/* 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([-2, 6, 0, -6, 0, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([6, 6, w^4 + w^3 - 6*w^2 - 5*w + 4]) primes_array = [ [2, 2, w],\ [3, 3, w^4 - 5*w^2 - w + 3],\ [23, 23, -w^2 + 3],\ [25, 5, w^3 - 5*w + 1],\ [29, 29, -w^2 + w + 1],\ [37, 37, -w^4 + 5*w^2 + 2*w - 1],\ [41, 41, w^3 - 4*w - 1],\ [41, 41, -2*w^4 - w^3 + 10*w^2 + 6*w - 5],\ [43, 43, 2*w^4 + w^3 - 11*w^2 - 5*w + 7],\ [47, 47, 2*w^4 + 2*w^3 - 11*w^2 - 9*w + 9],\ [53, 53, w^4 + w^3 - 6*w^2 - 6*w + 5],\ [53, 53, 2*w^4 - 11*w^2 - 2*w + 5],\ [53, 53, -2*w^3 + w^2 + 9*w - 5],\ [59, 59, -w^4 + 6*w^2 + w - 3],\ [61, 61, w^4 + w^3 - 7*w^2 - 5*w + 7],\ [73, 73, -4*w^4 - w^3 + 23*w^2 + 6*w - 19],\ [79, 79, -2*w^4 - w^3 + 12*w^2 + 4*w - 9],\ [81, 3, w^4 - w^3 - 5*w^2 + 2*w + 1],\ [83, 83, -w^3 - w^2 + 4*w + 3],\ [83, 83, -2*w^4 + 11*w^2 - 7],\ [89, 89, w^4 - 4*w^2 - 1],\ [89, 89, 2*w^4 - 11*w^2 - w + 9],\ [97, 97, 2*w^4 + 2*w^3 - 11*w^2 - 9*w + 7],\ [101, 101, 4*w^4 + 3*w^3 - 23*w^2 - 16*w + 17],\ [101, 101, -3*w^4 - w^3 + 16*w^2 + 7*w - 11],\ [107, 107, -w^3 + w^2 + 5*w - 1],\ [125, 5, w^4 + 2*w^3 - 5*w^2 - 8*w + 3],\ [131, 131, 2*w^3 - 2*w^2 - 8*w + 3],\ [131, 131, -w^4 - w^3 + 7*w^2 + 4*w - 5],\ [131, 131, -2*w^4 + 10*w^2 + w - 5],\ [137, 137, -3*w^4 - w^3 + 16*w^2 + 7*w - 9],\ [139, 139, -w^4 - w^3 + 7*w^2 + 7*w - 5],\ [151, 151, 2*w^4 + 2*w^3 - 12*w^2 - 10*w + 9],\ [163, 163, w^4 - w^3 - 4*w^2 + 2*w - 1],\ [163, 163, w^4 + w^3 - 6*w^2 - 3*w + 7],\ [173, 173, -w^2 - 1],\ [181, 181, w^4 - 5*w^2 - 1],\ [181, 181, w^4 + w^3 - 5*w^2 - 5*w - 1],\ [197, 197, -2*w^4 + 10*w^2 - 3],\ [211, 211, w^4 + w^3 - 5*w^2 - 5*w + 5],\ [223, 223, w^4 + 2*w^3 - 7*w^2 - 6*w + 7],\ [223, 223, -w^4 - 2*w^3 + 7*w^2 + 8*w - 7],\ [227, 227, -w^4 - w^3 + 4*w^2 + 5*w - 1],\ [227, 227, -2*w^4 - w^3 + 12*w^2 + 7*w - 11],\ [229, 229, -w^4 + w^3 + 5*w^2 - 2*w - 3],\ [229, 229, w^4 - 6*w^2 - 2*w + 7],\ [233, 233, 3*w^4 - 17*w^2 - 3*w + 13],\ [239, 239, w^4 - w^3 - 4*w^2 + 4*w + 1],\ [241, 241, w^4 + w^3 - 5*w^2 - 3*w + 1],\ [257, 257, -w^4 + 5*w^2 + w + 1],\ [271, 271, -4*w^4 - w^3 + 24*w^2 + 7*w - 21],\ [277, 277, 2*w^4 + w^3 - 10*w^2 - 6*w + 7],\ [283, 283, -2*w^4 + 9*w^2 + 2*w - 3],\ [289, 17, 3*w^4 + w^3 - 17*w^2 - 6*w + 11],\ [293, 293, -w^4 - 2*w^3 + 6*w^2 + 10*w - 7],\ [293, 293, 2*w^4 + 2*w^3 - 12*w^2 - 11*w + 11],\ [293, 293, -6*w^4 - 3*w^3 + 34*w^2 + 17*w - 27],\ [307, 307, 3*w^4 - w^3 - 15*w^2 + 3*w + 3],\ [307, 307, -3*w^4 - w^3 + 19*w^2 + 6*w - 21],\ [307, 307, w^2 - 2*w - 5],\ [331, 331, 2*w^4 - 11*w^2 - 3*w + 7],\ [337, 337, -w^4 + 6*w^2 - w - 5],\ [337, 337, 2*w^4 + 2*w^3 - 12*w^2 - 9*w + 11],\ [347, 347, w^4 - w^3 - 4*w^2 + 3*w - 3],\ [347, 347, w^4 - w^3 - 6*w^2 + 3*w + 5],\ [359, 359, -2*w^4 - w^3 + 11*w^2 + 4*w - 7],\ [359, 359, -3*w^4 - 2*w^3 + 14*w^2 + 9*w - 5],\ [367, 367, w^4 + 2*w^3 - 6*w^2 - 8*w + 7],\ [367, 367, -3*w^4 - w^3 + 15*w^2 + 5*w - 7],\ [373, 373, w^4 + w^3 - 5*w^2 - 4*w - 1],\ [389, 389, w^4 - 2*w^3 - 4*w^2 + 7*w + 1],\ [389, 389, -w^4 + 7*w^2 - w - 9],\ [397, 397, w^3 - 2*w^2 - 6*w + 7],\ [401, 401, -w^4 + w^3 + 7*w^2 - 3*w - 9],\ [419, 419, 2*w^3 - 9*w + 1],\ [421, 421, 2*w^4 + w^3 - 11*w^2 - 4*w + 5],\ [421, 421, w^2 + w - 5],\ [431, 431, 3*w^4 + 3*w^3 - 18*w^2 - 13*w + 13],\ [431, 431, 2*w^2 + w - 5],\ [431, 431, 2*w^4 + 2*w^3 - 13*w^2 - 10*w + 13],\ [431, 431, w^3 + w^2 - 5*w - 3],\ [431, 431, -w^4 + 6*w^2 - w - 7],\ [443, 443, w^4 + 2*w^3 - 5*w^2 - 9*w + 5],\ [457, 457, -2*w^4 - w^3 + 10*w^2 + 7*w - 7],\ [457, 457, 2*w^4 - 11*w^2 - 3*w + 5],\ [463, 463, -3*w^4 - w^3 + 15*w^2 + 7*w - 7],\ [479, 479, 2*w^3 + w^2 - 7*w - 1],\ [487, 487, -4*w^4 - 2*w^3 + 22*w^2 + 13*w - 9],\ [491, 491, 2*w^4 + 2*w^3 - 11*w^2 - 10*w + 9],\ [499, 499, 3*w^4 - 17*w^2 - 3*w + 15],\ [509, 509, -w^4 + 7*w^2 - 7],\ [523, 523, 2*w^4 - w^3 - 9*w^2 + 5*w + 3],\ [541, 541, w^4 + w^3 - 6*w^2 - 7*w + 7],\ [541, 541, 3*w^4 + w^3 - 16*w^2 - 4*w + 11],\ [547, 547, -w^4 - w^3 + 5*w^2 + 3*w - 7],\ [547, 547, w^4 + w^3 - 7*w^2 - 6*w + 11],\ [547, 547, 6*w^4 + 4*w^3 - 35*w^2 - 21*w + 29],\ [557, 557, 6*w^4 + 2*w^3 - 34*w^2 - 13*w + 25],\ [557, 557, 5*w^4 + 4*w^3 - 30*w^2 - 21*w + 27],\ [569, 569, w^4 - w^3 - 3*w^2 + 4*w - 3],\ [577, 577, w^4 - w^3 - 4*w^2 + 2*w + 5],\ [587, 587, 2*w^4 + w^3 - 13*w^2 - 7*w + 9],\ [593, 593, 2*w^4 + 2*w^3 - 11*w^2 - 12*w + 9],\ [593, 593, w^4 + w^3 - 7*w^2 - 2*w + 7],\ [601, 601, 2*w^4 + w^3 - 9*w^2 - 7*w + 5],\ [601, 601, -4*w^4 - 2*w^3 + 24*w^2 + 11*w - 19],\ [607, 607, -2*w^4 - w^3 + 12*w^2 + 6*w - 7],\ [607, 607, -w^4 - 3*w^3 + 7*w^2 + 14*w - 11],\ [607, 607, w^4 + 2*w^3 - 5*w^2 - 8*w - 1],\ [613, 613, -4*w^4 - 2*w^3 + 22*w^2 + 13*w - 15],\ [617, 617, w^3 - 6*w - 1],\ [617, 617, 9*w^4 + 5*w^3 - 52*w^2 - 27*w + 43],\ [619, 619, 2*w^4 + 3*w^3 - 12*w^2 - 14*w + 13],\ [641, 641, 2*w^4 + w^3 - 11*w^2 - 8*w + 7],\ [641, 641, -2*w^4 - 2*w^3 + 14*w^2 + 10*w - 15],\ [643, 643, 2*w^3 - 3*w^2 - 9*w + 9],\ [643, 643, -3*w^4 + 16*w^2 - 11],\ [647, 647, -w^4 + w^3 + 6*w^2 - w - 7],\ [661, 661, -6*w^4 - 2*w^3 + 35*w^2 + 11*w - 31],\ [673, 673, w^4 - 2*w^3 - 3*w^2 + 5*w + 3],\ [701, 701, w^4 + w^3 - 8*w^2 - 6*w + 7],\ [709, 709, 6*w^4 + 3*w^3 - 33*w^2 - 17*w + 23],\ [709, 709, -3*w^4 - 2*w^3 + 17*w^2 + 11*w - 15],\ [719, 719, 2*w^2 - 7],\ [727, 727, 3*w^4 + 3*w^3 - 17*w^2 - 17*w + 7],\ [733, 733, 5*w^4 + 2*w^3 - 28*w^2 - 11*w + 19],\ [757, 757, -4*w^4 - 3*w^3 + 21*w^2 + 15*w - 13],\ [761, 761, -6*w^4 - 3*w^3 + 35*w^2 + 16*w - 27],\ [761, 761, 2*w^4 - 12*w^2 - w + 7],\ [761, 761, w^4 + 2*w^3 - 5*w^2 - 11*w - 1],\ [769, 769, -4*w^4 - 2*w^3 + 21*w^2 + 10*w - 13],\ [797, 797, -4*w^4 - 4*w^3 + 23*w^2 + 20*w - 17],\ [797, 797, 2*w^4 - 8*w^2 - w + 1],\ [809, 809, -w^4 - w^3 + 7*w^2 + 9*w - 3],\ [811, 811, 3*w^4 + w^3 - 18*w^2 - 5*w + 13],\ [823, 823, 3*w^4 + 2*w^3 - 19*w^2 - 10*w + 15],\ [823, 823, -2*w^4 - 3*w^3 + 11*w^2 + 13*w - 9],\ [827, 827, w^3 + w^2 - 6*w + 1],\ [827, 827, 2*w^4 + w^3 - 12*w^2 - 5*w + 7],\ [829, 829, -w^4 - 2*w^3 + 7*w^2 + 8*w - 11],\ [829, 829, 4*w^4 + w^3 - 22*w^2 - 8*w + 15],\ [829, 829, 3*w^4 + w^3 - 14*w^2 - 6*w + 5],\ [839, 839, -2*w^4 + 12*w^2 + 2*w - 7],\ [853, 853, -3*w^4 - 2*w^3 + 16*w^2 + 9*w - 7],\ [857, 857, 2*w^4 + w^3 - 9*w^2 - 6*w + 1],\ [857, 857, 2*w^4 + 2*w^3 - 11*w^2 - 7*w + 5],\ [859, 859, w^4 - 2*w^3 - 5*w^2 + 8*w + 3],\ [859, 859, w^4 + w^3 - 6*w^2 - 5*w + 9],\ [877, 877, -4*w^4 - 3*w^3 + 24*w^2 + 15*w - 19],\ [883, 883, -w^4 - w^3 + 8*w^2 + 4*w - 11],\ [907, 907, 3*w^4 + w^3 - 18*w^2 - 7*w + 13],\ [907, 907, -3*w^4 + 16*w^2 + 3*w - 9],\ [907, 907, -2*w^4 + 9*w^2 + 1],\ [911, 911, w^4 - w^3 - 3*w^2 + 4*w - 5],\ [911, 911, -4*w^4 - 3*w^3 + 21*w^2 + 14*w - 11],\ [919, 919, 4*w^4 + w^3 - 21*w^2 - 8*w + 11],\ [929, 929, w^3 + w^2 - 3*w - 5],\ [929, 929, -3*w^4 - 2*w^3 + 17*w^2 + 9*w - 11],\ [937, 937, 4*w^4 + w^3 - 21*w^2 - 9*w + 13],\ [941, 941, -w^4 + w^3 + 5*w^2 - 4*w - 5],\ [953, 953, 2*w^3 - 2*w^2 - 6*w - 1],\ [953, 953, 3*w^4 + 2*w^3 - 19*w^2 - 11*w + 15],\ [961, 31, -5*w^4 - 2*w^3 + 27*w^2 + 12*w - 17],\ [967, 967, -w^4 - w^3 + 8*w^2 + 5*w - 13],\ [967, 967, 4*w^4 + 3*w^3 - 23*w^2 - 14*w + 15],\ [977, 977, w^3 + w^2 - 5*w - 1],\ [977, 977, -3*w^4 - 3*w^3 + 16*w^2 + 13*w - 11],\ [991, 991, -w^4 + w^3 + 5*w^2 - 6*w - 1]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x K = QQ e = 1 hecke_eigenvalues_array = [1, -1, -3, -8, 3, -5, -3, 0, 10, 3, -3, 6, -9, 3, -8, -5, 7, 8, -3, 12, 6, 12, 11, 15, -6, 6, 12, 21, 0, 15, 18, -23, -4, -1, -2, 18, -11, 2, 24, 22, 8, 16, 0, 0, 14, 4, -18, -30, -14, 18, 16, -25, -26, 2, -6, -18, 24, 4, 26, 11, -28, 28, -16, -18, 12, 36, 36, -28, -22, -22, 0, -6, 1, -6, -15, 22, -13, 33, 18, -27, 0, -15, -6, -26, -17, -32, -6, 35, -42, -25, 0, -14, 20, -1, -26, -8, -19, 30, -30, -27, -22, -24, -6, 6, -40, -22, 8, -13, 32, -28, 6, -18, 1, -30, 3, -22, -19, -42, 22, 34, 33, 1, 2, -51, 44, 46, -20, 6, 33, -27, -26, 30, 0, 3, -16, 31, -32, -18, -30, -25, 20, -34, -45, 26, 36, 18, 4, 16, 22, 26, 55, 37, -43, -9, 27, -13, -21, -42, 44, 45, -6, -18, -52, 53, 29, -33, 12, 20] 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 AL_eigenvalues[ZF.ideal([3, 3, w^4 - 5*w^2 - w + 3])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]