/* 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([9, 3, -w^2 + 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^4 - 10*x^2 + 8 K. = NumberField(heckePol) hecke_eigenvalues_array = [e, -1/4*e^3 + 7/2*e, -1, -1/2*e^3 + 4*e, -1/2*e^3 + 4*e, 1/2*e^3 - 5*e, 1/2*e^3 - 5*e, -e^3 + 6*e, -e^3 + 6*e, -2*e^2 + 8, -2*e^2 + 8, 3*e^2 - 16, 3*e^2 - 16, -e^3 + 9*e, -e^3 + 9*e, -e^2 + 12, -e^2 + 12, 3*e^2 - 20, 3*e^2 - 20, -e^2 + 18, -1/2*e^3 + 9*e, -1/2*e^3 + 9*e, -e^2 + 18, -3/2*e^3 + 9*e, -3/2*e^3 + 9*e, -1/2*e^3 + 4*e, -1/2*e^3 + 4*e, 2*e^3 - 21*e, 2*e^3 - 21*e, -e^2 + 10, 1/2*e^3 - 9*e, 1/2*e^3 - 9*e, -e^2 + 10, -e^2 + 16, -e^2 + 16, -e^2 - 8, -e^2 - 8, -e^2 + 14, 7/2*e^3 - 27*e, 7/2*e^3 - 27*e, -e^2 + 14, e^3 - 11*e, e^3 - 11*e, 3*e^3 - 25*e, 3*e^3 - 25*e, 2*e^2 - 2, 2*e^2 - 2, 2*e^2 - 26, 2*e^2 - 26, -e^3 + 4*e, 6*e^2 - 28, 6*e^2 - 28, -e^3 + 4*e, 6*e, 6*e, -5*e^2 + 22, -e^2 - 14, -e^2 - 14, -5*e^2 + 22, e, e, e^2 + 22, 9/2*e^3 - 34*e, -9/2*e^3 + 38*e, -9/2*e^3 + 38*e, 9/2*e^3 - 34*e, -7/2*e^3 + 26*e, -3/2*e^3 + 16*e, -3/2*e^3 + 16*e, -7/2*e^3 + 26*e, 0, 0, -1/2*e^3 + 12*e, 9/2*e^3 - 40*e, 9/2*e^3 - 40*e, -1/2*e^3 + 12*e, e^3 - 8*e, e^3 - 8*e, -9/2*e^3 + 47*e, e^2 + 2, e^2 + 2, -9/2*e^3 + 47*e, e^2 - 12, e^2 - 12, -3*e^2 + 34, 3/2*e^3 - 5*e, 3/2*e^3 - 5*e, -3*e^2 + 34, 8, 2*e^3 - 6*e, 2*e^3 - 6*e, 8, 9/2*e^3 - 47*e, 9/2*e^3 - 47*e, 3*e^2 - 28, 3*e^2 - 28, 42, 7*e^2 - 18, 3*e^2 - 36, 3*e^2 - 36, -e^2 + 10, 11/2*e^3 - 49*e, 11/2*e^3 - 49*e, -e^2 + 10, -4*e^2 - 6, -4*e^2 - 6, e^3 - 19*e, e^3 - 19*e, 2*e^3 - 22*e, -10*e^2 + 44, -10*e^2 + 44, 2*e^3 - 22*e, 1/2*e^3 - 8*e, 11/2*e^3 - 36*e, 11/2*e^3 - 36*e, 1/2*e^3 - 8*e, -7*e^2 + 36, -7*e^2 + 36, 14, 3/2*e^3 - 9*e, 3/2*e^3 - 9*e, 6*e^2 - 32, 3*e^3 - 40*e, 3*e^3 - 40*e, 6*e^2 - 32, e^3 - 22*e, e^3 - 22*e, -5*e^2 + 44, -5*e^2 + 44, e^3, e^3, 6*e^3 - 41*e, 6*e^3 - 41*e, 5*e^2 - 20, 5*e^2 - 20, -e^3 - 4*e, -e^3 - 4*e, -13/2*e^3 + 53*e, -13/2*e^3 + 53*e, -4*e^2, -4*e^2, -e^2 + 58, -8*e^2 + 34, -8*e^2 + 34, e^3 + 3*e, e^3 + 3*e, 8*e^2 - 54, 8*e^2 - 54, 6*e^3 - 37*e, 6*e^3 - 37*e, 3*e^3 - 13*e, 3*e^3 - 13*e, -2*e^3 + 9*e, -2*e^3 + 9*e, -2*e^2 - 14, -2*e^2 - 14, -4*e^2 + 16, -4*e^2 + 16, e^2 - 30, 4*e^2 + 14, 4*e^2 + 14, e^2 - 30] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([9, 3, -w^2 + 2])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]