/* 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([9, 6, -7, -2, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([7, 7, 1/3*w^3 - 2/3*w^2 - 7/3*w + 1]) primes_array = [ [4, 2, -1/3*w^3 + 2/3*w^2 + 4/3*w - 1],\ [7, 7, 1/3*w^3 - 2/3*w^2 - 7/3*w + 1],\ [7, 7, w - 2],\ [7, 7, w - 1],\ [9, 3, w],\ [9, 3, -1/3*w^3 + 2/3*w^2 + 7/3*w - 2],\ [23, 23, -1/3*w^3 + 2/3*w^2 + 1/3*w - 2],\ [23, 23, -2/3*w^3 + 4/3*w^2 + 11/3*w - 4],\ [31, 31, -2/3*w^3 + 1/3*w^2 + 14/3*w + 2],\ [41, 41, -1/3*w^3 + 5/3*w^2 + 4/3*w - 7],\ [41, 41, -1/3*w^3 + 5/3*w^2 + 4/3*w - 4],\ [47, 47, 1/3*w^3 - 5/3*w^2 - 1/3*w + 5],\ [47, 47, w^2 - w - 4],\ [73, 73, -w^3 + 3*w^2 + 4*w - 8],\ [73, 73, -2/3*w^3 + 7/3*w^2 - 1/3*w - 4],\ [73, 73, -w^3 + w^2 + 7*w + 1],\ [73, 73, 2/3*w^3 - 4/3*w^2 - 14/3*w + 5],\ [79, 79, w^2 - 5],\ [79, 79, -2/3*w^3 + 7/3*w^2 + 8/3*w - 6],\ [97, 97, -2/3*w^3 + 7/3*w^2 + 5/3*w - 4],\ [97, 97, 1/3*w^3 + 1/3*w^2 - 7/3*w - 5],\ [103, 103, w^3 - 3*w^2 - 2*w + 5],\ [103, 103, -4/3*w^3 + 11/3*w^2 + 13/3*w - 10],\ [113, 113, -1/3*w^3 + 5/3*w^2 + 4/3*w - 3],\ [113, 113, w^3 - 2*w^2 - 5*w + 1],\ [113, 113, -1/3*w^3 + 5/3*w^2 + 4/3*w - 1],\ [113, 113, -1/3*w^3 + 5/3*w^2 + 4/3*w - 10],\ [127, 127, -2/3*w^3 + 1/3*w^2 + 14/3*w + 1],\ [127, 127, -2/3*w^3 + 7/3*w^2 + 2/3*w - 6],\ [137, 137, -1/3*w^3 + 5/3*w^2 - 2/3*w - 6],\ [137, 137, 1/3*w^3 + 1/3*w^2 - 10/3*w - 1],\ [151, 151, -w^3 + 2*w^2 + 3*w - 5],\ [151, 151, 1/3*w^3 + 1/3*w^2 - 10/3*w - 7],\ [169, 13, -2/3*w^3 + 7/3*w^2 + 5/3*w - 2],\ [169, 13, 1/3*w^3 + 1/3*w^2 - 7/3*w - 7],\ [191, 191, 1/3*w^3 - 2/3*w^2 + 2/3*w - 2],\ [191, 191, 2*w - 5],\ [191, 191, -w^3 + 2*w^2 + 6*w - 2],\ [191, 191, 2/3*w^3 - 4/3*w^2 - 14/3*w - 1],\ [239, 239, 1/3*w^3 - 2/3*w^2 - 7/3*w - 3],\ [239, 239, w - 5],\ [241, 241, 1/3*w^3 + 1/3*w^2 - 10/3*w],\ [241, 241, 4/3*w^3 - 17/3*w^2 + 2/3*w + 9],\ [241, 241, -1/3*w^3 + 5/3*w^2 - 2/3*w - 7],\ [241, 241, -4/3*w^3 + 14/3*w^2 + 13/3*w - 14],\ [257, 257, -w - 4],\ [257, 257, -1/3*w^3 + 2/3*w^2 + 7/3*w - 6],\ [263, 263, -4/3*w^3 + 11/3*w^2 + 16/3*w - 8],\ [263, 263, 2/3*w^3 - 1/3*w^2 - 8/3*w - 3],\ [281, 281, -1/3*w^3 - 1/3*w^2 + 10/3*w - 2],\ [281, 281, -1/3*w^3 + 5/3*w^2 - 5/3*w - 5],\ [281, 281, -1/3*w^3 + 5/3*w^2 - 2/3*w - 9],\ [281, 281, -2/3*w^3 + 1/3*w^2 + 17/3*w],\ [289, 17, -w^3 + 2*w^2 + 4*w - 4],\ [289, 17, w^3 - 2*w^2 - 4*w + 2],\ [311, 311, -2/3*w^3 + 10/3*w^2 - 4/3*w - 9],\ [311, 311, -2/3*w^3 - 2/3*w^2 + 20/3*w + 5],\ [313, 313, -1/3*w^3 + 5/3*w^2 - 2/3*w - 8],\ [313, 313, -1/3*w^3 - 1/3*w^2 + 10/3*w - 1],\ [313, 313, w^3 - 4*w^2 - 2*w + 10],\ [313, 313, 1/3*w^3 + 4/3*w^2 - 10/3*w - 8],\ [337, 337, -4/3*w^3 + 5/3*w^2 + 19/3*w + 4],\ [337, 337, 5/3*w^3 - 13/3*w^2 - 17/3*w + 5],\ [353, 353, -4/3*w^3 + 8/3*w^2 + 25/3*w - 3],\ [353, 353, -1/3*w^3 - 1/3*w^2 + 16/3*w + 6],\ [353, 353, -2/3*w^3 + 10/3*w^2 + 2/3*w - 13],\ [353, 353, -5/3*w^3 + 7/3*w^2 + 29/3*w - 3],\ [367, 367, -5/3*w^3 + 4/3*w^2 + 35/3*w + 4],\ [367, 367, 2/3*w^3 - 7/3*w^2 - 8/3*w + 4],\ [367, 367, -4/3*w^3 + 14/3*w^2 + 1/3*w - 8],\ [367, 367, w^2 - 7],\ [383, 383, 4/3*w^3 - 14/3*w^2 - 10/3*w + 15],\ [383, 383, -5/3*w^3 + 13/3*w^2 + 20/3*w - 12],\ [383, 383, 2*w^2 - 2*w - 13],\ [383, 383, w^3 - w^2 - 8*w - 1],\ [401, 401, -4/3*w^3 + 11/3*w^2 + 22/3*w - 11],\ [401, 401, 5/3*w^3 - 10/3*w^2 - 29/3*w + 5],\ [409, 409, -w^3 + 2*w^2 + 3*w - 2],\ [409, 409, -1/3*w^3 + 8/3*w^2 - 2/3*w - 6],\ [409, 409, 1/3*w^3 - 8/3*w^2 + 2/3*w + 12],\ [409, 409, 4/3*w^3 - 8/3*w^2 - 19/3*w + 4],\ [433, 433, -w^3 + 3*w^2 - 1],\ [433, 433, -1/3*w^3 + 5/3*w^2 - 8/3*w - 5],\ [433, 433, -w^3 + w^2 + 8*w - 2],\ [433, 433, 5/3*w^3 - 7/3*w^2 - 32/3*w - 2],\ [439, 439, w^2 - 10],\ [439, 439, -2/3*w^3 + 4/3*w^2 + 17/3*w - 7],\ [449, 449, -2/3*w^3 + 4/3*w^2 + 11/3*w - 8],\ [449, 449, -4/3*w^3 + 5/3*w^2 + 22/3*w + 2],\ [457, 457, 2/3*w^3 - 7/3*w^2 - 14/3*w + 6],\ [457, 457, 2/3*w^3 - 7/3*w^2 - 14/3*w + 9],\ [463, 463, w^3 - 3*w^2 - 6*w + 11],\ [463, 463, -1/3*w^3 + 5/3*w^2 + 10/3*w - 4],\ [479, 479, -1/3*w^3 + 5/3*w^2 - 2/3*w + 1],\ [479, 479, 5/3*w^3 - 10/3*w^2 - 23/3*w + 9],\ [487, 487, -2/3*w^3 + 7/3*w^2 + 2/3*w - 8],\ [487, 487, -2/3*w^3 + 1/3*w^2 + 14/3*w - 1],\ [503, 503, w^3 - 4*w^2 - 2*w + 16],\ [503, 503, -1/3*w^3 + 5/3*w^2 - 2/3*w - 11],\ [521, 521, -4/3*w^3 + 11/3*w^2 + 10/3*w - 7],\ [521, 521, 4/3*w^3 - 5/3*w^2 - 22/3*w],\ [529, 23, -1/3*w^3 + 2/3*w^2 + 4/3*w - 6],\ [569, 569, -1/3*w^3 + 8/3*w^2 + 1/3*w - 10],\ [569, 569, -2/3*w^3 + 10/3*w^2 + 5/3*w - 10],\ [577, 577, -2/3*w^3 + 10/3*w^2 + 5/3*w - 12],\ [577, 577, -1/3*w^3 + 8/3*w^2 + 1/3*w - 8],\ [593, 593, w^3 - w^2 - 5*w - 5],\ [593, 593, 4/3*w^3 - 11/3*w^2 - 13/3*w + 4],\ [599, 599, 2*w^2 - 3*w - 7],\ [599, 599, 2/3*w^3 - 1/3*w^2 - 8/3*w - 4],\ [599, 599, -1/3*w^3 + 8/3*w^2 - 5/3*w - 9],\ [599, 599, -4/3*w^3 + 11/3*w^2 + 16/3*w - 7],\ [607, 607, 4/3*w^3 - 8/3*w^2 - 22/3*w + 9],\ [607, 607, -2/3*w^3 + 4/3*w^2 + 2/3*w - 5],\ [625, 5, -5],\ [631, 631, -1/3*w^3 + 2/3*w^2 + 13/3*w - 5],\ [631, 631, -2/3*w^3 + 4/3*w^2 + 17/3*w - 1],\ [641, 641, -2/3*w^3 + 7/3*w^2 + 14/3*w - 7],\ [641, 641, 2/3*w^3 - 7/3*w^2 - 14/3*w + 8],\ [647, 647, -2/3*w^3 - 5/3*w^2 + 26/3*w + 8],\ [647, 647, -1/3*w^3 + 8/3*w^2 + 1/3*w - 9],\ [647, 647, -2/3*w^3 + 10/3*w^2 + 5/3*w - 11],\ [647, 647, -2/3*w^3 + 13/3*w^2 - 10/3*w - 13],\ [673, 673, -4/3*w^3 + 5/3*w^2 + 25/3*w - 2],\ [673, 673, -w^3 + 3*w^2 + w - 7],\ [727, 727, -2/3*w^3 + 7/3*w^2 + 2/3*w - 11],\ [727, 727, -2/3*w^3 + 1/3*w^2 + 14/3*w - 4],\ [743, 743, w^2 - w - 10],\ [743, 743, 1/3*w^3 - 5/3*w^2 - 1/3*w - 1],\ [769, 769, -w^3 + 2*w^2 + 7*w - 4],\ [769, 769, 3*w - 2],\ [809, 809, -1/3*w^3 + 8/3*w^2 - 2/3*w - 8],\ [809, 809, -1/3*w^3 + 8/3*w^2 - 2/3*w - 10],\ [823, 823, -w^3 + 2*w^2 + 7*w - 5],\ [823, 823, 3*w - 1],\ [839, 839, 4/3*w^3 - 2/3*w^2 - 25/3*w - 8],\ [839, 839, 2/3*w^3 - 7/3*w^2 - 14/3*w + 12],\ [857, 857, -1/3*w^3 + 5/3*w^2 - 5/3*w - 7],\ [857, 857, -2/3*w^3 + 1/3*w^2 + 17/3*w - 2],\ [863, 863, -2/3*w^3 + 7/3*w^2 - 7/3*w - 2],\ [863, 863, 5/3*w^3 - 7/3*w^2 - 35/3*w + 1],\ [881, 881, -2/3*w^3 + 1/3*w^2 + 14/3*w + 8],\ [881, 881, -2/3*w^3 + 13/3*w^2 - 4/3*w - 18],\ [881, 881, -5/3*w^3 + 7/3*w^2 + 35/3*w + 3],\ [881, 881, -2/3*w^3 + 7/3*w^2 + 2/3*w + 1],\ [887, 887, -1/3*w^3 + 11/3*w^2 - 8/3*w - 7],\ [887, 887, -1/3*w^3 + 11/3*w^2 - 8/3*w - 18],\ [911, 911, 5/3*w^3 - 4/3*w^2 - 29/3*w - 3],\ [911, 911, 2*w^3 - 6*w^2 - 5*w + 13],\ [929, 929, 2*w^3 - 4*w^2 - 9*w + 2],\ [929, 929, 5/3*w^3 - 10/3*w^2 - 17/3*w],\ [929, 929, -4/3*w^3 + 14/3*w^2 + 13/3*w - 11],\ [929, 929, 1/3*w^3 + 4/3*w^2 - 7/3*w - 9],\ [937, 937, -w^3 + 4*w^2 + w - 13],\ [937, 937, 2/3*w^3 + 2/3*w^2 - 17/3*w - 3],\ [953, 953, -2*w^3 + w^2 + 15*w + 8],\ [953, 953, -5/3*w^3 + 19/3*w^2 - 1/3*w - 11],\ [961, 31, 4/3*w^3 - 8/3*w^2 - 16/3*w + 5],\ [967, 967, -4/3*w^3 + 2/3*w^2 + 31/3*w + 11],\ [967, 967, 7/3*w^3 - 5/3*w^2 - 46/3*w - 9],\ [977, 977, 2*w^3 - 5*w^2 - 8*w + 10],\ [977, 977, -w^3 + 3*w^2 + 2*w - 14],\ [977, 977, -4/3*w^3 + 8/3*w^2 + 25/3*w - 11],\ [977, 977, -w^3 + w^2 + 6*w - 7],\ [983, 983, -5/3*w^3 + 16/3*w^2 + 8/3*w - 12],\ [983, 983, -1/3*w^3 + 11/3*w^2 - 11/3*w - 15],\ [983, 983, 3*w^2 - 2*w - 16],\ [983, 983, -5/3*w^3 + 16/3*w^2 + 11/3*w - 14],\ [991, 991, 4/3*w^3 - 8/3*w^2 - 22/3*w + 1],\ [991, 991, 2/3*w^3 - 4/3*w^2 - 2/3*w - 3]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^3 + 2*x^2 - 6*x - 11 K. = NumberField(heckePol) hecke_eigenvalues_array = [e, -1, -e^2 + 7, -e^2 + e + 8, -e^2 - e + 6, 2*e^2 + e - 9, -e^2 + e + 6, 2*e^2 - e - 11, -2*e^2 - e + 5, -2*e^2 + 4, 3*e^2 + e - 12, 4*e^2 - 20, -2*e^2 + 12, -3*e - 1, -2*e^2 - 3*e + 5, 6, -3*e^2 - 5*e + 14, e^2 - 3*e - 4, -e^2 - 2*e - 3, e^2 + 4*e - 7, 4*e^2 - 2*e - 26, -3*e^2 - 4*e + 15, 3*e^2 - 15, -2*e^2 + 22, 2*e^2 + e - 9, e^2 + 5*e - 4, -3*e^2 + 4*e + 27, 3*e^2 + 4*e - 23, e^2 - 2*e + 5, -5*e^2 + 4*e + 39, -7*e^2 + e + 38, 5*e^2 - 5*e - 38, 3*e^2 - 3*e - 6, 3*e^2 + 3*e - 2, -3*e^2 - e + 28, -7*e^2 - e + 42, e^2 - 3*e + 2, -e^2 - e + 10, 3*e^2 + 3, e^2 - 5*e - 6, e^2 - 5*e - 6, -4*e^2 - 6*e + 20, 5*e^2 + 3*e - 26, -2*e^2 - 8, e^2 - 2*e + 3, 5*e^2 + 3*e - 20, 2*e^2 - 36, -2*e^2 + 2*e + 8, -7*e^2 + 6*e + 57, 7*e^2 - 5*e - 42, -13*e^2 - 3*e + 64, 4*e + 10, 13*e^2 - 63, -3*e^2 + 7*e + 16, -5*e^2 - e + 22, 8*e^2 - 4*e - 60, e^2 + 5*e - 8, 2*e^2 - 10*e - 18, -4*e^2 - 6*e + 16, 4*e^2 - 4*e - 6, 4*e^2 + 7*e - 31, 7*e^2 - 10*e - 47, e^2 + 2*e - 9, 6*e^2 + 3*e - 15, 4*e^2 - 5*e - 23, 7*e^2 - 2*e - 33, -e^2 + 2*e - 3, e^2 - 6*e - 23, -7*e^2 - 2*e + 47, 6*e^2 + 11*e - 31, -5*e^2 + 4*e + 19, 7*e^2 - e - 52, -10*e^2 + 11*e + 65, e^2 + 7*e + 2, -3*e^2 + 5*e + 2, 4*e^2, 6*e^2 - 6*e - 34, 7*e^2 + e - 48, -11*e^2 - 7*e + 56, -e^2 - 10*e - 9, -9*e^2 + 47, 7*e^2 - 5*e - 30, 11*e^2 + 4*e - 53, -2*e^2 + e + 25, e^2 + 6*e - 23, 14*e^2 - 6*e - 86, -e^2 - 3, -15*e^2 + 3*e + 84, 7*e^2 + 8*e - 41, 6*e^2 - 5*e - 19, -5*e^2 - 10*e + 27, -2*e^2 + 6*e - 6, -2*e + 20, 7*e^2 - 27, 3*e^2 - 6*e - 29, -12*e^2 + 6*e + 74, e^2 - 7*e - 12, 9*e^2 - 8*e - 49, 3*e^2 + e + 18, -3*e^2 + 4*e + 5, 12*e^2 + 7*e - 43, 3*e^2 - 7*e - 32, 5*e^2 + 14*e - 31, 8*e^2 - 12*e - 60, -4*e^2 + 34, 3*e^2 - 17*e - 22, 10*e^2 + 7*e - 59, -10*e^2 - 8*e + 30, 2*e^2 - 10*e - 32, e^2 - 9*e - 6, 9*e^2 - 2*e - 45, e^2 - 9*e - 6, -4*e^2 + 4*e + 36, -6*e^2 + 6*e + 68, -10*e^2 - 4*e + 52, -11*e^2 + 10*e + 59, -3*e^2 + e + 22, e^2 - 11*e - 10, 7*e^2 + 2*e - 5, 3*e^2 - 7*e - 18, -e^2 + 5*e + 42, 5*e^2 + 4*e - 21, -3*e^2 + 5*e + 20, -2*e^2 + 6*e + 26, 3*e^2 + 10*e - 19, e^2 + 16*e - 5, 7*e^2 - 12*e - 51, 8*e^2 - 15*e - 59, 7*e + 15, -12*e^2 + 10*e + 94, -20*e^2 - 6*e + 88, 6*e^2 - 6*e - 68, -20*e^2 - 4*e + 92, 16*e^2 + 6*e - 64, -14*e^2 - 3*e + 61, -4*e^2 + 11*e + 41, -7*e^2 + 5*e + 54, 12*e^2 + 8*e - 70, e^2 + 3*e - 24, -7*e^2 + 2*e + 45, -9*e^2 + 3*e + 46, -6*e^2 - 4*e + 16, -17*e^2 - 7*e + 80, -11*e^2 - 3*e + 50, -5*e^2 + 4*e + 35, -3*e^2 + 11*e + 44, -12*e^2 + e + 87, 8*e^2 - 5*e - 33, 6*e^2 - 5*e - 63, -9*e^2 - 4*e + 3, -18*e^2 - 9*e + 79, -5*e^2 - 15*e + 32, 9*e^2 + 11*e - 50, 5*e - 9, -9*e^2 + 11*e + 42, -4*e^2 + 10*e + 16, 15*e^2 - 2*e - 71, 13*e^2 - 11*e - 74, 4*e^2 - 6*e - 30, 13*e^2 + 9*e - 40, 2*e^2 + 13*e - 31, 3*e^2 + e - 4, -11*e^2 - e + 72, 12*e^2 + 11*e - 47, -6*e^2 - 8*e + 38, 12*e^2 - 13*e - 83, 11*e - 7, 8*e^2 - 5*e - 25, -e^2 - 7*e - 10, -21*e^2 + 7*e + 124] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([7, 7, 1/3*w^3 - 2/3*w^2 - 7/3*w + 1])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]