/* 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, 3, 3, -4, -1, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([43, 43, -2*w^4 + w^3 + 6*w^2 - 2*w - 1]) primes_array = [ [11, 11, w^4 + w^3 - 4*w^2 - 3*w + 2],\ [23, 23, -w^4 + 3*w^2 + 1],\ [23, 23, -w^4 + 3*w^2 + w - 2],\ [23, 23, w^4 - w^3 - 3*w^2 + 3*w + 2],\ [23, 23, -w^4 + w^3 + 4*w^2 - 3*w - 1],\ [23, 23, -w^2 + w + 3],\ [32, 2, 2],\ [43, 43, -2*w^4 + w^3 + 6*w^2 - 2*w - 1],\ [43, 43, -w^4 + 2*w^2 + w + 1],\ [43, 43, w^3 + w^2 - 4*w - 2],\ [43, 43, 2*w^4 - w^3 - 7*w^2 + 3*w + 3],\ [43, 43, w^4 - w^3 - 4*w^2 + 4*w + 2],\ [67, 67, 2*w^4 - 7*w^2 + 2],\ [67, 67, w^4 - 2*w^3 - 3*w^2 + 6*w + 2],\ [67, 67, 2*w^4 - 7*w^2 - w + 4],\ [67, 67, w^4 - 2*w^3 - 4*w^2 + 6*w + 2],\ [67, 67, -w^4 + w^3 + 5*w^2 - 3*w - 3],\ [89, 89, w^3 + w^2 - 4*w - 1],\ [89, 89, -2*w^4 + w^3 + 7*w^2 - 3*w - 2],\ [89, 89, -w^4 + w^3 + 4*w^2 - 4*w - 3],\ [89, 89, -w^4 + 2*w^2 + w + 2],\ [89, 89, 2*w^4 - w^3 - 6*w^2 + 2*w + 2],\ [109, 109, -w^3 + 2*w^2 + 3*w - 3],\ [109, 109, w^4 - 4*w^2 - 2*w + 3],\ [109, 109, -2*w^4 + 2*w^3 + 7*w^2 - 4*w - 3],\ [109, 109, 2*w^3 - 5*w - 1],\ [109, 109, -w^4 - w^3 + 5*w^2 + 2*w - 4],\ [131, 131, w^4 - 3*w^3 - 2*w^2 + 7*w],\ [131, 131, -w^4 + 2*w^3 + 5*w^2 - 7*w - 5],\ [131, 131, 2*w^4 - 2*w^3 - 6*w^2 + 3*w + 2],\ [131, 131, w^4 - 2*w^3 - 3*w^2 + 7*w],\ [131, 131, 2*w^4 + w^3 - 8*w^2 - 3*w + 4],\ [197, 197, 3*w^4 - w^3 - 10*w^2 + w + 5],\ [197, 197, 2*w^4 - 7*w^2 + w + 1],\ [197, 197, -3*w^4 + 2*w^3 + 10*w^2 - 5*w - 5],\ [197, 197, -2*w^4 - w^3 + 9*w^2 + 2*w - 5],\ [197, 197, -3*w^4 + 3*w^3 + 10*w^2 - 6*w - 5],\ [199, 199, w^4 + 2*w^3 - 5*w^2 - 5*w + 3],\ [199, 199, -2*w^4 + w^3 + 7*w^2 - 4*w - 2],\ [199, 199, -w^4 + 2*w^3 + 4*w^2 - 7*w - 3],\ [199, 199, 2*w^4 - 8*w^2 + w + 4],\ [199, 199, 3*w^4 - w^3 - 10*w^2 + 2*w + 4],\ [241, 241, w^4 - 2*w^3 - w^2 + 3*w - 3],\ [241, 241, -w^4 + 3*w^3 + 2*w^2 - 9*w],\ [241, 241, 3*w^4 - 11*w^2 - 2*w + 6],\ [241, 241, -w^4 + 2*w^3 + 5*w^2 - 6*w - 4],\ [241, 241, -2*w^4 + 2*w^3 + 9*w^2 - 5*w - 6],\ [243, 3, -3],\ [263, 263, -w^4 + 4*w^2 + w + 1],\ [263, 263, 2*w^4 - w^3 - 8*w^2 + w + 2],\ [263, 263, 3*w^4 - 3*w^3 - 9*w^2 + 7*w + 2],\ [263, 263, -2*w^3 + w^2 + 4*w - 4],\ [263, 263, -3*w^2 + w + 5],\ [307, 307, 2*w^4 - w^3 - 6*w^2 + 3*w - 2],\ [307, 307, w^4 - 2*w^2 - 2*w - 4],\ [307, 307, -3*w^4 + 2*w^3 + 11*w^2 - 3*w - 6],\ [307, 307, -2*w^4 + 2*w^3 + 7*w^2 - 6*w - 6],\ [307, 307, -w^4 + 3*w^3 + 4*w^2 - 7*w - 3],\ [331, 331, -w^4 - 2*w^3 + 3*w^2 + 7*w - 1],\ [331, 331, -w^4 + 3*w^3 + 4*w^2 - 8*w - 4],\ [331, 331, w^4 - w^2 - 4],\ [331, 331, 3*w^4 - 2*w^3 - 9*w^2 + 5*w + 2],\ [331, 331, -3*w^4 + 2*w^3 + 11*w^2 - 6*w - 5],\ [353, 353, -2*w^4 + w^3 + 9*w^2 - 3*w - 5],\ [353, 353, -2*w^4 - w^3 + 7*w^2 + 4*w - 4],\ [353, 353, -w^4 + 3*w^3 + 3*w^2 - 9*w - 3],\ [353, 353, -3*w^4 + 11*w^2 + w - 7],\ [353, 353, w^4 - w^3 - 6*w^2 + 3*w + 4],\ [373, 373, w^3 + 2*w^2 - 5*w - 3],\ [373, 373, 2*w^4 - 3*w^3 - 6*w^2 + 6*w],\ [373, 373, -w^4 + w^3 + 3*w^2 - 3*w + 3],\ [373, 373, -2*w^4 + 2*w^3 + 8*w^2 - 7*w - 5],\ [373, 373, -w^4 + w^3 + 4*w^2 - 3*w - 6],\ [397, 397, -w^4 - w^3 + 6*w^2 + 3*w - 5],\ [397, 397, 2*w^4 - 2*w^3 - 7*w^2 + 3*w + 3],\ [397, 397, -w^4 + 5*w^2 + 2*w - 5],\ [397, 397, w^4 + 2*w^3 - 5*w^2 - 5*w + 4],\ [397, 397, -w^4 + 3*w^3 + 3*w^2 - 7*w - 2],\ [419, 419, 3*w^3 - w^2 - 8*w],\ [419, 419, 2*w^4 - 3*w^3 - 6*w^2 + 6*w + 1],\ [419, 419, w^4 - 2*w^3 - 5*w^2 + 7*w + 4],\ [419, 419, -2*w^4 - w^3 + 9*w^2 + 3*w - 7],\ [419, 419, -w^4 + 3*w^3 + w^2 - 8*w + 1],\ [439, 439, -w^4 + w^3 + 5*w^2 - 5*w - 5],\ [439, 439, 2*w^4 - 9*w^2 - 2*w + 8],\ [439, 439, -3*w^4 + 2*w^3 + 11*w^2 - 5*w - 3],\ [439, 439, 2*w^4 - 2*w^3 - 8*w^2 + 3*w + 4],\ [439, 439, -w^4 + 3*w^3 + 3*w^2 - 6*w - 3],\ [461, 461, -w^4 + 5*w^2 + 2*w - 6],\ [461, 461, -w^4 + 3*w^2 - w + 3],\ [461, 461, -w^4 - 2*w^3 + 5*w^2 + 5*w - 5],\ [461, 461, -2*w^4 + 7*w^2 + w - 6],\ [461, 461, -w^4 + 3*w^3 + 3*w^2 - 7*w - 3],\ [463, 463, 2*w^4 - 2*w^3 - 8*w^2 + 5*w + 3],\ [463, 463, w^4 + w^3 - 3*w^2 - 2*w - 2],\ [463, 463, 2*w^4 - 7*w^2 - 2*w + 5],\ [463, 463, w^4 - 2*w^3 - 2*w^2 + 6*w + 1],\ [463, 463, -w^3 + 2*w^2 + w - 5],\ [571, 571, w^4 - w^3 - 3*w^2 + 2*w - 3],\ [571, 571, 3*w^4 - 2*w^3 - 10*w^2 + 6*w + 2],\ [571, 571, -2*w^4 + 2*w^3 + 8*w^2 - 7*w - 6],\ [571, 571, 2*w^4 - 5*w^2 - 2*w - 2],\ [571, 571, 3*w^4 - w^3 - 9*w^2 + 2*w + 3],\ [593, 593, 3*w^4 - w^3 - 9*w^2 + w + 3],\ [593, 593, -w^4 + 2*w^3 + 5*w^2 - 7*w - 3],\ [593, 593, -2*w^4 + 2*w^3 + 7*w^2 - 7*w - 4],\ [593, 593, w^4 - w^2 - 2*w - 4],\ [593, 593, 3*w^4 - w^3 - 10*w^2 + 3*w + 2],\ [617, 617, -w^4 - w^3 + 6*w^2 + 4*w - 5],\ [617, 617, 3*w^3 - w^2 - 7*w],\ [617, 617, 2*w^4 - w^3 - 8*w^2 + 6],\ [617, 617, 2*w^4 - 3*w^3 - 7*w^2 + 6*w + 3],\ [617, 617, -w^4 - 2*w^3 + 6*w^2 + 5*w - 6],\ [659, 659, -w^4 + w^3 + 4*w^2 - 5*w - 4],\ [659, 659, 2*w^3 + w^2 - 7*w],\ [659, 659, -3*w^4 + 2*w^3 + 9*w^2 - 4*w - 4],\ [659, 659, 3*w^4 - w^3 - 11*w^2 + 3*w + 3],\ [659, 659, w^4 - w^2 - w - 5],\ [661, 661, -5*w^4 + 2*w^3 + 18*w^2 - 4*w - 9],\ [661, 661, -w^4 + 3*w^3 + 6*w^2 - 9*w - 8],\ [661, 661, w^4 + 2*w^3 - 2*w^2 - 7*w - 2],\ [661, 661, -w^4 + w^3 + 5*w^2 - 2*w - 9],\ [661, 661, -3*w^4 + 2*w^3 + 12*w^2 - 2*w - 8],\ [683, 683, -w^4 + w^3 + 4*w^2 - 5*w - 3],\ [683, 683, 2*w^3 + w^2 - 7*w - 1],\ [683, 683, w^4 - w^2 - w - 4],\ [683, 683, 3*w^4 - 2*w^3 - 9*w^2 + 4*w + 3],\ [683, 683, 3*w^4 - w^3 - 11*w^2 + 3*w + 4],\ [727, 727, -5*w^4 + 3*w^3 + 19*w^2 - 7*w - 9],\ [727, 727, -2*w^4 - 2*w^3 + 7*w^2 + 9*w - 4],\ [727, 727, 3*w^4 + 2*w^3 - 13*w^2 - 6*w + 6],\ [727, 727, w^4 - 2*w^2 + 3*w - 4],\ [727, 727, -2*w^4 + 5*w^3 + 6*w^2 - 14*w - 4],\ [769, 769, -2*w^4 - w^3 + 9*w^2 + 2*w - 7],\ [769, 769, -w^4 + 4*w^2 + 3*w - 4],\ [769, 769, -w^3 + 3*w^2 + 3*w - 4],\ [769, 769, 3*w^4 - 3*w^3 - 10*w^2 + 6*w + 3],\ [769, 769, 3*w^3 - 8*w - 2],\ [857, 857, w^4 - 5*w^3 - 2*w^2 + 14*w - 1],\ [857, 857, 4*w^4 - w^3 - 13*w^2 + 4],\ [857, 857, -2*w^4 + 4*w^3 + 6*w^2 - 10*w - 5],\ [857, 857, 4*w^4 + w^3 - 14*w^2 - 5*w + 6],\ [857, 857, 4*w^4 + w^3 - 16*w^2 - 4*w + 8],\ [859, 859, 3*w^4 - w^3 - 11*w^2 + 5*w + 5],\ [859, 859, -2*w^4 + 5*w^3 + 6*w^2 - 13*w - 1],\ [859, 859, -2*w^4 - 2*w^3 + 5*w^2 + 7*w + 1],\ [859, 859, -w^3 + w^2 + 6*w + 1],\ [859, 859, 4*w^4 - w^3 - 13*w^2 + 3*w + 4],\ [881, 881, 2*w^4 + w^3 - 11*w^2 - 3*w + 11],\ [881, 881, w^4 - 6*w^2 - 3*w + 5],\ [881, 881, -2*w^4 + 5*w^3 + 6*w^2 - 12*w - 1],\ [881, 881, -2*w^4 - 3*w^3 + 9*w^2 + 8*w - 4],\ [881, 881, 3*w^4 + w^3 - 12*w^2 - 3*w + 7],\ [947, 947, 2*w^4 + 3*w^3 - 7*w^2 - 8*w],\ [947, 947, 2*w^4 - 3*w^3 - 6*w^2 + 4*w],\ [947, 947, 2*w^4 + 2*w^3 - 7*w^2 - 8*w + 4],\ [947, 947, -w^4 + 2*w^3 - 7*w + 3],\ [947, 947, -4*w^4 + 5*w^3 + 14*w^2 - 12*w - 5],\ [967, 967, 2*w^4 - w^3 - 9*w^2 + 3*w + 4],\ [967, 967, -2*w^4 - w^3 + 7*w^2 + 4*w - 5],\ [967, 967, w^4 - 2*w^2 + w - 4],\ [967, 967, w^4 - w^3 - 2*w^2 - 3],\ [967, 967, -2*w^4 + 3*w^3 + 6*w^2 - 8*w - 4],\ [991, 991, -w^4 + 4*w^3 + 2*w^2 - 12*w - 1],\ [991, 991, w^4 + 3*w^3 - 3*w^2 - 11*w - 1],\ [991, 991, 2*w^3 - 2*w^2 - 5*w + 8],\ [991, 991, -2*w^4 + 3*w^3 + 4*w^2 - 5*w + 3],\ [991, 991, 4*w^4 - 4*w^3 - 11*w^2 + 9*w]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x K = QQ e = 1 hecke_eigenvalues_array = [5, -4, -4, -4, 3, 3, -9, 1, -5, -12, 9, 9, -2, 12, 5, -2, -2, -8, 6, -1, 13, -1, -2, -2, -2, 19, -16, 6, -1, -15, 20, -8, -12, 2, 16, 2, -12, -24, 4, 25, 4, 4, 18, -3, -10, -10, -31, 20, 19, -23, -2, 12, -23, -7, -7, -21, -28, 28, 17, 24, -18, 17, -32, -10, 32, -3, -24, -24, -4, -11, -18, 10, -11, -8, 13, -22, -8, -15, 0, 0, 35, -14, -28, -8, -8, 20, -15, -22, 0, 21, 14, -28, 21, 30, 16, 2, -5, -19, -2, -30, 19, -16, -37, 20, -29, 34, 34, -43, 2, -47, -33, -12, -26, -12, -26, 9, -12, 30, 18, -3, 32, -31, -17, 19, 12, 19, -37, -2, 28, 21, -28, -42, 14, -21, 35, 35, 14, 42, -24, -24, -3, -24, -10, -50, -15, -8, -15, 13, 42, 21, -42, 42, 56, 3, -25, -4, -39, 24, 37, -26, -12, 23, 16, 47, 40, 33, 19, -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([43, 43, -2*w^4 + w^3 + 6*w^2 - 2*w - 1])] = -1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]