/* 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([-1, 4, 5, -4, -2, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([21, 21, w^4 - 3*w^3 - w^2 + 6*w - 3]) primes_array = [ [3, 3, w - 1],\ [5, 5, w^2 - w - 2],\ [7, 7, w^4 - 2*w^3 - 3*w^2 + 4*w + 2],\ [13, 13, w^3 - 2*w^2 - 2*w + 2],\ [29, 29, -w^4 + 3*w^3 + 2*w^2 - 7*w - 1],\ [29, 29, -w^2 + 2*w + 3],\ [31, 31, w^4 - 2*w^3 - 3*w^2 + 5*w],\ [31, 31, w^3 - 2*w^2 - 3*w + 2],\ [32, 2, 2],\ [43, 43, -w^2 - w + 4],\ [53, 53, -w^4 + w^3 + 6*w^2 - 2*w - 5],\ [53, 53, -w^4 + 2*w^3 + 4*w^2 - 5*w - 2],\ [73, 73, w^4 - w^3 - 6*w^2 + 2*w + 6],\ [73, 73, w^3 - w^2 - 4*w + 2],\ [81, 3, 2*w^4 - 5*w^3 - 4*w^2 + 9*w + 2],\ [83, 83, w^3 - w^2 - 5*w],\ [89, 89, -w^4 + 2*w^3 + 4*w^2 - 4*w - 2],\ [97, 97, w^4 - 3*w^3 - 2*w^2 + 6*w + 2],\ [101, 101, -w^4 + 3*w^3 + 3*w^2 - 7*w - 3],\ [103, 103, -w^4 + w^3 + 6*w^2 - w - 7],\ [103, 103, 2*w^4 - 4*w^3 - 6*w^2 + 8*w + 1],\ [103, 103, w^4 - w^3 - 6*w^2 + 3*w + 4],\ [107, 107, -w^4 + 2*w^3 + 3*w^2 - 5*w + 2],\ [109, 109, w^3 - 4*w - 1],\ [109, 109, -w^2 + 2*w + 4],\ [131, 131, w^4 - w^3 - 6*w^2 + w + 6],\ [137, 137, w^3 - w^2 - 5*w + 3],\ [149, 149, w^4 - 3*w^3 - 3*w^2 + 9*w + 6],\ [149, 149, -w^4 + 6*w^2 + 3*w],\ [163, 163, 2*w^2 - w - 5],\ [167, 167, -2*w^4 + 3*w^3 + 8*w^2 - 4*w - 3],\ [169, 13, w^4 - 3*w^3 + 6*w - 3],\ [169, 13, w^3 - 4*w - 4],\ [173, 173, w^4 - w^3 - 4*w^2 + 2*w + 1],\ [173, 173, -w^4 + 3*w^3 + 3*w^2 - 7*w - 5],\ [173, 173, w^4 - w^3 - 4*w^2 + w - 1],\ [179, 179, w^2 - 3*w - 2],\ [191, 191, -w^4 + 3*w^3 + w^2 - 6*w - 1],\ [193, 193, -w^3 + 2*w^2 + w - 4],\ [193, 193, -w^4 + 3*w^3 + 3*w^2 - 8*w - 1],\ [199, 199, w^4 - 3*w^3 - w^2 + 7*w],\ [199, 199, -2*w^4 + 5*w^3 + 5*w^2 - 10*w - 2],\ [199, 199, w^4 - 3*w^3 + 5*w - 4],\ [211, 211, 2*w^3 - 2*w^2 - 7*w - 1],\ [211, 211, -2*w^3 + 3*w^2 + 5*w - 2],\ [223, 223, -2*w^4 + 3*w^3 + 10*w^2 - 7*w - 9],\ [227, 227, w^3 - w^2 - 4*w + 3],\ [229, 229, w^4 - 4*w^3 + 10*w - 5],\ [229, 229, -2*w^4 + 4*w^3 + 7*w^2 - 9*w - 8],\ [239, 239, -2*w^4 + 3*w^3 + 7*w^2 - 3*w - 3],\ [241, 241, w^3 - w^2 - 2*w - 2],\ [257, 257, -w^4 + w^3 + 6*w^2 - 4*w - 4],\ [263, 263, 2*w^3 - 4*w^2 - 5*w + 6],\ [269, 269, w^3 - 2*w^2 - 4*w + 4],\ [269, 269, -2*w^4 + 2*w^3 + 11*w^2 - 3*w - 10],\ [271, 271, w^3 - 5*w],\ [281, 281, w^4 - w^3 - 5*w^2 + 3*w + 3],\ [281, 281, -2*w^4 + 4*w^3 + 6*w^2 - 6*w - 3],\ [281, 281, w^3 - 5*w - 1],\ [283, 283, -2*w^4 + 4*w^3 + 7*w^2 - 8*w - 3],\ [289, 17, -2*w^4 + 3*w^3 + 8*w^2 - 5*w - 2],\ [293, 293, 2*w^4 - 5*w^3 - 7*w^2 + 14*w + 9],\ [311, 311, -w^4 + 2*w^3 + 2*w^2 - 4*w + 3],\ [313, 313, w^3 - 6*w],\ [331, 331, w^4 - 2*w^3 - 2*w^2 + w - 2],\ [359, 359, w^4 - w^3 - 5*w^2 + 2*w - 1],\ [373, 373, -w^4 + 3*w^3 + 2*w^2 - 8*w - 3],\ [373, 373, 2*w^3 - 4*w^2 - 3*w + 3],\ [379, 379, -w^4 + 4*w^3 - w^2 - 9*w + 3],\ [379, 379, -w^4 + 3*w^3 + w^2 - 6*w - 2],\ [379, 379, w^4 + w^3 - 7*w^2 - 6*w + 3],\ [383, 383, -w^4 + 2*w^3 + 3*w^2 - 6*w - 2],\ [383, 383, w^4 - 5*w^2 - 3*w + 3],\ [383, 383, 2*w^4 - 5*w^3 - 3*w^2 + 8*w],\ [389, 389, 3*w^4 - 7*w^3 - 8*w^2 + 16*w + 1],\ [397, 397, -2*w^4 + 3*w^3 + 8*w^2 - 3*w - 7],\ [397, 397, -w^4 + 2*w^3 + 3*w^2 - 4*w - 5],\ [397, 397, -2*w^4 + 3*w^3 + 7*w^2 - 4*w - 3],\ [409, 409, -w^4 + w^3 + 7*w^2 - 5*w - 9],\ [419, 419, w^4 - 2*w^3 - w^2 + 2*w - 2],\ [419, 419, 2*w^4 - 3*w^3 - 7*w^2 + 5*w + 2],\ [419, 419, -w^4 + 4*w^3 - 9*w - 2],\ [433, 433, -w^4 + 3*w^3 - 8*w + 4],\ [433, 433, w^4 - 7*w^2 - w + 6],\ [439, 439, 3*w^4 - 6*w^3 - 10*w^2 + 11*w + 7],\ [449, 449, -2*w^4 + 5*w^3 + 5*w^2 - 11*w - 4],\ [457, 457, -2*w^4 + 5*w^3 + 4*w^2 - 10*w - 2],\ [463, 463, 2*w^4 - 5*w^3 - 4*w^2 + 8*w + 1],\ [467, 467, 2*w^4 - 4*w^3 - 6*w^2 + 6*w + 1],\ [479, 479, 3*w^4 - 4*w^3 - 12*w^2 + 4*w + 7],\ [479, 479, 2*w^4 - 6*w^3 - 4*w^2 + 14*w + 1],\ [491, 491, w^4 - 6*w^2 - w + 4],\ [503, 503, 2*w^4 - w^3 - 11*w^2 - 3*w + 8],\ [503, 503, 2*w^3 - 5*w^2 - 2*w + 6],\ [509, 509, w^4 - 5*w^2 - 6*w],\ [509, 509, w^4 - 3*w^3 - 3*w^2 + 9*w + 1],\ [523, 523, 3*w^4 - 7*w^3 - 7*w^2 + 12*w + 3],\ [523, 523, 3*w^4 - 8*w^3 - 7*w^2 + 17*w + 3],\ [523, 523, w^4 - 7*w^2 - 2*w + 4],\ [529, 23, -w^4 + w^3 + 7*w^2 - 2*w - 9],\ [541, 541, w^4 - 3*w^3 - 3*w^2 + 5*w + 2],\ [547, 547, -w^3 + 3*w^2 + 3*w - 4],\ [557, 557, 2*w^3 - 4*w^2 - 6*w + 9],\ [577, 577, 2*w^4 - 2*w^3 - 11*w^2 + 5*w + 10],\ [577, 577, w^4 - 4*w^3 - w^2 + 9*w],\ [577, 577, 2*w^4 - 3*w^3 - 9*w^2 + 4*w + 8],\ [577, 577, w^4 - 7*w^2 - 3*w + 7],\ [577, 577, -3*w^4 + 3*w^3 + 13*w^2 - w - 4],\ [593, 593, 3*w - 2],\ [601, 601, 2*w^4 - 4*w^3 - 7*w^2 + 5*w + 6],\ [601, 601, -w^4 + 3*w^3 + w^2 - 7*w - 1],\ [607, 607, w^3 - w^2 - 2*w - 3],\ [625, 5, 2*w^4 - 5*w^3 - 8*w^2 + 14*w + 11],\ [631, 631, -w^2 + w - 2],\ [631, 631, w^4 - 7*w^2 - w + 8],\ [643, 643, -2*w^4 + 4*w^3 + 9*w^2 - 10*w - 8],\ [647, 647, 3*w^4 - 6*w^3 - 10*w^2 + 12*w + 8],\ [653, 653, -3*w^4 + 6*w^3 + 11*w^2 - 12*w - 9],\ [653, 653, w^4 - w^3 - 2*w^2 - 3*w - 3],\ [653, 653, -2*w^4 + 4*w^3 + 7*w^2 - 6*w - 8],\ [701, 701, w^4 - w^3 - 2*w^2 - w - 5],\ [719, 719, 2*w^4 - 5*w^3 - 6*w^2 + 14*w + 2],\ [727, 727, 2*w^4 - 5*w^3 - 5*w^2 + 9*w + 3],\ [733, 733, 2*w^4 - 5*w^3 - 5*w^2 + 13*w + 3],\ [739, 739, -2*w^3 + 4*w^2 + 4*w - 5],\ [743, 743, w^4 - 2*w^3 - 3*w^2 + 6*w - 3],\ [743, 743, 2*w^4 - 2*w^3 - 11*w^2 + 2*w + 10],\ [757, 757, -w^4 + 2*w^3 + 5*w^2 - 6*w - 10],\ [757, 757, 2*w^4 - 6*w^3 - 2*w^2 + 11*w - 4],\ [761, 761, 3*w^3 - 4*w^2 - 9*w + 3],\ [769, 769, 3*w^3 - 5*w^2 - 9*w + 7],\ [769, 769, -2*w^4 + 4*w^3 + 7*w^2 - 11*w - 2],\ [773, 773, 2*w^4 - 5*w^3 - 6*w^2 + 12*w + 1],\ [773, 773, 3*w^4 - 4*w^3 - 12*w^2 + 3*w + 5],\ [773, 773, 2*w^3 - w^2 - 10*w - 4],\ [797, 797, -w^4 + 4*w^3 + 2*w^2 - 12*w - 3],\ [811, 811, w^4 - 2*w^3 - 6*w^2 + 8*w + 4],\ [823, 823, w^4 - 4*w^3 - w^2 + 10*w + 1],\ [823, 823, 2*w^4 - 4*w^3 - 4*w^2 + 4*w - 3],\ [827, 827, -w^4 + 2*w^3 + 2*w^2 - w + 3],\ [827, 827, 2*w^4 - 4*w^3 - 5*w^2 + 7*w + 2],\ [829, 829, -w^4 + 4*w^3 + 2*w^2 - 10*w - 2],\ [829, 829, 3*w^3 - 3*w^2 - 11*w - 2],\ [829, 829, w^4 - 2*w^3 - 3*w^2 + 2*w - 3],\ [839, 839, -w^4 + 5*w^3 - 3*w^2 - 10*w + 5],\ [839, 839, -w^4 + w^3 + 5*w^2 - 9],\ [853, 853, 2*w^4 - 6*w^3 - 5*w^2 + 15*w + 4],\ [859, 859, -3*w^4 + 8*w^3 + 6*w^2 - 18*w + 5],\ [863, 863, w^4 - w^3 - 6*w^2 + 5*w + 2],\ [877, 877, 3*w^3 - 4*w^2 - 8*w + 1],\ [881, 881, 3*w^4 - 7*w^3 - 9*w^2 + 15*w + 3],\ [881, 881, -w^3 + 7*w + 1],\ [883, 883, -2*w^4 + 6*w^3 + 3*w^2 - 15*w],\ [887, 887, w^4 - w^3 - 5*w^2 + w - 1],\ [907, 907, 3*w^4 - 5*w^3 - 11*w^2 + 9*w + 5],\ [919, 919, w^4 - 2*w^3 - 6*w^2 + 8*w + 7],\ [937, 937, 2*w^4 - 2*w^3 - 9*w^2 + 2*w + 3],\ [937, 937, -3*w^4 + 8*w^3 + 6*w^2 - 16*w + 1],\ [937, 937, 3*w^4 - 5*w^3 - 11*w^2 + 7*w + 4],\ [947, 947, w^3 - w^2 - 7*w + 2],\ [947, 947, -w^4 + 2*w^3 + 4*w^2 - 7*w - 3],\ [967, 967, -w^4 + 4*w^3 - 11*w],\ [971, 971, 3*w^4 - 6*w^3 - 8*w^2 + 10*w],\ [971, 971, w^4 - 8*w^2 + 2*w + 9],\ [971, 971, w^4 - 4*w^3 + 2*w^2 + 7*w - 4],\ [977, 977, -w^4 + 4*w^3 - 12*w + 5],\ [983, 983, -w^4 + w^3 + 4*w^2 + 3*w - 2],\ [983, 983, -2*w^4 + 3*w^3 + 8*w^2 - 6*w - 4],\ [997, 997, 2*w^4 - 4*w^3 - 5*w^2 + 5*w - 3],\ [997, 997, w^4 + w^3 - 5*w^2 - 9*w - 2]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x K = QQ e = 1 hecke_eigenvalues_array = [1, -1, -1, -5, -6, 0, 10, 5, 9, 2, -3, 4, 10, -7, -14, -17, 13, -3, -13, -6, -6, 9, -20, 12, 1, 3, -6, 8, 14, 12, 13, -5, -8, 2, 10, 4, -6, 21, 18, 5, 8, -9, 1, 4, 23, -17, -2, -10, 7, 6, -1, 3, 3, 18, -8, 13, 30, 0, -5, -5, 10, 11, -13, 11, -20, -5, -26, -14, 31, -6, 6, -1, 4, -7, 1, 36, 25, -22, -21, 28, 10, -5, -36, 14, -1, -18, 10, -1, -38, 1, -9, 3, -4, -11, 8, 19, -13, -5, 5, 10, 20, -12, 30, -7, -14, -2, 14, 10, 6, -2, 42, 25, 29, -20, -5, 2, 6, 2, -30, 24, 10, -2, -44, 52, -18, 18, -24, 2, -20, -23, -22, 40, 40, -25, -24, 50, -8, -32, 9, 12, 39, -55, -50, 10, 35, 23, -5, -14, 24, -38, 37, 49, 30, 38, 20, -43, -22, -48, 46, -18, -18, -40, 18, -27, -44, 38, -18, 14, 34, -32] 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])] = -1 AL_eigenvalues[ZF.ideal([7, 7, w^4 - 2*w^3 - 3*w^2 + 4*w + 2])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]