/* 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, -4, -8, -1, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([9, 3, w^3 - 2*w^2 - 5*w]) primes_array = [ [3, 3, w],\ [3, 3, w^3 - 2*w^2 - 6*w + 1],\ [9, 3, w^3 - 2*w^2 - 5*w + 1],\ [16, 2, 2],\ [17, 17, w^3 - w^2 - 8*w - 5],\ [17, 17, -2*w^3 + 4*w^2 + 11*w - 2],\ [17, 17, -w^3 + 3*w^2 + 2*w - 2],\ [17, 17, w^3 - 2*w^2 - 4*w + 1],\ [25, 5, w^3 - 3*w^2 - 3*w + 1],\ [25, 5, w^2 - 2*w - 7],\ [29, 29, w^3 - 2*w^2 - 6*w - 1],\ [29, 29, w - 2],\ [53, 53, w^3 - 3*w^2 - 4*w + 4],\ [53, 53, -w^3 + 3*w^2 + 4*w - 5],\ [61, 61, -w^3 + 2*w^2 + 7*w - 4],\ [61, 61, -4*w^3 + 11*w^2 + 14*w - 8],\ [101, 101, -w^3 + 2*w^2 + 7*w - 1],\ [103, 103, -w^3 + 3*w^2 + 2*w - 5],\ [103, 103, -w^3 + w^2 + 8*w + 2],\ [113, 113, w^3 - 3*w^2 - 3*w + 7],\ [113, 113, w^2 - 2*w - 1],\ [127, 127, w^3 - 2*w^2 - 4*w - 2],\ [127, 127, -2*w^3 + 4*w^2 + 11*w + 1],\ [131, 131, -3*w^3 + 7*w^2 + 14*w - 8],\ [131, 131, -w^2 + w + 8],\ [131, 131, 2*w^3 - 5*w^2 - 9*w + 1],\ [131, 131, -3*w^3 + 6*w^2 + 16*w + 1],\ [139, 139, -2*w^3 + 5*w^2 + 9*w - 2],\ [139, 139, -w^2 + w + 7],\ [157, 157, -2*w^3 + 5*w^2 + 7*w - 5],\ [157, 157, -2*w^3 + 3*w^2 + 13*w + 2],\ [157, 157, -2*w^3 + 5*w^2 + 6*w - 2],\ [157, 157, -3*w^3 + 5*w^2 + 19*w + 4],\ [169, 13, -2*w^3 + 4*w^2 + 10*w - 1],\ [173, 173, 3*w^3 - 8*w^2 - 10*w + 7],\ [173, 173, 2*w^3 - 2*w^2 - 15*w - 8],\ [179, 179, -3*w^2 + 6*w + 17],\ [179, 179, -2*w^3 + 3*w^2 + 12*w + 1],\ [179, 179, -3*w^3 + 7*w^2 + 13*w - 7],\ [179, 179, -2*w^3 + 3*w^2 + 14*w + 4],\ [191, 191, w^3 - 3*w^2 - w + 4],\ [191, 191, -2*w^3 + 3*w^2 + 14*w + 2],\ [199, 199, -w^3 + 8*w + 10],\ [199, 199, 5*w^3 - 15*w^2 - 14*w + 14],\ [211, 211, -w - 4],\ [211, 211, -w^3 + 2*w^2 + 6*w - 5],\ [211, 211, 2*w^3 - 3*w^2 - 12*w - 4],\ [211, 211, -3*w^3 + 7*w^2 + 13*w - 4],\ [257, 257, -3*w^3 + 8*w^2 + 10*w - 2],\ [257, 257, 2*w^3 - 2*w^2 - 16*w - 7],\ [263, 263, 3*w^3 - 6*w^2 - 17*w + 1],\ [263, 263, w^3 - 2*w^2 - 3*w - 1],\ [269, 269, 5*w^3 - 13*w^2 - 19*w + 7],\ [269, 269, -3*w^3 + 6*w^2 + 18*w - 1],\ [277, 277, w^3 - 4*w^2 - w + 5],\ [277, 277, -w^3 + 4*w^2 + w - 11],\ [283, 283, -w^3 + 5*w^2 - w - 14],\ [283, 283, 2*w^3 - 7*w^2 - 4*w + 10],\ [283, 283, -w^3 + w^2 + 9*w + 1],\ [283, 283, w^2 - 4*w - 5],\ [311, 311, -w^3 + w^2 + 9*w + 5],\ [311, 311, w^2 - 4*w - 1],\ [313, 313, -4*w^3 + 9*w^2 + 19*w - 10],\ [313, 313, -2*w^2 + 5*w + 11],\ [337, 337, -3*w^3 + 6*w^2 + 14*w + 2],\ [337, 337, -4*w^3 + 8*w^2 + 21*w + 1],\ [347, 347, -2*w^3 + 6*w^2 + 7*w - 10],\ [347, 347, -w^3 + 4*w^2 + 2*w - 7],\ [361, 19, -4*w^3 + 11*w^2 + 13*w - 10],\ [361, 19, -2*w^3 + 5*w^2 + 8*w - 10],\ [373, 373, 2*w^3 - 5*w^2 - 10*w + 2],\ [373, 373, -w^3 + 2*w^2 + 8*w - 2],\ [373, 373, -2*w^3 + 4*w^2 + 13*w - 1],\ [373, 373, -w^3 + 3*w^2 + 5*w - 8],\ [389, 389, -w^3 + 4*w^2 + 2*w - 10],\ [389, 389, -2*w^3 + 6*w^2 + 7*w - 7],\ [419, 419, -2*w^3 + 5*w^2 + 6*w - 1],\ [419, 419, -3*w^3 + 5*w^2 + 19*w + 5],\ [433, 433, w^2 - 4*w - 2],\ [433, 433, w^3 - w^2 - 9*w - 4],\ [439, 439, -w^3 + 4*w^2 + w - 2],\ [439, 439, w^3 - 4*w^2 - w + 14],\ [443, 443, -2*w^3 + 5*w^2 + 9*w + 1],\ [443, 443, -w^2 + w + 10],\ [467, 467, w^3 - w^2 - 8*w + 4],\ [467, 467, w^3 - 4*w^2 + 2],\ [491, 491, -4*w^3 + 12*w^2 + 10*w - 11],\ [491, 491, 2*w^3 - w^2 - 18*w - 11],\ [491, 491, 3*w^3 - 9*w^2 - 7*w + 11],\ [491, 491, 2*w^3 - 20*w - 19],\ [503, 503, w^3 - 3*w^2 - 4*w + 10],\ [503, 503, -w^3 + w^2 + 10*w - 2],\ [523, 523, -4*w^3 + 9*w^2 + 22*w - 7],\ [523, 523, w^3 - 3*w^2 - 7*w + 5],\ [529, 23, -2*w^3 + 4*w^2 + 10*w + 5],\ [529, 23, -2*w^3 + 4*w^2 + 10*w - 7],\ [547, 547, 2*w^3 - w^2 - 18*w - 14],\ [547, 547, 3*w^3 - 8*w^2 - 14*w + 8],\ [571, 571, 5*w^3 - 11*w^2 - 25*w + 5],\ [571, 571, -4*w^3 + 9*w^2 + 19*w - 11],\ [599, 599, -5*w^3 + 12*w^2 + 20*w - 4],\ [599, 599, 4*w^3 - 6*w^2 - 25*w - 11],\ [601, 601, -4*w^3 + 9*w^2 + 20*w - 5],\ [601, 601, w^3 - 9*w - 14],\ [601, 601, 3*w^3 - 8*w^2 - 11*w + 2],\ [601, 601, -2*w^3 + 3*w^2 + 14*w - 1],\ [641, 641, -2*w^3 + 3*w^2 + 15*w - 5],\ [641, 641, -4*w^3 + 8*w^2 + 23*w - 2],\ [647, 647, -3*w^3 + 8*w^2 + 13*w - 11],\ [647, 647, -w^3 + 4*w^2 + 3*w - 7],\ [659, 659, 3*w^3 - 7*w^2 - 14*w - 2],\ [659, 659, w^3 - w^2 - 6*w - 11],\ [673, 673, -3*w^3 + 7*w^2 + 11*w - 1],\ [673, 673, -4*w^3 + 7*w^2 + 24*w + 5],\ [677, 677, 5*w^3 - 12*w^2 - 21*w + 11],\ [677, 677, -2*w^3 + 6*w^2 + 5*w - 1],\ [677, 677, -2*w^3 + 6*w^2 + 8*w - 7],\ [677, 677, 2*w^3 - 6*w^2 - 8*w + 11],\ [719, 719, -2*w^3 + 6*w^2 + 10*w - 7],\ [719, 719, -4*w^3 + 10*w^2 + 20*w - 13],\ [727, 727, 3*w^3 - 10*w^2 - 8*w + 10],\ [727, 727, 3*w^3 - 8*w^2 - 13*w + 10],\ [727, 727, 2*w^3 - 8*w^2 - 3*w + 23],\ [727, 727, -w^3 + 4*w^2 + 3*w - 8],\ [751, 751, -7*w^3 + 18*w^2 + 25*w - 7],\ [751, 751, w^3 - 7*w^2 + 5*w + 26],\ [757, 757, -4*w^3 + 9*w^2 + 18*w - 7],\ [757, 757, -3*w^3 + 5*w^2 + 17*w + 1],\ [797, 797, 2*w^2 - 4*w - 5],\ [797, 797, 2*w^3 - 6*w^2 - 6*w + 11],\ [823, 823, -2*w^3 + 2*w^2 + 15*w + 14],\ [823, 823, -3*w^3 + 8*w^2 + 10*w - 1],\ [829, 829, -6*w^3 + 16*w^2 + 22*w - 7],\ [829, 829, 5*w^3 - 13*w^2 - 20*w + 8],\ [829, 829, -w^3 + 7*w + 4],\ [829, 829, -5*w^3 + 12*w^2 + 23*w - 14],\ [841, 29, -3*w^3 + 6*w^2 + 15*w - 2],\ [859, 859, -4*w^3 + 7*w^2 + 24*w - 1],\ [859, 859, -5*w^3 + 15*w^2 + 14*w - 13],\ [881, 881, 3*w^3 - 7*w^2 - 16*w + 4],\ [881, 881, -w^3 + 3*w^2 + 6*w - 7],\ [883, 883, -2*w^3 + 3*w^2 + 12*w + 10],\ [883, 883, -3*w^3 + 7*w^2 + 13*w + 2],\ [887, 887, 4*w^3 - 11*w^2 - 11*w + 11],\ [887, 887, 2*w^3 - 7*w^2 - 3*w + 7],\ [887, 887, 4*w^3 - 5*w^2 - 29*w - 10],\ [887, 887, 3*w^2 - 7*w - 16],\ [907, 907, -w^3 + 3*w^2 + 8*w - 5],\ [907, 907, -5*w^3 + 11*w^2 + 28*w - 8],\ [911, 911, -4*w^3 + 10*w^2 + 18*w - 17],\ [911, 911, 2*w^2 - 2*w - 1],\ [919, 919, 4*w^3 - 7*w^2 - 25*w - 4],\ [919, 919, 2*w^3 - 5*w^2 - 5*w + 1],\ [937, 937, -w^3 + 3*w^2 + w - 8],\ [937, 937, -2*w^3 + 3*w^2 + 14*w - 2],\ [953, 953, -2*w^3 + w^2 + 17*w + 19],\ [953, 953, 4*w^3 - 11*w^2 - 13*w + 4],\ [961, 31, 2*w^3 - 3*w^2 - 11*w - 8],\ [961, 31, w^3 - 4*w^2 + 2*w + 11],\ [971, 971, 2*w^2 - 5*w - 4],\ [971, 971, -w^3 + 4*w^2 - 11],\ [991, 991, -2*w^3 + 6*w^2 + 9*w - 8],\ [991, 991, -2*w^3 + 3*w^2 + 13*w - 1],\ [991, 991, 3*w^3 - 8*w^2 - 14*w + 11],\ [991, 991, -2*w^3 + 5*w^2 + 7*w - 8],\ [997, 997, -2*w^3 + 5*w^2 + 6*w + 1],\ [997, 997, -3*w^3 + 5*w^2 + 19*w + 7],\ [997, 997, -6*w^3 + 12*w^2 + 33*w - 4],\ [997, 997, 4*w^3 - 11*w^2 - 15*w + 11]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^4 - 11*x^3 + 29*x^2 + 8*x - 8 K. = NumberField(heckePol) hecke_eigenvalues_array = [1, 1, e, 1/2*e^3 - 3/2*e^2 - 15/2*e + 3, -e^2 + 6*e, -1/2*e^3 + 7/2*e^2 - 7/2*e, -e^2 + 6*e, -1/2*e^3 + 7/2*e^2 - 7/2*e, -e^3 + 7*e^2 - 8*e, -e^3 + 7*e^2 - 8*e, 1/2*e^3 - 5/2*e^2 - 3/2*e - 2, 1/2*e^3 - 5/2*e^2 - 3/2*e - 2, e^3 - 6*e^2 - e + 12, e^3 - 6*e^2 - e + 12, e^3 - 6*e^2 + e, e^3 - 6*e^2 + e, e^3 - 11*e^2 + 25*e + 10, 4*e^2 - 24*e + 2, 4*e^2 - 24*e + 2, 2*e^3 - 14*e^2 + 15*e + 2, 2*e^3 - 14*e^2 + 15*e + 2, 1/2*e^3 - 15/2*e^2 + 45/2*e + 8, 1/2*e^3 - 15/2*e^2 + 45/2*e + 8, e^3 - 5*e^2 - e - 10, -e^3 + 3*e^2 + 13*e + 6, -e^3 + 3*e^2 + 13*e + 6, e^3 - 5*e^2 - e - 10, -e^3 + 3*e^2 + 15*e - 6, -e^3 + 3*e^2 + 15*e - 6, -2*e^2 + 13*e - 4, -2*e^2 + 13*e - 4, e^3 - 8*e^2 + 13*e + 2, e^3 - 8*e^2 + 13*e + 2, 3*e^3 - 19*e^2 + 9*e + 24, -e^3 + 4*e^2 + 7*e + 8, -e^3 + 4*e^2 + 7*e + 8, -3*e^3 + 15*e^2 + 13*e - 16, 2*e^3 - 14*e^2 + 16*e, 2*e^3 - 14*e^2 + 16*e, -3*e^3 + 15*e^2 + 13*e - 16, 1/2*e^3 + 5/2*e^2 - 59/2*e - 2, 1/2*e^3 + 5/2*e^2 - 59/2*e - 2, 2*e^3 - 16*e^2 + 26*e + 4, 2*e^3 - 16*e^2 + 26*e + 4, -1/2*e^3 - 13/2*e^2 + 103/2*e + 6, -1/2*e^3 - 13/2*e^2 + 103/2*e + 6, -2*e^3 + 14*e^2 - 18*e + 6, -2*e^3 + 14*e^2 - 18*e + 6, -e^3 + 6*e^2 - 5*e + 12, -e^3 + 6*e^2 - 5*e + 12, -e^3 + 13*e^2 - 37*e - 14, -e^3 + 13*e^2 - 37*e - 14, -e^3 + 7*e^2 - 6*e - 8, -e^3 + 7*e^2 - 6*e - 8, -3*e^3 + 19*e^2 - 14*e + 10, -3*e^3 + 19*e^2 - 14*e + 10, e^3 - 9*e^2 + 19*e - 6, e^3 - 9*e^2 + 19*e - 6, 5/2*e^3 - 43/2*e^2 + 85/2*e + 2, 5/2*e^3 - 43/2*e^2 + 85/2*e + 2, -3/2*e^3 + 25/2*e^2 - 27/2*e - 28, -3/2*e^3 + 25/2*e^2 - 27/2*e - 28, 1/2*e^3 - 7/2*e^2 - 13/2*e + 26, 1/2*e^3 - 7/2*e^2 - 13/2*e + 26, -3/2*e^3 + 33/2*e^2 - 93/2*e - 14, -3/2*e^3 + 33/2*e^2 - 93/2*e - 14, -2*e^3 + 12*e^2 - 12, -2*e^3 + 12*e^2 - 12, -2*e^3 + 14*e^2 - 13*e - 10, -2*e^3 + 14*e^2 - 13*e - 10, -7/2*e^3 + 57/2*e^2 - 87/2*e - 20, -1/2*e^3 + 19/2*e^2 - 77/2*e - 10, -1/2*e^3 + 19/2*e^2 - 77/2*e - 10, -7/2*e^3 + 57/2*e^2 - 87/2*e - 20, -4*e^2 + 27*e - 2, -4*e^2 + 27*e - 2, 2*e^3 - 10*e^2 - 6*e + 2, 2*e^3 - 10*e^2 - 6*e + 2, e^3 - 6*e^2 - e - 14, e^3 - 6*e^2 - e - 14, -2*e^3 + 12*e^2 - 10*e + 6, -2*e^3 + 12*e^2 - 10*e + 6, -2*e^3 + 12*e^2 - 2*e - 12, -2*e^3 + 12*e^2 - 2*e - 12, 5/2*e^3 - 31/2*e^2 + 29/2*e - 18, 5/2*e^3 - 31/2*e^2 + 29/2*e - 18, 3/2*e^3 - 29/2*e^2 + 67/2*e + 20, -8*e^2 + 42*e + 2, -8*e^2 + 42*e + 2, 3/2*e^3 - 29/2*e^2 + 67/2*e + 20, -e^3 + 3*e^2 + 9*e + 12, -e^3 + 3*e^2 + 9*e + 12, 3*e^3 - 9*e^2 - 39*e - 26, 3*e^3 - 9*e^2 - 39*e - 26, e^3 - 4*e^2 - 11*e + 40, -3*e^2 + 10*e + 34, 4*e^3 - 32*e^2 + 48*e + 22, 4*e^3 - 32*e^2 + 48*e + 22, -7/2*e^3 + 25/2*e^2 + 85/2*e - 10, -7/2*e^3 + 25/2*e^2 + 85/2*e - 10, -3*e^3 + 23*e^2 - 33*e + 6, -3*e^3 + 23*e^2 - 33*e + 6, -5/2*e^3 + 45/2*e^2 - 85/2*e - 28, 1/2*e^3 + 11/2*e^2 - 79/2*e - 24, 1/2*e^3 + 11/2*e^2 - 79/2*e - 24, -5/2*e^3 + 45/2*e^2 - 85/2*e - 28, 4*e^3 - 20*e^2 - 24*e + 42, 4*e^3 - 20*e^2 - 24*e + 42, 2*e^3 - 16*e^2 + 22*e + 28, 2*e^3 - 16*e^2 + 22*e + 28, -5*e^3 + 33*e^2 - 25*e, -5*e^3 + 33*e^2 - 25*e, 5*e^3 - 28*e^2 - 3*e + 12, 5*e^3 - 28*e^2 - 3*e + 12, -e^3 + 27*e + 16, -e^3 + 27*e + 16, e^3 - e^2 - 22*e - 2, e^3 - e^2 - 22*e - 2, -2*e^3 + 16*e^2 - 24*e - 32, -2*e^3 + 16*e^2 - 24*e - 32, 4*e^3 - 36*e^2 + 66*e + 16, 7/2*e^3 - 45/2*e^2 + 27/2*e + 10, 4*e^3 - 36*e^2 + 66*e + 16, 7/2*e^3 - 45/2*e^2 + 27/2*e + 10, -4*e^3 + 24*e^2 - 26, -4*e^3 + 24*e^2 - 26, -2*e^3 + 7*e^2 + 24*e + 20, -2*e^3 + 7*e^2 + 24*e + 20, -11*e^2 + 68*e, -11*e^2 + 68*e, -3*e^3 + 5*e^2 + 63*e + 26, -3*e^3 + 5*e^2 + 63*e + 26, 5*e^3 - 29*e^2 + 6*e + 4, 5*e^3 - 29*e^2 + 6*e + 4, e^3 - 13*e^2 + 42*e + 14, e^3 - 13*e^2 + 42*e + 14, 3*e^3 - 23*e^2 + 33*e + 44, -1/2*e^3 + 7/2*e^2 - 1/2*e, -1/2*e^3 + 7/2*e^2 - 1/2*e, 2*e^3 - 10*e^2 - 9*e + 30, 2*e^3 - 10*e^2 - 9*e + 30, 7/2*e^3 - 41/2*e^2 + 3/2*e + 28, 7/2*e^3 - 41/2*e^2 + 3/2*e + 28, -5/2*e^3 + 43/2*e^2 - 69/2*e - 12, -3*e^3 + 19*e^2 - 5*e - 50, -5/2*e^3 + 43/2*e^2 - 69/2*e - 12, -3*e^3 + 19*e^2 - 5*e - 50, 3*e^3 - 7*e^2 - 57*e - 10, 3*e^3 - 7*e^2 - 57*e - 10, -1/2*e^3 + 11/2*e^2 - 45/2*e + 14, -1/2*e^3 + 11/2*e^2 - 45/2*e + 14, 3/2*e^3 - 29/2*e^2 + 59/2*e + 24, 3/2*e^3 - 29/2*e^2 + 59/2*e + 24, 2*e^3 - 8*e^2 - 23*e + 26, 2*e^3 - 8*e^2 - 23*e + 26, e^3 - 27*e - 8, e^3 - 27*e - 8, -e^3 + 2*e^2 + 15*e - 12, -e^3 + 2*e^2 + 15*e - 12, 1/2*e^3 - 7/2*e^2 + 37/2*e - 30, 1/2*e^3 - 7/2*e^2 + 37/2*e - 30, -e^3 + 17*e^2 - 65*e + 14, -3*e^3 + 25*e^2 - 41*e - 12, -e^3 + 17*e^2 - 65*e + 14, -3*e^3 + 25*e^2 - 41*e - 12, -4*e^3 + 29*e^2 - 40*e + 4, -4*e^3 + 29*e^2 - 40*e + 4, 6*e^3 - 46*e^2 + 69*e + 10, 6*e^3 - 46*e^2 + 69*e + 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([3, 3, w])] = -1 AL_eigenvalues[ZF.ideal([3, 3, w^3 - 2*w^2 - 6*w + 1])] = -1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]