/* 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, 5, -4, -2, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([23, 23, -w^3 + 4*w + 1]) primes_array = [ [3, 3, w],\ [3, 3, w - 2],\ [3, 3, w - 1],\ [16, 2, 2],\ [23, 23, -w^3 + 4*w + 1],\ [25, 5, w^3 - 2*w^2 - 2*w + 2],\ [25, 5, w^3 - w^2 - 3*w + 1],\ [29, 29, w^3 - 2*w^2 - 4*w + 4],\ [29, 29, w^3 - w^2 - 5*w + 1],\ [43, 43, -w^3 + 2*w^2 + 3*w - 2],\ [43, 43, w^3 - w^2 - 4*w + 2],\ [61, 61, -w^2 + 2*w + 4],\ [61, 61, w^2 - 5],\ [79, 79, 3*w^3 - 7*w^2 - 8*w + 16],\ [79, 79, -w^3 + w^2 + 6*w - 4],\ [101, 101, -w^3 + 3*w^2 + w - 7],\ [101, 101, w^3 - 4*w - 4],\ [103, 103, 2*w^3 - w^2 - 8*w - 1],\ [103, 103, 2*w^3 - 5*w^2 - 4*w + 8],\ [107, 107, 2*w^2 - 3*w - 4],\ [107, 107, 3*w - 5],\ [107, 107, w^3 - 4*w^2 + 10],\ [107, 107, 2*w^2 - w - 5],\ [113, 113, -w^3 + 4*w^2 + w - 11],\ [113, 113, -2*w^3 + 4*w^2 + 5*w - 5],\ [113, 113, 2*w^3 - 2*w^2 - 7*w + 2],\ [113, 113, w^3 + w^2 - 6*w - 7],\ [121, 11, w^3 - 3*w^2 - 4*w + 8],\ [121, 11, w^3 - 7*w - 2],\ [127, 127, w^2 - 4*w + 2],\ [127, 127, w^3 - w^2 - 2*w - 2],\ [127, 127, -w^3 + 2*w^2 + w - 4],\ [127, 127, -2*w^3 + 5*w^2 + 3*w - 8],\ [131, 131, 2*w^3 - 2*w^2 - 9*w + 2],\ [131, 131, -w^3 - w^2 + 7*w + 5],\ [157, 157, w^3 - w^2 - 7*w + 8],\ [157, 157, w^3 - w^2 - 5*w + 4],\ [169, 13, 2*w^2 - 2*w - 5],\ [173, 173, -w^3 + 7*w - 1],\ [173, 173, -w^3 - w^2 + 6*w + 4],\ [179, 179, -2*w^3 + 4*w^2 + 5*w - 8],\ [179, 179, -2*w^3 + 2*w^2 + 7*w + 1],\ [181, 181, -w^3 + 4*w^2 - 7],\ [181, 181, w^2 - 4*w + 5],\ [191, 191, 3*w^3 - 7*w^2 - 9*w + 20],\ [191, 191, -w^3 + 4*w + 5],\ [191, 191, -w^3 + 3*w^2 + w - 8],\ [191, 191, 3*w^3 - 3*w^2 - 12*w + 2],\ [199, 199, 2*w^2 - 7],\ [199, 199, 3*w^3 - 6*w^2 - 9*w + 13],\ [233, 233, w^3 - 3*w^2 - 4*w + 11],\ [233, 233, 3*w - 7],\ [251, 251, 3*w^3 - 7*w^2 - 7*w + 13],\ [251, 251, 4*w^3 - w^2 - 20*w - 11],\ [251, 251, -w^3 + w^2 + 5*w - 7],\ [251, 251, -2*w^3 + 4*w^2 + 8*w - 11],\ [257, 257, w^2 - 2*w - 7],\ [257, 257, w^2 - 8],\ [269, 269, 2*w^3 - 3*w^2 - 8*w + 2],\ [269, 269, 2*w^3 - 3*w^2 - 8*w + 7],\ [283, 283, -2*w^3 + 4*w^2 + 4*w - 7],\ [283, 283, 2*w^3 - 2*w^2 - 6*w - 1],\ [289, 17, -w^2 + w - 2],\ [289, 17, w^2 - w - 7],\ [311, 311, -w^3 + 2*w^2 + 3*w - 8],\ [311, 311, -w^3 + w^2 + 4*w + 4],\ [313, 313, -w^3 + 2*w^2 + 5*w - 5],\ [313, 313, w^3 - w^2 - 6*w + 1],\ [337, 337, 3*w - 1],\ [337, 337, 3*w - 2],\ [347, 347, -2*w^3 - w^2 + 11*w + 11],\ [347, 347, -2*w^3 + 7*w^2 + 3*w - 19],\ [367, 367, 4*w^3 - 8*w^2 - 13*w + 19],\ [367, 367, 4*w^3 - 4*w^2 - 17*w - 2],\ [373, 373, w^2 - 3*w - 5],\ [373, 373, w^2 + w - 7],\ [433, 433, 2*w^3 - 3*w^2 - 5*w + 1],\ [433, 433, -2*w^3 + 3*w^2 + 5*w - 5],\ [443, 443, w^3 - 7*w + 2],\ [443, 443, 3*w^3 - 6*w^2 - 12*w + 20],\ [491, 491, -2*w^3 + 3*w^2 + 8*w - 8],\ [491, 491, 2*w^3 - 3*w^2 - 11*w + 14],\ [521, 521, w^3 + w^2 - 8*w - 7],\ [521, 521, w^3 - 5*w^2 + 2*w + 10],\ [521, 521, w^3 + 2*w^2 - 5*w - 8],\ [521, 521, w^3 - 4*w^2 - 3*w + 13],\ [523, 523, 3*w^3 - 5*w^2 - 11*w + 11],\ [523, 523, w^3 + w^2 - 6*w - 1],\ [529, 23, 3*w^2 - 3*w - 10],\ [547, 547, w^3 - 3*w^2 - 4*w + 14],\ [547, 547, -2*w^3 + 2*w^2 + 7*w - 5],\ [547, 547, w^3 - 4*w + 4],\ [547, 547, 3*w^3 - 9*w^2 - 6*w + 22],\ [563, 563, w^3 - 2*w^2 - 6*w + 8],\ [563, 563, -3*w^3 + 4*w^2 + 11*w - 7],\ [563, 563, 3*w^3 - 5*w^2 - 10*w + 5],\ [563, 563, 3*w^3 - 8*w^2 - 3*w + 10],\ [571, 571, 2*w^3 - 3*w^2 - 5*w + 4],\ [571, 571, -2*w^3 + 3*w^2 + 5*w - 2],\ [599, 599, 2*w^3 - 2*w^2 - 5*w + 1],\ [599, 599, -2*w^3 + 4*w^2 + 3*w - 4],\ [641, 641, -2*w^2 + 7*w - 7],\ [641, 641, 2*w^2 + w - 8],\ [641, 641, 2*w^2 - 5*w - 5],\ [641, 641, w^3 - 4*w^2 - 2*w + 16],\ [647, 647, -w^3 + 3*w^2 - 2*w - 4],\ [647, 647, w^3 - w - 4],\ [653, 653, -2*w^3 + 4*w^2 + 9*w - 13],\ [653, 653, 3*w^3 - 7*w^2 - 6*w + 11],\ [673, 673, w^3 - 2*w^2 - 5*w + 11],\ [673, 673, -2*w^3 + 2*w^2 + 10*w - 5],\ [673, 673, 2*w^3 - 4*w^2 - 8*w + 5],\ [673, 673, 2*w^3 - 7*w^2 - w + 13],\ [677, 677, -w^3 + 2*w^2 + 6*w - 2],\ [677, 677, 3*w^3 - 7*w^2 - 8*w + 13],\ [677, 677, 3*w^3 - 2*w^2 - 13*w - 1],\ [677, 677, -w^3 + w^2 + 7*w - 5],\ [701, 701, w^2 - 5*w - 1],\ [701, 701, -w^3 + 4*w^2 + 3*w - 7],\ [701, 701, -w^3 - w^2 + 8*w + 1],\ [701, 701, w^2 + 3*w - 5],\ [719, 719, -2*w^3 + w^2 + 7*w + 5],\ [719, 719, 2*w^3 - 5*w^2 - 3*w + 11],\ [727, 727, 2*w^2 - 4*w - 11],\ [727, 727, -3*w^3 + w^2 + 10*w + 5],\ [751, 751, 4*w^3 - 12*w^2 - 7*w + 28],\ [751, 751, 2*w^3 - 6*w^2 - 5*w + 20],\ [757, 757, -w^3 + 2*w^2 - 5],\ [757, 757, w^3 - w^2 - w - 4],\ [797, 797, -4*w^3 + 7*w^2 + 13*w - 11],\ [797, 797, 4*w^3 - 5*w^2 - 17*w + 2],\ [797, 797, -4*w^3 + 7*w^2 + 15*w - 16],\ [797, 797, 4*w^3 - 5*w^2 - 15*w + 5],\ [809, 809, -w^3 + 4*w^2 + 3*w - 16],\ [809, 809, 3*w^3 - 8*w^2 - 10*w + 26],\ [841, 29, 3*w^2 - 3*w - 8],\ [857, 857, 4*w^3 - 2*w^2 - 21*w - 7],\ [857, 857, -w^3 + 2*w^2 - w + 4],\ [881, 881, -w^3 + 2*w^2 + 7*w - 10],\ [881, 881, w^3 - 5*w^2 - w + 13],\ [881, 881, w^3 + 2*w^2 - 8*w - 8],\ [881, 881, -w^3 + w^2 + 8*w + 2],\ [887, 887, w^3 - w^2 - 5*w - 5],\ [887, 887, -w^3 + 2*w^2 + 4*w - 10],\ [907, 907, -3*w^3 + 3*w^2 + 9*w - 5],\ [907, 907, 3*w^3 - 6*w^2 - 6*w + 4],\ [919, 919, 4*w^3 - 3*w^2 - 16*w - 2],\ [919, 919, -4*w^3 + 9*w^2 + 10*w - 17],\ [937, 937, -3*w^3 + w^2 + 12*w + 7],\ [937, 937, -3*w^3 + 8*w^2 + 5*w - 17],\ [953, 953, -2*w^3 + 6*w^2 + 2*w - 13],\ [953, 953, -w^3 + 3*w^2 + 4*w - 2],\ [953, 953, w^3 - 7*w + 4],\ [953, 953, 2*w^3 - 8*w - 7],\ [961, 31, 3*w^3 - 4*w^2 - 13*w + 1],\ [961, 31, 3*w^3 - 5*w^2 - 12*w + 13],\ [991, 991, 3*w^3 - w^2 - 15*w - 4],\ [991, 991, -w^3 - 3*w^2 + 7*w + 5],\ [991, 991, w^3 - 6*w^2 + 2*w + 8],\ [991, 991, 3*w^3 - 8*w^2 - 8*w + 17]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x K = QQ e = 1 hecke_eigenvalues_array = [-2, 0, -2, -7, -1, -6, -6, -4, -4, 4, 4, -2, -2, -2, -2, -14, -14, 0, 0, -8, -4, -4, -8, -2, -8, -8, -2, 4, 4, -12, -10, -10, -12, 16, 16, -14, -14, -22, 18, 18, 0, 0, -10, -10, -16, -6, -6, -16, 2, 2, -6, -6, 6, -4, -4, 6, -18, -18, 26, 26, -6, -6, 30, 2, -24, -24, 28, 28, -4, -4, -12, -12, 4, 4, -20, -20, -26, -26, 12, 12, -20, -20, 28, 18, 18, 28, 0, 0, -30, -20, -12, -12, -20, 24, -24, -24, 24, -4, -4, 16, 16, -42, 0, 0, -42, -32, -32, 24, 24, 26, 26, 26, 26, -38, -38, -38, -38, -24, 26, 26, -24, -18, -18, -38, -38, 16, 16, 22, 22, 52, 26, 26, 52, -12, -12, -42, -30, -30, 30, -46, -46, 30, 32, 32, -8, -8, 60, 60, 34, 34, -30, -6, -6, -30, -54, -54, 40, 12, 12, 40] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([23, 23, -w^3 + 4*w + 1])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]