/* 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, 7, 6, -6, -1, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([41, 41, w^5 - 6*w^3 + w^2 + 7*w - 2]) primes_array = [ [9, 3, -w^2 + 2],\ [11, 11, w^4 + w^3 - 4*w^2 - 3*w + 2],\ [11, 11, -w^5 + 6*w^3 - w^2 - 7*w + 1],\ [19, 19, -w^5 - w^4 + 5*w^3 + 4*w^2 - 5*w - 3],\ [29, 29, -2*w^4 - w^3 + 9*w^2 + 2*w - 5],\ [31, 31, -w^5 + w^4 + 6*w^3 - 5*w^2 - 7*w + 1],\ [41, 41, 2*w^5 + w^4 - 10*w^3 - 3*w^2 + 7*w + 2],\ [41, 41, w^5 - w^4 - 6*w^3 + 5*w^2 + 6*w - 3],\ [41, 41, w^5 - 6*w^3 + w^2 + 7*w - 2],\ [59, 59, -w^5 - w^4 + 4*w^3 + 4*w^2 - 2*w - 3],\ [59, 59, w^5 - w^4 - 5*w^3 + 6*w^2 + 2*w - 4],\ [59, 59, -w^5 + 4*w^3 - w^2 - w + 1],\ [61, 61, -w^4 + 4*w^2 - 2*w - 2],\ [64, 2, -2],\ [71, 71, 2*w^5 + w^4 - 10*w^3 - 3*w^2 + 9*w + 3],\ [81, 3, w^5 + 2*w^4 - 4*w^3 - 8*w^2 + 2*w + 3],\ [89, 89, -w^5 + 6*w^3 - 7*w],\ [89, 89, w^5 + w^4 - 5*w^3 - 4*w^2 + 6*w + 2],\ [101, 101, w^5 - 4*w^3 + 2*w^2 + w - 3],\ [101, 101, w^5 + 2*w^4 - 4*w^3 - 8*w^2 + 3*w + 4],\ [121, 11, -2*w^5 - w^4 + 9*w^3 + w^2 - 5*w + 1],\ [121, 11, -w^5 - w^4 + 4*w^3 + 3*w^2 - 2],\ [125, 5, w^5 - 5*w^3 + w^2 + 4*w - 3],\ [131, 131, w^4 - w^3 - 5*w^2 + 4*w + 3],\ [131, 131, -w^4 + 4*w^2 - w - 3],\ [139, 139, 2*w^5 - w^4 - 10*w^3 + 7*w^2 + 6*w - 3],\ [139, 139, -w^5 - w^4 + 5*w^3 + 2*w^2 - 5*w + 2],\ [139, 139, -2*w^5 + 11*w^3 - 2*w^2 - 12*w + 2],\ [149, 149, -w^5 - w^4 + 4*w^3 + w^2 - w + 3],\ [151, 151, -w^5 + 6*w^3 - 6*w - 2],\ [169, 13, -w^5 - 2*w^4 + 5*w^3 + 8*w^2 - 5*w - 4],\ [179, 179, -w^4 + 5*w^2 - 2*w - 4],\ [179, 179, w^5 + 3*w^4 - 3*w^3 - 12*w^2 - w + 7],\ [181, 181, -w^5 + 5*w^3 - w^2 - 5*w - 2],\ [191, 191, w^5 + w^4 - 4*w^3 - 3*w^2 + 2*w - 1],\ [191, 191, 2*w^5 - 12*w^3 + 2*w^2 + 15*w - 2],\ [199, 199, -2*w^4 - w^3 + 9*w^2 + w - 4],\ [199, 199, -3*w^5 - w^4 + 15*w^3 + w^2 - 14*w],\ [211, 211, w^5 - 5*w^3 + 2*w + 2],\ [211, 211, w^5 - 2*w^4 - 7*w^3 + 8*w^2 + 9*w - 1],\ [229, 229, -3*w^5 - w^4 + 16*w^3 + 2*w^2 - 16*w],\ [229, 229, -w^4 - w^3 + 6*w^2 + 3*w - 4],\ [241, 241, w^5 + 3*w^4 - 4*w^3 - 13*w^2 + 3*w + 8],\ [241, 241, 2*w^5 - 10*w^3 + 3*w^2 + 7*w - 2],\ [241, 241, -2*w^4 - 2*w^3 + 9*w^2 + 4*w - 5],\ [241, 241, w^5 - 6*w^3 + 2*w^2 + 9*w - 4],\ [251, 251, -w^5 + 5*w^3 - w^2 - 5*w + 4],\ [251, 251, w^4 + 2*w^3 - 4*w^2 - 6*w + 2],\ [251, 251, 2*w^4 + 2*w^3 - 7*w^2 - 6*w + 1],\ [269, 269, 2*w^5 + 2*w^4 - 10*w^3 - 6*w^2 + 9*w - 1],\ [269, 269, -2*w^5 - 2*w^4 + 8*w^3 + 6*w^2 - 3*w - 2],\ [269, 269, -2*w^5 + 12*w^3 - w^2 - 14*w + 1],\ [269, 269, -2*w^5 - w^4 + 10*w^3 + 2*w^2 - 7*w - 2],\ [271, 271, -w^5 + w^4 + 5*w^3 - 6*w^2 - 4*w + 3],\ [281, 281, -3*w^5 + 16*w^3 - 2*w^2 - 14*w],\ [289, 17, -w^4 - w^3 + 2*w^2 + 3*w + 3],\ [289, 17, -2*w^5 + w^4 + 10*w^3 - 6*w^2 - 7*w + 4],\ [289, 17, -2*w^5 + 11*w^3 - 3*w^2 - 12*w + 2],\ [311, 311, -2*w^5 + w^4 + 11*w^3 - 7*w^2 - 9*w + 4],\ [311, 311, -2*w^4 - w^3 + 7*w^2 + w - 1],\ [311, 311, 2*w^5 - 9*w^3 + 2*w^2 + 5*w - 2],\ [311, 311, -2*w^4 + 10*w^2 - w - 7],\ [331, 331, w^5 - 4*w^3 + 4],\ [331, 331, 2*w^5 - 10*w^3 + 4*w^2 + 7*w - 6],\ [331, 331, -3*w^5 + 15*w^3 - 4*w^2 - 13*w + 4],\ [331, 331, -3*w^5 - w^4 + 14*w^3 + w^2 - 10*w + 1],\ [349, 349, -w^5 + 6*w^3 + w^2 - 7*w - 4],\ [359, 359, -2*w^5 - 2*w^4 + 9*w^3 + 7*w^2 - 6*w - 6],\ [359, 359, w^5 + w^4 - 5*w^3 - 3*w^2 + 7*w],\ [361, 19, -2*w^5 + 12*w^3 - w^2 - 14*w],\ [379, 379, -w^5 - w^4 + 6*w^3 + 5*w^2 - 7*w - 4],\ [389, 389, w^5 - w^4 - 6*w^3 + 5*w^2 + 5*w - 2],\ [389, 389, -2*w^5 - w^4 + 8*w^3 + 2*w^2 - w - 2],\ [389, 389, -2*w^5 + 10*w^3 - w^2 - 7*w + 1],\ [389, 389, 2*w^5 - w^4 - 10*w^3 + 7*w^2 + 6*w - 6],\ [401, 401, -w^4 - 2*w^3 + 5*w^2 + 5*w - 3],\ [401, 401, 2*w^5 - w^4 - 10*w^3 + 8*w^2 + 6*w - 7],\ [401, 401, w^5 - 2*w^4 - 7*w^3 + 9*w^2 + 8*w - 2],\ [409, 409, -2*w^5 - 2*w^4 + 9*w^3 + 7*w^2 - 5*w - 5],\ [419, 419, w^5 - w^4 - 7*w^3 + 4*w^2 + 9*w - 2],\ [419, 419, w^5 - 6*w^3 - w^2 + 6*w + 3],\ [421, 421, -2*w^5 - w^4 + 8*w^3 + 2*w^2 - 2*w - 1],\ [431, 431, -3*w^5 - w^4 + 16*w^3 + w^2 - 16*w + 2],\ [439, 439, w^5 - w^4 - 5*w^3 + 4*w^2 + 2*w + 2],\ [439, 439, -2*w^5 + 10*w^3 - w^2 - 8*w - 3],\ [439, 439, -2*w^5 - 3*w^4 + 9*w^3 + 11*w^2 - 6*w - 6],\ [439, 439, 3*w^5 + w^4 - 14*w^3 - 2*w^2 + 9*w + 2],\ [449, 449, 2*w^5 + w^4 - 11*w^3 - 4*w^2 + 13*w + 4],\ [449, 449, -w^5 + 2*w^4 + 6*w^3 - 10*w^2 - 6*w + 3],\ [449, 449, 2*w^5 + w^4 - 11*w^3 - 2*w^2 + 11*w - 2],\ [449, 449, -w^5 + w^4 + 5*w^3 - 6*w^2 - 3*w + 1],\ [461, 461, -2*w^5 + w^4 + 10*w^3 - 6*w^2 - 8*w + 3],\ [461, 461, -2*w^5 - 2*w^4 + 10*w^3 + 6*w^2 - 9*w - 3],\ [479, 479, w^4 - 4*w^2 + 3*w + 2],\ [479, 479, 3*w^5 + w^4 - 15*w^3 - w^2 + 13*w - 2],\ [479, 479, -2*w^4 - w^3 + 9*w^2 + 3*w - 3],\ [479, 479, -w^5 - w^4 + 4*w^3 + 3*w^2 - 4],\ [491, 491, -2*w^5 + 11*w^3 - 11*w - 4],\ [491, 491, 3*w^5 + w^4 - 14*w^3 - w^2 + 9*w - 2],\ [491, 491, -2*w^5 - 2*w^4 + 10*w^3 + 8*w^2 - 10*w - 7],\ [491, 491, -w^5 - w^4 + 6*w^3 + 4*w^2 - 6*w - 3],\ [499, 499, 2*w^5 + w^4 - 10*w^3 - 4*w^2 + 8*w + 4],\ [509, 509, -3*w^5 - w^4 + 16*w^3 + w^2 - 15*w + 1],\ [509, 509, w^5 - 5*w^3 + 2*w^2 + 2*w - 3],\ [509, 509, 2*w^4 + w^3 - 9*w^2 + 3],\ [509, 509, 2*w^3 + 2*w^2 - 6*w - 3],\ [521, 521, w^5 - 6*w^3 + w^2 + 9*w - 1],\ [521, 521, -w^5 - 2*w^4 + 3*w^3 + 7*w^2 - 1],\ [529, 23, -w^5 + 5*w^3 - 3*w^2 - 4*w + 7],\ [541, 541, w^3 + w^2 - 5*w - 4],\ [541, 541, -w^4 - 2*w^3 + 5*w^2 + 7*w - 4],\ [541, 541, -3*w^5 - w^4 + 15*w^3 + w^2 - 11*w - 1],\ [541, 541, 2*w^5 - w^4 - 12*w^3 + 5*w^2 + 14*w - 2],\ [601, 601, 2*w^2 + w - 3],\ [619, 619, -2*w^5 - 2*w^4 + 8*w^3 + 7*w^2 - 3*w - 4],\ [631, 631, w^5 + w^4 - 6*w^3 - 5*w^2 + 9*w + 3],\ [631, 631, -3*w^5 + 16*w^3 - 4*w^2 - 15*w + 3],\ [641, 641, w^5 - 3*w^3 + 2*w^2 - 3*w - 2],\ [641, 641, -2*w^5 + w^4 + 12*w^3 - 7*w^2 - 12*w + 3],\ [661, 661, -2*w^5 + w^4 + 11*w^3 - 5*w^2 - 9*w],\ [661, 661, -3*w^5 - w^4 + 14*w^3 + w^2 - 10*w],\ [691, 691, -2*w^5 - 3*w^4 + 9*w^3 + 11*w^2 - 7*w - 6],\ [701, 701, 2*w^5 - 9*w^3 + 2*w^2 + 3*w - 2],\ [709, 709, 2*w^5 + 2*w^4 - 10*w^3 - 7*w^2 + 11*w + 3],\ [719, 719, 3*w^4 + 2*w^3 - 13*w^2 - 4*w + 6],\ [719, 719, -w^3 - 2*w^2 + 2*w + 5],\ [739, 739, -2*w^5 - 3*w^4 + 10*w^3 + 11*w^2 - 11*w - 4],\ [739, 739, -3*w^5 + w^4 + 16*w^3 - 7*w^2 - 15*w + 2],\ [751, 751, -w^5 + 4*w^3 - 3*w^2 + 2],\ [761, 761, 2*w^5 + 2*w^4 - 8*w^3 - 7*w^2 + 3*w + 3],\ [761, 761, -w^5 + w^4 + 6*w^3 - 4*w^2 - 5*w + 2],\ [761, 761, 2*w^5 + 3*w^4 - 9*w^3 - 10*w^2 + 6*w + 2],\ [761, 761, -w^5 - 3*w^4 + 3*w^3 + 12*w^2 + w - 4],\ [769, 769, 2*w^5 - 2*w^4 - 12*w^3 + 11*w^2 + 12*w - 6],\ [809, 809, w^5 + w^4 - 3*w^3 - 2*w^2 - 3*w + 1],\ [809, 809, 2*w^5 + w^4 - 9*w^3 - w^2 + 3*w - 2],\ [809, 809, w^5 + 3*w^4 - 4*w^3 - 12*w^2 + 2*w + 5],\ [811, 811, 2*w^5 - 10*w^3 + 3*w^2 + 8*w],\ [821, 821, w^4 - w^3 - 5*w^2 + 4*w + 6],\ [821, 821, -2*w^5 + 11*w^3 - 2*w^2 - 9*w + 2],\ [829, 829, 2*w^5 - 12*w^3 + w^2 + 13*w],\ [839, 839, -2*w^5 - 2*w^4 + 11*w^3 + 8*w^2 - 12*w - 5],\ [841, 29, -3*w^5 - 4*w^4 + 13*w^3 + 13*w^2 - 9*w - 6],\ [859, 859, -3*w^5 - 2*w^4 + 14*w^3 + 6*w^2 - 11*w - 4],\ [859, 859, -3*w^5 + 15*w^3 - 3*w^2 - 13*w + 3],\ [881, 881, 3*w^5 - 16*w^3 + 4*w^2 + 15*w - 2],\ [881, 881, 2*w^5 + 2*w^4 - 10*w^3 - 7*w^2 + 11*w + 1],\ [911, 911, w^5 + w^4 - 5*w^3 - 4*w^2 + 3*w + 6],\ [911, 911, -w^5 - 2*w^4 + 5*w^3 + 9*w^2 - 7*w - 7],\ [911, 911, -w^4 - 2*w^3 + 3*w^2 + 6*w - 2],\ [919, 919, 3*w^5 + w^4 - 14*w^3 + 8*w - 3],\ [919, 919, w^5 - 4*w^3 + w^2 - w - 2],\ [929, 929, -2*w^5 + 2*w^4 + 12*w^3 - 10*w^2 - 13*w + 5],\ [941, 941, -2*w^5 - 3*w^4 + 9*w^3 + 10*w^2 - 6*w - 3],\ [941, 941, 2*w^4 - w^3 - 10*w^2 + 5*w + 5],\ [961, 31, -2*w^5 - 2*w^4 + 9*w^3 + 6*w^2 - 5*w - 4],\ [971, 971, -2*w^4 + 9*w^2 - 3*w - 6],\ [971, 971, 2*w^5 + w^4 - 9*w^3 + 5*w - 5],\ [971, 971, 3*w^4 + w^3 - 13*w^2 - w + 5],\ [971, 971, -4*w^5 - 2*w^4 + 18*w^3 + 4*w^2 - 9*w],\ [991, 991, -w^4 - w^3 + 6*w^2 + 4*w - 5],\ [991, 991, 4*w^5 - 21*w^3 + 4*w^2 + 19*w - 2]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x K = QQ e = 1 hecke_eigenvalues_array = [3, 4, -1, 6, -2, 5, 0, -12, -1, -4, 11, 11, -1, 1, -12, 5, 9, -3, 6, -7, 14, 7, 18, 17, 0, -11, -3, 14, -20, -6, 15, -22, 16, -8, 0, -20, -11, 7, 15, 5, 21, -9, 22, -10, -10, -12, 16, 1, 21, 7, -3, -25, 18, 21, -18, -23, -6, -9, 12, 16, -8, -32, -28, 24, -23, -3, 14, 14, -9, -27, 10, -9, 11, -2, 33, -9, -10, -28, 2, -28, -28, 13, 24, -32, 2, -13, -28, 13, 10, -4, 0, 2, -18, -12, -37, 0, -35, -18, 0, 24, -13, 18, 38, 13, 29, -6, 0, 18, -27, 10, -14, -4, 38, 19, 25, -18, 13, 0, 0, 29, -43, 23, 34, 38, 24, -3, -43, -8, 44, 12, 10, 9, -21, -18, -40, 25, -53, 15, -50, -6, 52, -24, -32, -7, -1, 24, 33, -27, -12, -13, 55, 4, -24, 24, 22, 38, 14, -28, -22, -28, -47, -37] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([41, 41, w^5 - 6*w^3 + w^2 + 7*w - 2])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]