/* 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, 2, -5, -2, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([13, 13, -w + 2]) primes_array = [ [3, 3, w],\ [4, 2, -w^3 + 3*w^2 + w - 2],\ [11, 11, -w^3 + 3*w^2 + 2*w - 5],\ [13, 13, w^3 - 2*w^2 - 4*w + 2],\ [13, 13, -w + 2],\ [17, 17, w^3 - 3*w^2 - 2*w + 2],\ [19, 19, -w^2 + w + 4],\ [19, 19, w^3 - 2*w^2 - 3*w + 2],\ [23, 23, w^2 - 2*w - 1],\ [27, 3, w^3 - 2*w^2 - 5*w - 1],\ [29, 29, -w^3 + 2*w^2 + 5*w - 1],\ [37, 37, -w^3 + 2*w^2 + 5*w - 4],\ [41, 41, 2*w^3 - 5*w^2 - 6*w + 4],\ [53, 53, w^3 - 2*w^2 - 3*w - 2],\ [59, 59, w - 4],\ [67, 67, 2*w^2 - 3*w - 8],\ [73, 73, 2*w^3 - 5*w^2 - 6*w + 5],\ [89, 89, -2*w^3 + 6*w^2 + 5*w - 7],\ [97, 97, -2*w^3 + 6*w^2 + 3*w - 7],\ [97, 97, -w^3 + 3*w^2 + 3*w - 1],\ [103, 103, w^2 - 4*w - 4],\ [113, 113, 2*w^3 - 5*w^2 - 4*w + 1],\ [113, 113, -w^3 + 3*w^2 + w - 5],\ [139, 139, 2*w^2 - 3*w - 4],\ [139, 139, w^3 - w^2 - 6*w - 1],\ [149, 149, 2*w^3 - 5*w^2 - 5*w + 2],\ [149, 149, w^3 - 4*w^2 + 10],\ [151, 151, -3*w - 1],\ [157, 157, w^2 - w - 8],\ [157, 157, w^3 - 7*w - 8],\ [163, 163, -3*w^3 + 8*w^2 + 8*w - 10],\ [163, 163, w^3 - 3*w^2 - 4*w + 7],\ [163, 163, -w^3 + 4*w^2 + w - 8],\ [163, 163, -2*w^3 + 5*w^2 + 7*w - 4],\ [167, 167, -2*w^3 + 5*w^2 + 8*w - 4],\ [169, 13, -2*w^3 + 4*w^2 + 7*w - 5],\ [173, 173, 2*w^3 - 4*w^2 - 9*w + 4],\ [173, 173, -w^3 + 5*w^2 - 2*w - 5],\ [181, 181, 3*w^3 - 7*w^2 - 12*w + 7],\ [191, 191, w^3 - 3*w^2 - 3*w - 1],\ [199, 199, 2*w^3 - 3*w^2 - 10*w - 1],\ [199, 199, 2*w^3 - 6*w^2 - 6*w + 13],\ [223, 223, -2*w^3 + 6*w^2 + 4*w - 11],\ [223, 223, 3*w^3 - 7*w^2 - 12*w + 8],\ [223, 223, -2*w^3 + 5*w^2 + 6*w - 1],\ [229, 229, -w^3 + 3*w^2 + 2*w + 1],\ [229, 229, 3*w^3 - 4*w^2 - 15*w - 5],\ [233, 233, -3*w^3 + 7*w^2 + 11*w - 5],\ [233, 233, 2*w^3 - 5*w^2 - 4*w + 5],\ [241, 241, w^3 - 3*w^2 - 5*w + 7],\ [257, 257, w^3 - 3*w^2 - 4*w + 8],\ [263, 263, w^2 - 4*w - 5],\ [271, 271, -2*w^3 + 6*w^2 + 5*w - 13],\ [277, 277, -2*w^3 + 6*w^2 + 3*w - 8],\ [277, 277, w^3 - 2*w^2 - 2*w - 2],\ [281, 281, 2*w^3 - 5*w^2 - 4*w + 2],\ [283, 283, -3*w^2 + 7*w + 10],\ [283, 283, -w^3 + 4*w^2 - w - 7],\ [293, 293, 2*w^3 - 5*w^2 - 3*w + 4],\ [293, 293, w^3 - 8*w - 4],\ [307, 307, -w^3 + 2*w^2 + 6*w - 2],\ [311, 311, w^2 - 5],\ [317, 317, -w^3 + 5*w^2 - 2*w - 11],\ [331, 331, -w^3 + 3*w^2 - 5],\ [349, 349, -2*w^2 + 5*w + 2],\ [353, 353, -2*w^3 + 5*w^2 + 4*w - 4],\ [353, 353, 3*w^3 - 7*w^2 - 10*w + 2],\ [353, 353, -2*w^3 + 7*w^2 + 2*w - 10],\ [353, 353, w^3 + w^2 - 9*w - 13],\ [361, 19, -w^3 + w^2 + 5*w - 1],\ [373, 373, -w^3 + 3*w^2 + 4*w - 11],\ [373, 373, 2*w^3 - 5*w^2 - 5*w - 2],\ [383, 383, w^3 - 5*w^2 + w + 13],\ [389, 389, 3*w^3 - 10*w^2 - 4*w + 16],\ [397, 397, -w^3 + 2*w^2 + 3*w + 5],\ [397, 397, -3*w^3 + 7*w^2 + 10*w - 7],\ [419, 419, -w^3 + 2*w^2 + 4*w + 4],\ [419, 419, 3*w^3 - 7*w^2 - 12*w + 5],\ [421, 421, -w^2 + 5*w - 2],\ [421, 421, 3*w^3 - 7*w^2 - 10*w + 5],\ [431, 431, w^2 - 10],\ [439, 439, w^3 - w^2 - 7*w - 7],\ [461, 461, -2*w^3 + 7*w^2 + 5*w - 10],\ [467, 467, -3*w^3 + 6*w^2 + 14*w - 4],\ [479, 479, 3*w^2 - 6*w - 5],\ [479, 479, -4*w^3 + 11*w^2 + 10*w - 10],\ [491, 491, -w^3 + w^2 + 6*w - 1],\ [499, 499, -4*w^3 + 9*w^2 + 15*w - 10],\ [499, 499, 2*w^3 - 5*w^2 - 7*w + 2],\ [503, 503, -2*w^3 + 7*w^2 - 8],\ [521, 521, 2*w^2 - w - 7],\ [521, 521, -3*w^3 + 7*w^2 + 11*w - 11],\ [569, 569, 2*w^2 - 5*w - 1],\ [569, 569, -4*w - 1],\ [571, 571, -w^3 + 3*w^2 - 7],\ [577, 577, -w^3 + w^2 + 8*w - 4],\ [587, 587, -3*w^3 + 8*w^2 + 7*w - 8],\ [593, 593, 3*w^2 - 4*w - 11],\ [593, 593, -3*w^3 + 9*w^2 + 6*w - 14],\ [601, 601, -3*w^3 + 9*w^2 + 4*w - 10],\ [601, 601, 5*w^3 - 9*w^2 - 24*w - 2],\ [607, 607, 2*w^2 - 7*w - 5],\ [613, 613, w^3 - 6*w - 2],\ [613, 613, -w^2 + 3*w + 8],\ [617, 617, -w^2 + 4*w - 5],\ [619, 619, 3*w^3 - 6*w^2 - 13*w + 5],\ [625, 5, -5],\ [631, 631, -2*w^3 + 4*w^2 + 8*w - 7],\ [647, 647, -2*w^3 + 4*w^2 + 5*w - 1],\ [647, 647, w^3 - 3*w^2 - 2*w - 2],\ [653, 653, w^3 - 4*w^2 - w + 13],\ [653, 653, w^3 - 2*w^2 - 7*w + 5],\ [659, 659, -4*w^3 + 10*w^2 + 15*w - 14],\ [659, 659, 5*w^3 - 12*w^2 - 17*w + 14],\ [661, 661, w^2 - 5*w - 2],\ [661, 661, -w^3 + 5*w^2 - 2*w - 7],\ [673, 673, -w^3 + w^2 + 6*w - 2],\ [683, 683, 2*w^3 - 8*w^2 + w + 8],\ [683, 683, 5*w^3 - 12*w^2 - 19*w + 14],\ [701, 701, -2*w^3 + 6*w^2 + 5*w - 4],\ [709, 709, -w^2 + 3*w - 4],\ [709, 709, -2*w^3 + 5*w^2 + 9*w - 8],\ [719, 719, w^2 - 7],\ [719, 719, -2*w^3 + 3*w^2 + 8*w + 5],\ [733, 733, -2*w^3 + 7*w^2 + 3*w - 10],\ [739, 739, 4*w^3 - 11*w^2 - 13*w + 14],\ [739, 739, -5*w^3 + 10*w^2 + 21*w - 4],\ [739, 739, 3*w^3 - 8*w^2 - 6*w + 8],\ [739, 739, -w^3 + w^2 + 8*w - 5],\ [743, 743, -w - 5],\ [743, 743, w^3 - w^2 - 8*w - 8],\ [751, 751, -2*w^3 + 5*w^2 + 8*w - 2],\ [751, 751, -2*w^3 + 6*w^2 - 1],\ [757, 757, -2*w^3 + 6*w^2 + 3*w - 10],\ [757, 757, 5*w^2 - 9*w - 22],\ [761, 761, -2*w^3 + 4*w^2 + 9*w - 7],\ [761, 761, 2*w^3 - 3*w^2 - 12*w - 1],\ [769, 769, 5*w^3 - 13*w^2 - 14*w + 10],\ [773, 773, -3*w^3 + 5*w^2 + 16*w + 1],\ [773, 773, w^3 - 11*w + 2],\ [787, 787, -w^3 + 4*w^2 - w - 11],\ [797, 797, -5*w^3 + 12*w^2 + 17*w - 11],\ [809, 809, -2*w^3 + 7*w^2 + 4*w - 11],\ [809, 809, 4*w^3 - 9*w^2 - 14*w + 4],\ [821, 821, -w^3 + 5*w^2 - 2*w - 17],\ [827, 827, -3*w^3 + 8*w^2 + 9*w - 7],\ [829, 829, 2*w^3 - 6*w^2 - 5*w + 2],\ [829, 829, -w^3 + 4*w^2 - w - 10],\ [839, 839, w^2 - 8],\ [853, 853, -5*w^3 + 12*w^2 + 18*w - 16],\ [853, 853, w^3 - 6*w - 8],\ [863, 863, 5*w^3 - 11*w^2 - 19*w + 7],\ [881, 881, -3*w^3 + 9*w^2 + 9*w - 11],\ [907, 907, -w^3 + 2*w^2 + 7*w - 4],\ [907, 907, -2*w^3 + 3*w^2 + 7*w + 4],\ [929, 929, -w^3 + 5*w^2 - 3*w - 11],\ [937, 937, 2*w^3 - 5*w^2 - 6*w + 11],\ [937, 937, -2*w^3 + 6*w^2 + 3*w - 11],\ [941, 941, -3*w^3 + 8*w^2 + 11*w - 8],\ [953, 953, -4*w^3 + 10*w^2 + 10*w - 13],\ [953, 953, w^3 - 5*w^2 - w + 13],\ [967, 967, 3*w^3 - 7*w^2 - 9*w + 7],\ [967, 967, -2*w^3 + 5*w^2 + 8*w + 1],\ [967, 967, 4*w^3 - 13*w^2 - 6*w + 20],\ [967, 967, -2*w^2 + 3*w + 13],\ [971, 971, -6*w^3 + 13*w^2 + 25*w - 8],\ [977, 977, w^3 - w^2 - 4*w - 10],\ [977, 977, 2*w^3 - 6*w^2 - 7*w + 13],\ [983, 983, -4*w^3 + 10*w^2 + 11*w - 13],\ [983, 983, -w^3 + 5*w^2 - w - 7],\ [983, 983, 3*w^3 - 9*w^2 - 7*w + 19],\ [983, 983, -2*w^3 + 3*w^2 + 10*w - 1],\ [997, 997, 5*w^3 - 12*w^2 - 17*w + 13],\ [997, 997, -2*w^3 + 3*w^2 + 13*w - 4],\ [997, 997, w^3 - 7*w - 2],\ [997, 997, w^3 - 9*w - 2]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^4 - 8*x^2 + 9 K. = NumberField(heckePol) hecke_eigenvalues_array = [-1/3*e^3 + 5/3*e, e, -e^2 + 2, 1/3*e^3 - 11/3*e, 1, e^3 - 6*e, e^2 - 5, 2/3*e^3 - 10/3*e, -2/3*e^3 + 7/3*e, -1/3*e^3 + 11/3*e, -4/3*e^3 + 20/3*e, 2*e, -1/3*e^3 + 11/3*e, -e^2 - 5, 3*e^2 - 15, -2*e^2 - 1, 2*e^3 - 17*e, -2/3*e^3 + 13/3*e, e^2 - 4, 8/3*e^3 - 46/3*e, -4*e^2 + 15, -e^3 + 3*e, -e^2 - 1, 8*e^2 - 32, e^3 - 11*e, -5/3*e^3 + 40/3*e, -2*e^2 + 18, -1/3*e^3 - 7/3*e, 3*e^2 - 8, -3, 2*e, -2*e^2 + 4, 2*e^2 - 12, -8/3*e^3 + 52/3*e, -e^3 + 4*e, 5/3*e^3 - 37/3*e, -4*e^3 + 27*e, 4*e^2 - 24, -13/3*e^3 + 59/3*e, -11, -e^3 + 2*e, 4, -3*e^2 + 26, e^3 - 11*e, 5*e^3 - 27*e, 4*e^2 - 10, -8/3*e^3 + 58/3*e, -5/3*e^3 + 61/3*e, 5/3*e^3 - 7/3*e, -e^2 + 1, 7*e^2 - 36, -3*e^2 + 3, 4*e^2 - 10, 2*e^2 - 31, -2*e^2 - 1, 11/3*e^3 - 61/3*e, -10*e^2 + 37, 2*e^2 - 4, 0, -2/3*e^3 - 20/3*e, 1/3*e^3 + 10/3*e, -4*e^2 + 20, -5*e^2 + 37, -5*e^2 + 9, -1/3*e^3 + 11/3*e, 5/3*e^3 - 40/3*e, -17/3*e^3 + 97/3*e, 3*e^2 - 29, -e^2 + 29, 16/3*e^3 - 68/3*e, -e^2 - 14, -e^2 - 14, -3*e^2 - 3, -4*e^2, e^2 - 33, 3*e^3 - 15*e, e^2 - 18, 5/3*e^3 - 13/3*e, -8/3*e^3 + 64/3*e, e^3 - 5*e, -11*e^2 + 35, -e^2 - 11, 8*e^2 - 25, 11*e, 5*e^3 - 36*e, -1/3*e^3 + 47/3*e, 25/3*e^3 - 131/3*e, 8/3*e^3 - 40/3*e, -2*e^3 + 10*e, 14*e^2 - 56, -3*e^2 + 9, 10/3*e^3 - 32/3*e, 13/3*e^3 - 95/3*e, 4*e^3 - 21*e, 11*e^2 - 39, -2/3*e^3 + 16/3*e, 4*e^3 - 37*e, -5*e^2 + 33, 6, -e^2 - 3, -14/3*e^3 + 133/3*e, 3*e^3 - 15*e, -e^3 + 5*e, -2, -9*e^2 + 31, -4*e^3 + 21*e, -e^2 + 21, -11/3*e^3 + 118/3*e, 8/3*e^3 - 52/3*e, 3*e^2 - 8, e^2 - 43, 1/3*e^3 - 5/3*e, -5*e^3 + 47*e, -5/3*e^3 + 52/3*e, -9*e, -14*e^2 + 60, 7/3*e^3 - 38/3*e, 7*e^2 - 16, 14/3*e^3 - 130/3*e, -13/3*e^3 + 35/3*e, -14*e^2 + 52, -17/3*e^3 + 88/3*e, 6*e^2 - 39, -15*e^2 + 59, -5*e^2 + 47, 5*e^3 - 19*e, 6*e^3 - 30*e, 2*e^3 - 28*e, -e^3 + 12*e, -10*e^2 + 21, 2*e^2 - 6, 14/3*e^3 - 127/3*e, -6*e^2 + 20, -12, -3*e^2 + 14, 4/3*e^3 - 32/3*e, 29/3*e^3 - 139/3*e, e^3 - 11*e, -29/3*e^3 + 166/3*e, 11/3*e^3 - 85/3*e, 7*e^2 - 52, -23/3*e^3 + 106/3*e, 8*e^2 + 1, -23/3*e^3 + 151/3*e, 2*e^2 - 40, -1/3*e^3 + 8/3*e, -e^3 - 5*e, e^2 + 7, -8*e^2 + 46, -5*e^3 + 13*e, -10*e^2 + 61, -7/3*e^3 - 1/3*e, -19/3*e^3 + 83/3*e, 1/3*e^3 - 8/3*e, 8*e^2 - 62, 13*e^2 - 74, -8*e^2 + 32, 11*e^2 - 51, -20/3*e^3 + 130/3*e, 32/3*e^3 - 166/3*e, -15*e^2 + 48, -19/3*e^3 + 80/3*e, -19/3*e^3 + 101/3*e, 16, -2*e^2 + 10, -e^3 - 6*e, 5*e^2 - 43, 7*e^2 - 37, 7*e^3 - 30*e, -9*e^2 + 68, -2*e^2 - 16, -9*e^3 + 55*e, -6*e^3 + 36*e, 20/3*e^3 - 100/3*e, 5*e, -5*e^3 + 25*e] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([13, 13, -w + 2])] = -1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]