/* 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([3, 0, -5, 0, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([13, 13, w^3 - 4*w + 2]) primes_array = [ [3, 3, w],\ [4, 2, w^2 - w - 1],\ [9, 3, -w^2 + 2],\ [13, 13, w^3 - 4*w + 2],\ [13, 13, -w^3 + 4*w + 2],\ [17, 17, w^3 - 3*w - 1],\ [17, 17, -w^3 + 3*w - 1],\ [29, 29, w^3 + w^2 - 4*w - 2],\ [29, 29, w^3 - w^2 - 4*w + 2],\ [43, 43, w^2 + w - 4],\ [43, 43, w^2 - w - 4],\ [53, 53, w^3 - 2*w - 2],\ [53, 53, -w^3 + 2*w - 2],\ [79, 79, w^3 - w^2 - 4*w + 1],\ [79, 79, -w^3 - w^2 + 4*w + 1],\ [101, 101, -w^3 + w^2 + 3*w - 5],\ [101, 101, w^3 + w^2 - 3*w - 5],\ [103, 103, 2*w^2 - w - 4],\ [103, 103, 2*w^2 + w - 4],\ [107, 107, 2*w^2 + w - 5],\ [107, 107, 3*w^2 - 3*w - 4],\ [107, 107, 2*w^2 - 2*w - 1],\ [107, 107, 2*w^2 - w - 5],\ [113, 113, w^3 - w^2 - 5*w + 1],\ [113, 113, -w^3 - w^2 + 5*w + 1],\ [121, 11, w^3 - 2*w^2 - 3*w + 2],\ [121, 11, -2*w^3 + w^2 + 7*w - 4],\ [127, 127, -2*w^3 + 7*w - 2],\ [127, 127, 2*w^3 - 7*w - 2],\ [131, 131, w^3 - w^2 - 1],\ [131, 131, -w^3 + w^2 + 4*w + 1],\ [131, 131, -2*w^3 + 3*w^2 + 4*w - 2],\ [131, 131, -2*w^3 + w^2 + 4*w + 2],\ [139, 139, w^2 - 2*w - 4],\ [139, 139, w^2 + 2*w - 4],\ [173, 173, 2*w^2 - 3*w - 5],\ [173, 173, 2*w^2 + 3*w - 5],\ [179, 179, -w^2 - 3*w + 4],\ [179, 179, 2*w^3 - w^2 - 9*w + 2],\ [179, 179, -2*w^3 - w^2 + 9*w + 2],\ [179, 179, w^2 - 3*w - 4],\ [199, 199, -3*w^3 + w^2 + 12*w - 4],\ [199, 199, -3*w^3 - w^2 + 12*w + 4],\ [211, 211, 2*w^3 - 7*w + 1],\ [211, 211, -2*w^3 + 7*w + 1],\ [233, 233, -w^2 - w - 2],\ [233, 233, -w^2 + w - 2],\ [257, 257, -3*w^2 - w + 10],\ [257, 257, 3*w^2 - w - 10],\ [263, 263, -w^3 + w^2 + 6*w + 1],\ [263, 263, w^2 + 2*w - 5],\ [263, 263, w^2 - 2*w - 5],\ [263, 263, w^3 + w^2 - 6*w + 1],\ [269, 269, -2*w^3 - w^2 + 6*w + 1],\ [269, 269, 2*w^3 - w^2 - 6*w + 1],\ [277, 277, w^3 - w^2 - 2],\ [277, 277, -w^3 + 2*w^2 + 3*w - 8],\ [277, 277, w^3 + 2*w^2 - 3*w - 8],\ [277, 277, -w^3 - w^2 - 2],\ [283, 283, -2*w^3 + w^2 + 8*w - 1],\ [283, 283, 2*w^3 + w^2 - 8*w - 1],\ [289, 17, w^2 - 7],\ [313, 313, w^3 + 2*w^2 - 4*w - 2],\ [313, 313, 2*w^3 - w^2 - 6*w + 2],\ [313, 313, -2*w^3 - w^2 + 6*w + 2],\ [313, 313, -w^3 + 2*w^2 + 4*w - 2],\ [337, 337, -w^3 - 2*w^2 + 7*w + 2],\ [337, 337, -2*w^3 + 4*w + 1],\ [337, 337, 2*w^3 - 4*w + 1],\ [337, 337, -w^3 + 2*w^2 + 7*w - 2],\ [367, 367, -w^3 + 3*w^2 - 2*w - 5],\ [367, 367, 2*w^3 - 3*w^2 - 2*w + 4],\ [373, 373, -2*w^3 + 10*w + 1],\ [373, 373, -3*w^3 - w^2 + 14*w + 5],\ [373, 373, 3*w^3 - w^2 - 14*w + 5],\ [373, 373, 2*w^3 - 10*w + 1],\ [389, 389, 2*w^3 - 6*w - 1],\ [389, 389, -2*w^3 + 6*w - 1],\ [419, 419, 2*w^3 - 11*w - 4],\ [419, 419, 2*w^2 - 2*w - 7],\ [419, 419, 2*w^2 + 2*w - 7],\ [419, 419, -2*w^3 + 11*w - 4],\ [439, 439, 3*w^2 - w - 14],\ [439, 439, -3*w^2 - w + 14],\ [443, 443, w^3 + 3*w^2 - 4*w - 10],\ [443, 443, 3*w^3 + w^2 - 14*w - 4],\ [443, 443, 3*w^3 - w^2 - 14*w + 4],\ [443, 443, -w^3 + 3*w^2 + 4*w - 10],\ [503, 503, -w - 5],\ [503, 503, 3*w^3 + w^2 - 12*w - 1],\ [503, 503, 3*w^3 - w^2 - 12*w + 1],\ [503, 503, w - 5],\ [521, 521, w^3 + 2*w^2 - 5*w - 2],\ [521, 521, -w^3 + 2*w^2 + 5*w - 2],\ [523, 523, 3*w^2 - 2*w - 10],\ [523, 523, w^3 + 3*w^2 - 6*w - 11],\ [529, 23, 2*w^2 - 11],\ [529, 23, 3*w^2 - 10],\ [547, 547, -2*w^3 + w^2 + 6*w - 7],\ [547, 547, 2*w^3 + w^2 - 6*w - 7],\ [563, 563, -2*w^3 + w^2 + 4*w - 4],\ [563, 563, -3*w^3 + 10*w - 4],\ [563, 563, 3*w^3 - 10*w - 4],\ [563, 563, 2*w^3 + w^2 - 4*w - 4],\ [569, 569, w^3 + 2*w^2 - 6*w - 10],\ [569, 569, -w^3 + 2*w^2 + 6*w - 10],\ [571, 571, -2*w^3 + 3*w^2 + 5*w - 2],\ [571, 571, -2*w^3 + 2*w^2 + 6*w + 1],\ [599, 599, 3*w^3 - 11*w - 2],\ [599, 599, 2*w^3 + w^2 - 6*w - 8],\ [599, 599, 2*w^3 - w^2 - 6*w + 8],\ [599, 599, 3*w^3 - 11*w + 2],\ [601, 601, -w^3 - 2*w^2 + 6*w + 2],\ [601, 601, 4*w^3 - w^2 - 16*w + 5],\ [601, 601, 4*w^3 + w^2 - 16*w - 5],\ [601, 601, w^3 - 2*w^2 - 6*w + 2],\ [607, 607, -w^3 + w^2 + 4*w - 8],\ [607, 607, -w^3 - w^2 + 4*w + 8],\ [625, 5, -5],\ [641, 641, w^2 + 4*w - 1],\ [641, 641, w^2 - 4*w - 1],\ [647, 647, w^3 + 3*w^2 - 4*w - 8],\ [647, 647, -w^3 + 7*w + 2],\ [647, 647, w^3 - 7*w + 2],\ [647, 647, -w^3 + 3*w^2 + 4*w - 8],\ [653, 653, 2*w^3 - 3*w^2 - 12*w + 10],\ [653, 653, 2*w^3 + 3*w^2 - 12*w - 10],\ [677, 677, 3*w^2 + w - 8],\ [677, 677, 3*w^2 - w - 8],\ [701, 701, 2*w^3 - 2*w^2 - 7*w + 2],\ [701, 701, -2*w^3 - 2*w^2 + 7*w + 2],\ [727, 727, -2*w^3 - 2*w^2 + 11*w + 7],\ [727, 727, -2*w^3 + 2*w^2 + 11*w - 7],\ [751, 751, -4*w^2 + 3*w + 7],\ [751, 751, w^3 - 2*w^2 - w - 2],\ [797, 797, 3*w^3 - 11*w + 1],\ [797, 797, 3*w^3 - 11*w - 1],\ [809, 809, -w^3 + 7*w - 1],\ [809, 809, w^3 - 7*w - 1],\ [823, 823, -2*w^3 + 4*w^2 + 9*w - 14],\ [823, 823, 2*w^3 + 4*w^2 - 9*w - 14],\ [841, 29, 3*w^2 - 7],\ [857, 857, w^3 + w^2 - w - 5],\ [857, 857, -w^3 + w^2 + w - 5],\ [859, 859, 5*w^3 + w^2 - 20*w - 2],\ [859, 859, -5*w^3 + w^2 + 20*w - 2],\ [881, 881, w^3 + 3*w^2 - 2*w - 10],\ [881, 881, -w^3 + 3*w^2 + 2*w - 10],\ [883, 883, 2*w^3 - w^2 - 8*w - 2],\ [883, 883, 2*w^3 + w^2 - 8*w + 2],\ [907, 907, 3*w^3 - 2*w^2 - 13*w + 5],\ [907, 907, 3*w^3 + 2*w^2 - 13*w - 5],\ [919, 919, 3*w^3 - w^2 - 15*w + 1],\ [919, 919, 3*w^3 + 3*w^2 - 10*w - 5],\ [953, 953, w^3 + 2*w^2 - 2*w - 8],\ [953, 953, -w^3 + 2*w^2 + 2*w - 8],\ [991, 991, -w^3 - w^2 + 4*w - 4],\ [991, 991, w^3 - w^2 - 4*w - 4],\ [997, 997, w^3 + 4*w^2 - 3*w - 16],\ [997, 997, -3*w^3 + 3*w^2 + 11*w - 13],\ [997, 997, 3*w^3 + 3*w^2 - 11*w - 13],\ [997, 997, w^3 - 4*w^2 - 3*w + 16]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^3 - x^2 - 4*x + 3 K. = NumberField(heckePol) hecke_eigenvalues_array = [e, e + 1, e - 1, -1, -e + 4, 2*e^2 + e - 4, -e^2 - 3*e + 4, -e^2 + 4, -e^2 + 4, -3*e, -2*e^2 + 4*e + 4, -4*e^2 - 3*e + 18, -2*e - 1, -4*e^2 - 3*e + 9, 4*e^2 - 2*e - 11, -6*e^2 + 2*e + 15, 4*e^2 - 4*e - 9, -3*e^2 + 6*e + 9, 3*e^2 - 14, -4*e^2 - 5*e + 17, -4*e^2 - 5*e + 17, 4*e^2 - 4*e - 3, -2*e^2 + 3*e + 2, e^2 - 3*e + 5, -4*e^2 + 17, -7*e^2 + 5*e + 18, 2*e^2 + 3*e - 13, -2*e^2 + 7*e + 6, 2, 3*e^2 - 5*e - 8, -4*e^2 - e + 18, 5*e^2 - 3*e - 13, -7*e^2 + e + 16, -e^2 + 10*e + 4, -5*e^2 - 5*e + 16, -5*e^2 - 3*e + 21, 6*e^2 + 3*e - 25, 3*e^2 - 8, 6*e + 9, -3*e^2 + 2*e + 17, 7*e^2 - 9, -5*e^2 - 3*e + 19, 3*e^2 - 2*e - 1, -7*e^2 + 7*e + 23, 3*e^2 + e - 1, 5*e^2 - 9*e - 13, 5*e^2 - 9*e - 13, 4*e^2 + 3*e - 14, -4*e^2 + 2*e + 6, -e^2 - e - 17, -e^2 + e - 11, e^2 - 7*e + 3, -2*e^2 + e + 12, 2*e^2 + 7*e - 7, -4*e^2 - 9, 11*e^2 - 5*e - 25, -2*e^2 + 12*e + 10, 6*e^2 - e - 17, -10*e + 3, 4*e^2 + 14*e - 18, 2*e^2 + 7*e - 21, -6*e^2 + 11*e + 20, -3*e^2 - e + 2, 11*e^2 + 3*e - 42, 2*e^2 + 4*e + 7, 2*e + 12, 2*e^2 - 4*e, -9*e^2 + e + 37, 5*e^2 - 4*e - 6, -10*e^2 - 6*e + 39, -2*e^2 + e - 24, 6*e^2 + e - 26, 12*e^2 + 5*e - 41, -6*e^2 - 6*e + 32, -11*e^2 - 2*e + 26, 11*e^2 - 7*e - 19, 8*e^2 - 8*e - 14, 7*e^2 - 3*e - 39, 4*e + 5, -4*e^2 + 3*e + 10, 4*e^2 + 5*e - 28, 13*e^2 + 3*e - 45, e^2 - 3*e - 26, -e^2 + e + 32, 6*e^2 - 9, -11*e^2 - 5*e + 45, -9*e^2 + 4*e + 12, 14*e^2 + e - 29, 4*e^2 - 9*e - 28, 7*e^2 - 17*e - 30, -8*e^2 + 6*e + 13, 3*e^2 - 7*e + 1, 7*e^2 - 10*e - 12, -10*e^2 + 7*e + 10, -7*e^2 + 2*e + 5, 13*e^2 + 4*e - 36, -9*e^2 + 2*e + 15, -10*e^2 - 2*e + 19, e^2 - 2*e - 13, 14*e^2 - 4*e - 45, 9*e^2 - 13*e - 32, 6*e^2 - 5*e - 9, -2*e^2 + 8*e + 18, e^2 - e + 13, 14*e^2 - 2*e - 31, 7*e^2 + 9*e - 33, -6*e^2 - 7*e + 17, -4*e^2 + 15*e + 9, -12*e^2 + 29, -15*e^2 + 9*e + 27, 5*e^2 - 5*e + 15, 14*e^2 - 18*e - 42, 19*e^2 - 6*e - 35, -16*e^2 - 5*e + 52, -2*e - 6, 3*e + 2, 10*e^2 - 5*e - 43, 5*e^2 + 12*e - 24, -4*e^2 - 7*e + 4, 16*e^2 + e - 38, 17*e^2 - 2*e - 49, 11*e^2 - 6*e - 15, -2*e^2 + e + 25, 7*e^2 - e - 6, -2*e^2 + 12*e - 12, e^2 - 2*e - 26, 12*e^2 - 13*e - 25, 2*e^2 + e - 44, -e^2 - 5*e, 6*e^2 - 6*e, 13*e^2 - e - 27, -14*e^2 - 8*e + 46, 4*e^2 - 13*e + 2, -11*e^2 - 5*e + 38, -12*e^2 - e + 31, -7*e^2 - 4*e + 58, 2*e^2 - 2*e - 45, -7*e^2 + 19*e + 27, 4*e^2 + 9*e + 10, 18*e^2 - 13*e - 56, -9*e^2 + 8*e + 26, 9*e^2 - 10*e - 4, 9*e^2 + e - 32, 2*e^2 - 18*e - 12, -6*e^2 - 12*e + 52, -e^2 + 2*e - 7, 11*e^2 + 2*e - 16, 23*e^2 - 10*e - 62, 5*e^2 - 10*e - 17, 10*e^2 + 16*e - 33, -11*e^2 + 22*e + 44, 10*e^2 - 7*e - 22, 12*e^2 - 3*e - 9, -4*e^2 + 11*e + 2, 2*e^2 - 23*e + 6, -3*e^2 - 5*e + 7, -3*e^2 + 18*e + 2, 7*e^2 - 4*e + 7, e^2 + 3*e + 18, e + 28, -18*e^2 + 8*e + 36, -10*e^2 + 10] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([13, 13, w^3 - 4*w + 2])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]