^5 + 5*w^3 - 4*w - 2],\ [709, 709, -w^5 + w^4 + 7*w^3 - 6*w^2 - 11*w + 5],\ [719, 719, -2*w^5 + 3*w^4 + 9*w^3 - 12*w^2 - 7*w + 4],\ [751, 751, w^4 + w^3 - 5*w^2 - 6*w + 4],\ [769, 769, 4*w^5 - 5*w^4 - 19*w^3 + 18*w^2 + 18*w - 10],\ [769, 769, 4*w^5 - 6*w^4 - 17*w^3 + 22*w^2 + 13*w - 11],\ [809, 809, w^5 - w^4 - 4*w^3 + 5*w^2 + w - 7],\ [809, 809, 2*w^5 - 5*w^4 - 9*w^3 + 19*w^2 + 10*w - 7],\ [809, 809, -w^5 + 5*w^3 + w^2 - 7*w],\ [809, 809, -3*w^5 + 4*w^4 + 15*w^3 - 16*w^2 - 15*w + 11],\ [811, 811, -2*w^5 + 4*w^4 + 8*w^3 - 13*w^2 - 6*w + 5],\ [811, 811, -w^5 + 3*w^4 + 3*w^3 - 9*w^2 - w],\ [811, 811, 2*w^5 - 4*w^4 - 9*w^3 + 16*w^2 + 10*w - 9],\ [811, 811, 2*w^5 - 4*w^4 - 10*w^3 + 16*w^2 + 10*w - 7],\ [821, 821, -w^5 + 2*w^4 + 5*w^3 - 6*w^2 - 5*w - 2],\ [829, 829, 3*w^5 - 5*w^4 - 13*w^3 + 19*w^2 + 12*w - 9],\ [839, 839, 2*w^5 - 4*w^4 - 7*w^3 + 14*w^2 + w - 6],\ [839, 839, 4*w^5 - 5*w^4 - 18*w^3 + 19*w^2 + 14*w - 10],\ [841, 29, -2*w^5 + 4*w^4 + 10*w^3 - 15*w^2 - 10*w + 5],\ [841, 29, -w^5 + 3*w^4 + 6*w^3 - 12*w^2 - 10*w + 6],\ [859, 859, 3*w^5 - 3*w^4 - 15*w^3 + 10*w^2 + 14*w - 2],\ [881, 881, w^5 - 2*w^4 - 6*w^3 + 7*w^2 + 10*w - 4],\ [911, 911, -4*w^5 + 5*w^4 + 19*w^3 - 20*w^2 - 17*w + 11],\ [919, 919, 2*w^5 - 3*w^4 - 10*w^3 + 13*w^2 + 9*w - 10],\ [929, 929, -w^5 + w^4 + 5*w^3 - 2*w^2 - 8*w - 1],\ [941, 941, -5*w^5 + 7*w^4 + 23*w^3 - 26*w^2 - 20*w + 11],\ [961, 31, -3*w^5 + 4*w^4 + 16*w^3 - 16*w^2 - 20*w + 8],\ [971, 971, w^5 - 5*w^3 + 5*w - 4],\ [991, 991, 3*w^5 - 5*w^4 - 12*w^3 + 18*w^2 + 8*w - 6],\ [991, 991, 3*w^5 - 5*w^4 - 14*w^3 + 18*w^2 + 12*w - 7]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x K = QQ e = 1 hecke_eigenvalues_array = [5, -5, 7, -3, 6, 7, -3, 1, 6, 3, -1, 7, -1, -9, -8, -10, 1, 2, -6, 0, 9, 4, -10, -12, 12, -12, 3, 0, -1, 18, -6, -16, -20, -16, 4, -16, 12, -8, -7, -18, -3, 24, 6, 16, -10, -20, -8, 10, -14, 8, -15, 8, 19, 3, 2, 20, -4, 20, -24, -26, -11, 16, -18, 25, 16, 11, 16, 0, 30, 30, 30, 15, -12, -4, -10, 15, 9, 9, -13, 0, -18, 0, -34, 21, -24, -15, -24, -9, -24, 3, -28, -9, 27, 18, 30, -34, 25, -33, 14, 36, 28, -10, 11, -32, -14, 26, 23, 41, 3, -30, 36, 29, -22, 10, 10, 23, 27, 45, -37, -5, 16, 21, -23, -32, 4, 24, 3, 42, -3, -38, -25, -10, -23, 42, -38, 12, 0, 13, 10, -2, 3, -24, 49, 24, -9, 19, 27, -16, 16] 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, 2*w^5 - 3*w^4 - 9*w^3 + 11*w^2 + 9*w - 4])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]