/* 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([4, 3, -5, -1, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([4, 2, -w^2 + 3]) primes_array = [ [2, 2, w^3 - 4*w - 2],\ [2, 2, w - 1],\ [11, 11, -w^3 + 5*w + 1],\ [17, 17, -w^3 + 3*w + 1],\ [23, 23, w^3 + w^2 - 4*w - 5],\ [29, 29, -w^3 + w^2 + 2*w - 1],\ [29, 29, w^3 + w^2 - 6*w - 5],\ [29, 29, w^3 - w^2 - 4*w + 1],\ [47, 47, w^3 - 3*w - 3],\ [53, 53, 2*w^2 - 2*w - 5],\ [53, 53, -w^3 + 2*w^2 + 5*w - 7],\ [59, 59, 2*w^3 - 8*w - 3],\ [59, 59, w^3 - w^2 - 2*w + 3],\ [67, 67, 2*w - 1],\ [71, 71, 2*w^2 - 5],\ [73, 73, -w^3 + w^2 + 4*w + 1],\ [73, 73, -2*w^3 + 8*w + 1],\ [73, 73, -w^3 + 5*w - 1],\ [79, 79, w^2 - w + 1],\ [79, 79, w^3 + w^2 - 4*w - 7],\ [81, 3, -3],\ [83, 83, 2*w^3 + w^2 - 9*w - 5],\ [83, 83, w^2 + w - 5],\ [97, 97, 4*w^3 + w^2 - 19*w - 11],\ [97, 97, -2*w^3 + 3*w^2 + 7*w - 7],\ [101, 101, -w^3 + 2*w^2 + 3*w - 3],\ [103, 103, w^3 + w^2 - 4*w - 1],\ [103, 103, -2*w^3 + w^2 + 7*w - 5],\ [109, 109, w^3 - w^2 - 2*w - 3],\ [113, 113, 2*w^2 - 9],\ [131, 131, -2*w^2 + 2*w + 7],\ [137, 137, -2*w^3 + w^2 + 9*w - 3],\ [137, 137, 2*w^3 - w^2 - 9*w - 1],\ [149, 149, 2*w^3 - w^2 - 7*w - 1],\ [157, 157, w^3 - 5*w + 3],\ [163, 163, -w^3 + 3*w^2 + 4*w - 7],\ [173, 173, w^3 + 2*w^2 - 3*w - 7],\ [179, 179, w^3 - w^2 - 2*w + 5],\ [179, 179, w^3 - 5*w - 5],\ [191, 191, 2*w^2 + 2*w - 7],\ [197, 197, 3*w^2 - w - 5],\ [197, 197, -2*w^3 + w^2 + 7*w - 7],\ [223, 223, 2*w^3 - 2*w^2 - 8*w + 3],\ [227, 227, w^2 + 3*w - 5],\ [233, 233, -3*w^3 - w^2 + 12*w + 9],\ [233, 233, 3*w^2 - w - 11],\ [239, 239, w^3 - 5*w + 5],\ [239, 239, w^3 + 2*w^2 - 5*w - 3],\ [241, 241, -w^3 + w^2 + 6*w - 5],\ [241, 241, w^3 + 2*w^2 - 5*w - 7],\ [251, 251, 3*w^3 - 13*w - 3],\ [251, 251, w^3 - 2*w^2 - 5*w + 3],\ [257, 257, -2*w^3 + w^2 + 7*w - 1],\ [271, 271, -2*w^3 + 2*w^2 + 6*w - 5],\ [277, 277, w^3 - w - 3],\ [277, 277, -w^3 - w^2 + 6*w + 1],\ [283, 283, -w^3 + 4*w^2 + w - 7],\ [293, 293, 4*w^3 - 18*w - 9],\ [293, 293, 3*w^3 - w^2 - 12*w - 1],\ [307, 307, 3*w^3 - 15*w - 7],\ [313, 313, 2*w^3 - 6*w - 1],\ [313, 313, w^3 + w^2 - 4*w - 9],\ [317, 317, -3*w^2 - w + 5],\ [317, 317, 2*w^3 - w^2 - 7*w - 5],\ [331, 331, 2*w^2 - 2*w - 9],\ [337, 337, w^3 - w^2 - 6*w + 1],\ [347, 347, -2*w^3 + 2*w^2 + 6*w - 3],\ [349, 349, 2*w^3 - 2*w^2 - 4*w + 3],\ [373, 373, 6*w^3 + 2*w^2 - 26*w - 15],\ [379, 379, 3*w^3 - w^2 - 12*w - 5],\ [389, 389, -2*w^3 + 4*w^2 + 4*w - 9],\ [389, 389, 3*w^3 + 2*w^2 - 13*w - 13],\ [397, 397, -3*w^3 + 2*w^2 + 11*w - 1],\ [401, 401, 2*w^3 - w^2 - 9*w - 3],\ [409, 409, w^3 + w^2 - 2*w - 5],\ [419, 419, 3*w^3 - 2*w^2 - 11*w - 1],\ [421, 421, -w^3 + 3*w^2 + 2*w - 11],\ [439, 439, 2*w^3 - 6*w - 5],\ [449, 449, -w^3 + w^2 + 6*w - 3],\ [457, 457, w^3 - w^2 - 3],\ [461, 461, 2*w^3 - 2*w^2 - 10*w + 1],\ [461, 461, -2*w^3 - 2*w^2 + 6*w + 7],\ [463, 463, -2*w^3 + 3*w^2 + 5*w - 1],\ [467, 467, 3*w^3 - 6*w^2 - 13*w + 19],\ [487, 487, w^2 - 3*w - 5],\ [491, 491, -2*w^3 + w^2 + 5*w - 1],\ [499, 499, -w^3 + 3*w^2 + 6*w - 9],\ [503, 503, -2*w^3 + 3*w^2 + 7*w - 13],\ [503, 503, 4*w^3 + 2*w^2 - 18*w - 11],\ [509, 509, -3*w^3 + 3*w^2 + 12*w - 11],\ [523, 523, 3*w^3 - 11*w - 5],\ [541, 541, 3*w^2 - w - 9],\ [541, 541, -w^3 - 2*w^2 + 7*w + 9],\ [541, 541, -3*w^3 + 2*w^2 + 11*w - 5],\ [541, 541, -w^3 + 4*w^2 + 3*w - 11],\ [569, 569, -3*w^3 + 3*w^2 + 12*w - 13],\ [571, 571, 4*w^3 - w^2 - 17*w - 5],\ [571, 571, w^2 + 3*w - 3],\ [577, 577, -w^3 - 2*w^2 + 7*w + 3],\ [593, 593, -2*w^3 + w^2 + 9*w - 5],\ [601, 601, -2*w^3 + 6*w - 5],\ [607, 607, 2*w^3 - 2*w^2 - 10*w + 3],\ [607, 607, 2*w^3 - 3*w^2 - 5*w + 9],\ [613, 613, 2*w^3 - w^2 - 5*w - 5],\ [613, 613, w^2 + w - 9],\ [617, 617, 6*w^3 - 28*w - 13],\ [619, 619, -w^3 + 2*w^2 + 5*w + 1],\ [619, 619, 3*w^2 - 5*w - 7],\ [625, 5, -5],\ [653, 653, 2*w^3 - 5*w^2 - 9*w + 13],\ [653, 653, 2*w^3 + w^2 - 7*w + 1],\ [659, 659, -3*w^3 + 2*w^2 + 9*w - 7],\ [659, 659, -3*w^3 - 2*w^2 + 17*w + 11],\ [661, 661, -2*w^3 + 8*w - 5],\ [677, 677, -w^3 - 3*w^2 + 6*w + 7],\ [677, 677, 3*w^2 + w - 11],\ [677, 677, -w^3 - 2*w^2 + 9*w + 3],\ [677, 677, 2*w^3 - 4*w^2 - 1],\ [719, 719, -w^3 + 3*w^2 - 7],\ [719, 719, -w^2 + w - 3],\ [727, 727, -w^3 + 5*w^2 + 4*w - 13],\ [727, 727, 3*w^3 - 11*w - 3],\ [739, 739, 2*w^3 + 2*w^2 - 8*w - 11],\ [739, 739, -w^3 + 3*w^2 + 4*w - 5],\ [743, 743, w^3 + 3*w^2 - 4*w - 7],\ [751, 751, -2*w^3 + 2*w^2 + 8*w - 1],\ [757, 757, w^3 + 3*w^2 - 6*w - 11],\ [757, 757, -4*w - 1],\ [769, 769, 2*w^3 - 2*w^2 - 8*w + 11],\ [773, 773, -w^3 + 7*w - 1],\ [787, 787, w^2 - 3*w - 7],\ [797, 797, 4*w^3 - 16*w - 7],\ [797, 797, w^3 - 7*w - 1],\ [809, 809, 3*w^3 - 3*w^2 - 12*w + 5],\ [809, 809, w^3 - 5*w^2 - 2*w + 17],\ [811, 811, w^3 - 2*w^2 - 7*w + 9],\ [823, 823, -2*w^3 + 3*w^2 + w + 1],\ [827, 827, -3*w^3 + w^2 + 12*w - 3],\ [827, 827, 8*w^3 - 36*w - 15],\ [829, 829, -w^2 - w - 3],\ [853, 853, 4*w^3 + w^2 - 17*w - 7],\ [853, 853, 4*w - 3],\ [857, 857, 2*w^3 + 3*w^2 - 11*w - 11],\ [859, 859, w^3 - 5*w^2 - 4*w + 15],\ [877, 877, -4*w^3 + w^2 + 15*w - 3],\ [881, 881, -4*w^2 + 6*w + 9],\ [881, 881, -w^3 + 3*w - 5],\ [881, 881, 3*w^2 - 3*w - 11],\ [881, 881, -w^3 + 4*w^2 + w - 13],\ [883, 883, 2*w^3 - 3*w^2 - 3*w - 3],\ [883, 883, w^3 - 5*w^2 + 13],\ [887, 887, -w^3 + 2*w^2 + w - 7],\ [911, 911, -4*w^3 + 4*w^2 + 14*w - 13],\ [911, 911, 2*w^2 + 2*w - 9],\ [919, 919, -w^2 - 5*w + 9],\ [919, 919, 3*w^3 - 15*w - 5],\ [937, 937, -3*w^3 + 3*w^2 + 10*w - 5],\ [953, 953, -8*w^3 - w^2 + 35*w + 17],\ [961, 31, 3*w^3 - w^2 - 14*w - 5],\ [961, 31, 2*w^3 - 5*w^2 - 5*w + 9],\ [967, 967, 5*w^2 + w - 9],\ [967, 967, -5*w^3 + 23*w + 13],\ [977, 977, -w^3 - 2*w^2 + 7*w + 11],\ [977, 977, -3*w^3 + 2*w^2 + 13*w - 7],\ [983, 983, 2*w^3 - 6*w - 7],\ [991, 991, 6*w^2 - 11],\ [991, 991, 3*w^3 - 5*w^2 - 12*w + 13],\ [997, 997, w^3 + w^2 - 8*w - 3],\ [997, 997, -3*w^3 + 4*w^2 + 9*w - 9]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x K = QQ e = 1 hecke_eigenvalues_array = [-1, 0, -4, -2, 0, -6, 2, -6, -8, -2, 10, -12, -4, -12, -16, 2, 6, -14, 0, -8, 14, -12, 12, 14, 2, 10, -8, 0, -2, 2, -12, -6, 2, -10, 2, -20, 2, -12, 12, 8, -22, -2, -24, -20, 6, 26, 24, 24, 14, -2, 12, -12, 6, 8, 10, -26, 12, -6, 22, -28, 6, -2, 34, 10, -4, -2, 4, 2, -26, -4, -18, -10, 30, -18, 30, 36, 6, 32, 18, 26, 30, 18, -24, -28, -40, -28, -20, -40, 0, -34, -20, 34, -10, -18, -6, 22, -20, -28, -38, -18, 26, -8, 24, 26, 2, 6, -20, 12, 2, -30, 18, -4, -4, 22, 38, 6, -18, 6, 24, 40, 40, 8, -52, 36, -24, 8, 22, -22, 30, 14, -12, -50, 18, 30, -26, 4, 40, -12, -44, -42, 42, -38, -54, 12, -50, -14, -30, 50, -30, -4, -52, 8, -8, 32, 16, 16, -2, -38, 34, -34, 48, -8, 14, 6, -48, 32, -8, -54, -10] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([2, 2, w - 1])] = -1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]