/* 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([28, 14, w^2 - w - 5]) 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 K = QQ e = 1 hecke_eigenvalues_array = [-1, 1, -2, -1, -1, 2, -6, 3, 5, -6, -3, -3, 6, 14, -11, 10, 8, -2, -14, 13, -17, -14, -8, -12, 9, 6, -3, 16, -8, 15, 15, -8, 16, -16, -1, 0, 21, -15, 0, 24, 12, -10, -10, 17, -10, -18, -6, -30, 24, 18, 33, 24, 9, -29, 16, 6, -24, -14, 19, -8, -26, 8, -34, 30, -3, -30, 36, -5, -17, -32, -14, -15, 21, -12, -27, -30, -6, 22, 23, 5, -20, 4, 25, -38, -20, -14, -17, 30, 9, -26, -2, -34, -22, 27, 24, 38, -7, -36, -6, -18, -18, -26, 6, 18, -19, -13, -18, 36, 0, -12, -24, -33, 5, -7, 38, -5, 25, -45, 30, 3, -33, 18, 0, 23, -13, -40, -19, -6, -3, 2, 23, 24, -21, 14, -34, -33, -12, 42, 3, 9, -39, -15, -15, -42, -54, 0, 24, -36, 27, 0, -30, 30, 30, -22, -7, 12, 48, -4, -16, 5, -24, -6, 42, 54, -6, -15, 6, -6, 14, 56] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([4, 2, -1/3*w^3 + 2/3*w^2 + 4/3*w - 1])] = 1 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]