/* 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, 4, 3, -6, -1, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([16, 2, w^4 - w^3 - 6*w^2 + 3*w + 4]) primes_array = [ [2, 2, w],\ [7, 7, -2*w^4 + w^3 + 12*w^2 + w - 5],\ [8, 2, w^4 - 7*w^2 - 3*w + 5],\ [19, 19, -2*w^4 + w^3 + 12*w^2 - 5],\ [19, 19, -w^3 + w^2 + 5*w - 1],\ [29, 29, 2*w^4 - 2*w^3 - 11*w^2 + 4*w + 3],\ [31, 31, -2*w^4 + w^3 + 12*w^2 + 2*w - 5],\ [37, 37, -2*w^4 + 13*w^2 + 5*w - 7],\ [53, 53, 3*w^4 - 2*w^3 - 18*w^2 + 3*w + 9],\ [59, 59, 2*w^4 - 13*w^2 - 6*w + 5],\ [61, 61, -3*w^4 + 2*w^3 + 18*w^2 - 2*w - 11],\ [61, 61, w^2 - w - 3],\ [61, 61, 6*w^4 - 2*w^3 - 38*w^2 - 6*w + 23],\ [67, 67, -w^4 + 7*w^2 + 4*w - 5],\ [67, 67, -w^4 + 6*w^2 + 2*w - 1],\ [71, 71, -w^2 + 3],\ [73, 73, 3*w^4 - w^3 - 19*w^2 - 3*w + 9],\ [79, 79, 3*w^4 - w^3 - 19*w^2 - 4*w + 11],\ [83, 83, -w^4 - w^3 + 7*w^2 + 8*w - 3],\ [97, 97, -2*w^4 + w^3 + 12*w^2 + 2*w - 3],\ [97, 97, w^4 - w^3 - 5*w^2 + 2*w + 3],\ [103, 103, -4*w^4 + w^3 + 25*w^2 + 7*w - 13],\ [103, 103, 7*w^4 - 3*w^3 - 43*w^2 - 4*w + 23],\ [107, 107, 4*w^4 - 2*w^3 - 24*w^2 - w + 9],\ [113, 113, 5*w^4 - 3*w^3 - 30*w^2 + w + 15],\ [137, 137, -3*w^4 + 2*w^3 + 17*w^2 - 7],\ [139, 139, w^2 - 5],\ [149, 149, 4*w^4 - w^3 - 26*w^2 - 5*w + 17],\ [151, 151, -4*w^4 + 2*w^3 + 24*w^2 + w - 13],\ [157, 157, -4*w^4 + w^3 + 26*w^2 + 5*w - 15],\ [157, 157, 5*w^4 - 2*w^3 - 31*w^2 - 4*w + 15],\ [157, 157, -3*w^4 + 2*w^3 + 19*w^2 - 2*w - 15],\ [169, 13, -w^4 + 2*w^3 + 5*w^2 - 8*w - 3],\ [173, 173, w^4 - w^3 - 5*w^2 + w - 1],\ [179, 179, -2*w^4 + 2*w^3 + 12*w^2 - 5*w - 7],\ [191, 191, -4*w^4 + 2*w^3 + 24*w^2 + 2*w - 11],\ [193, 193, 2*w^4 - 2*w^3 - 11*w^2 + 5*w + 5],\ [197, 197, w^4 - 6*w^2 - w + 1],\ [197, 197, -w^4 + 8*w^2 + 2*w - 11],\ [199, 199, -3*w^4 + w^3 + 20*w^2 + 3*w - 17],\ [211, 211, 4*w^4 - 2*w^3 - 25*w^2 + 13],\ [223, 223, w^4 - 6*w^2 - 3*w - 1],\ [223, 223, 2*w^4 - 13*w^2 - 4*w + 9],\ [223, 223, -2*w^4 + 2*w^3 + 11*w^2 - 3*w - 3],\ [227, 227, 2*w^4 - 12*w^2 - 7*w + 5],\ [227, 227, -w^4 + 8*w^2 + w - 7],\ [229, 229, -2*w^4 - w^3 + 14*w^2 + 11*w - 11],\ [229, 229, -3*w^4 + 19*w^2 + 8*w - 11],\ [233, 233, -w^4 + 8*w^2 + 2*w - 7],\ [233, 233, 3*w^4 - w^3 - 20*w^2 - w + 11],\ [239, 239, -w^4 + 6*w^2 + 5*w - 3],\ [243, 3, -3],\ [251, 251, -w^4 + w^3 + 5*w^2 - 3*w - 3],\ [263, 263, 3*w^4 - 2*w^3 - 18*w^2 + 2*w + 7],\ [269, 269, -3*w^4 + 20*w^2 + 8*w - 11],\ [271, 271, 4*w^4 - 2*w^3 - 25*w^2 - w + 13],\ [283, 283, -w^2 + 3*w + 1],\ [283, 283, 6*w^4 - 4*w^3 - 36*w^2 + 3*w + 19],\ [283, 283, -6*w^4 + 3*w^3 + 36*w^2 + 2*w - 17],\ [289, 17, -8*w^4 + 2*w^3 + 49*w^2 + 12*w - 21],\ [293, 293, -3*w^4 + 2*w^3 + 17*w^2 - 2*w - 7],\ [293, 293, 2*w^4 - 2*w^3 - 12*w^2 + 4*w + 7],\ [307, 307, -2*w^4 + 13*w^2 + 7*w - 9],\ [311, 311, -w^4 + 5*w^2 + 4*w + 3],\ [311, 311, 3*w^4 - w^3 - 18*w^2 - 2*w + 7],\ [317, 317, -w^4 + w^3 + 6*w^2 - 4*w - 3],\ [337, 337, 4*w^4 - w^3 - 25*w^2 - 7*w + 15],\ [347, 347, -2*w^4 + w^3 + 13*w^2 + w - 11],\ [347, 347, -3*w^4 + w^3 + 20*w^2 + 2*w - 17],\ [349, 349, -w^3 + w^2 + 3*w + 3],\ [359, 359, 3*w^4 - w^3 - 18*w^2 - 3*w + 11],\ [367, 367, -w^4 - w^3 + 8*w^2 + 6*w - 7],\ [367, 367, 4*w^4 - 3*w^3 - 24*w^2 + 3*w + 11],\ [373, 373, -w^3 + 6*w + 1],\ [379, 379, 2*w^4 - 14*w^2 - 5*w + 9],\ [397, 397, -2*w^4 - w^3 + 15*w^2 + 11*w - 13],\ [397, 397, w^4 - 6*w^2 - 2*w - 1],\ [397, 397, -w^4 - w^3 + 8*w^2 + 8*w - 5],\ [401, 401, -2*w^4 + 14*w^2 + 7*w - 13],\ [409, 409, -4*w^4 + 2*w^3 + 24*w^2 - 11],\ [421, 421, 3*w^4 - 2*w^3 - 18*w^2 + 9],\ [433, 433, -2*w^4 + 2*w^3 + 12*w^2 - 5*w - 5],\ [433, 433, 6*w^4 - w^3 - 38*w^2 - 12*w + 21],\ [439, 439, -5*w^4 + 2*w^3 + 30*w^2 + 3*w - 15],\ [443, 443, 4*w^4 - 2*w^3 - 26*w^2 - w + 15],\ [443, 443, -w^4 + w^3 + 5*w^2 - w - 5],\ [443, 443, -w^4 + w^3 + 6*w^2 - 4*w - 5],\ [443, 443, -w^4 + 6*w^2 + 5*w - 1],\ [443, 443, w^4 + w^3 - 7*w^2 - 7*w + 1],\ [461, 461, 2*w^4 - 14*w^2 - 6*w + 9],\ [461, 461, -w^4 + w^3 + 4*w^2 - 2*w + 5],\ [463, 463, w^4 - 7*w^2 - 2*w + 1],\ [479, 479, -6*w^4 + 3*w^3 + 37*w^2 + w - 19],\ [499, 499, -4*w^4 + 25*w^2 + 11*w - 13],\ [499, 499, 5*w^4 - 3*w^3 - 31*w^2 + 17],\ [503, 503, -4*w^4 + 3*w^3 + 23*w^2 - 3*w - 7],\ [503, 503, -3*w^4 + w^3 + 18*w^2 + w - 9],\ [503, 503, 2*w^4 - w^3 - 14*w^2 - w + 11],\ [509, 509, 8*w^4 - 2*w^3 - 52*w^2 - 11*w + 35],\ [521, 521, -2*w^4 + 2*w^3 + 13*w^2 - 5*w - 11],\ [521, 521, 10*w^4 - 4*w^3 - 63*w^2 - 6*w + 39],\ [529, 23, -w^4 - w^3 + 7*w^2 + 7*w - 3],\ [547, 547, 9*w^4 - 5*w^3 - 55*w^2 + w + 29],\ [547, 547, -w^4 + 2*w^3 + 6*w^2 - 9*w - 7],\ [557, 557, -5*w^4 + w^3 + 32*w^2 + 6*w - 17],\ [563, 563, 4*w^4 - w^3 - 26*w^2 - 4*w + 19],\ [587, 587, -2*w^3 + 3*w^2 + 10*w - 5],\ [587, 587, 7*w^4 - w^3 - 45*w^2 - 15*w + 25],\ [593, 593, -9*w^4 + 3*w^3 + 57*w^2 + 10*w - 33],\ [599, 599, w^4 + w^3 - 6*w^2 - 7*w + 3],\ [601, 601, w^4 - 2*w^3 - 4*w^2 + 9*w - 1],\ [601, 601, 3*w^4 - 3*w^3 - 16*w^2 + 5*w + 3],\ [607, 607, 6*w^4 - 2*w^3 - 38*w^2 - 5*w + 23],\ [607, 607, 4*w^4 - 2*w^3 - 24*w^2 - 3*w + 11],\ [613, 613, -w^4 + w^3 + 7*w^2 - 3*w - 5],\ [641, 641, 2*w^4 - 2*w^3 - 11*w^2 + 4*w + 7],\ [641, 641, -w^3 + 6*w - 1],\ [647, 647, 7*w^4 - 3*w^3 - 45*w^2 - 2*w + 31],\ [659, 659, -3*w^4 + w^3 + 19*w^2 + w - 9],\ [661, 661, -2*w^4 + 15*w^2 + 5*w - 13],\ [683, 683, -w^4 + 8*w^2 + 2*w - 9],\ [683, 683, -4*w^4 + 3*w^3 + 24*w^2 - 4*w - 17],\ [683, 683, 6*w^4 - 3*w^3 - 36*w^2 - w + 15],\ [691, 691, -3*w^4 + w^3 + 18*w^2 + 4*w - 11],\ [709, 709, w^4 - w^3 - 4*w^2 + 4*w - 3],\ [733, 733, -5*w^4 + 2*w^3 + 30*w^2 + 2*w - 15],\ [743, 743, -9*w^4 + 5*w^3 + 55*w^2 - 31],\ [751, 751, -4*w^4 + 2*w^3 + 23*w^2 + 2*w - 9],\ [751, 751, 5*w^4 - w^3 - 32*w^2 - 8*w + 19],\ [761, 761, w^4 + w^3 - 7*w^2 - 6*w + 3],\ [761, 761, -6*w^4 + w^3 + 38*w^2 + 11*w - 19],\ [787, 787, -w^3 + w^2 + 3*w - 5],\ [787, 787, -6*w^4 + 3*w^3 + 35*w^2 + 3*w - 13],\ [809, 809, -7*w^4 + 5*w^3 + 42*w^2 - 6*w - 23],\ [821, 821, 3*w^4 - 2*w^3 - 18*w^2 + 2*w + 5],\ [827, 827, 7*w^4 - 3*w^3 - 43*w^2 - 3*w + 23],\ [829, 829, 4*w^4 - w^3 - 26*w^2 - 7*w + 19],\ [829, 829, 5*w^4 - 2*w^3 - 30*w^2 - 4*w + 17],\ [841, 29, w^4 - w^3 - 5*w^2 - 1],\ [841, 29, 6*w^4 - 2*w^3 - 38*w^2 - 6*w + 19],\ [853, 853, 3*w^4 - 20*w^2 - 7*w + 13],\ [857, 857, 2*w^3 - 3*w^2 - 8*w + 3],\ [859, 859, -5*w^4 + 2*w^3 + 33*w^2 + 2*w - 27],\ [863, 863, -3*w^4 + 2*w^3 + 17*w^2 - 3],\ [881, 881, -5*w^4 + w^3 + 32*w^2 + 10*w - 21],\ [883, 883, w^2 - 2*w - 5],\ [887, 887, 3*w^4 + w^3 - 21*w^2 - 12*w + 13],\ [911, 911, -4*w^4 + w^3 + 26*w^2 + 7*w - 17],\ [911, 911, 6*w^4 - 2*w^3 - 37*w^2 - 5*w + 19],\ [919, 919, 14*w^4 - 3*w^3 - 86*w^2 - 26*w + 37],\ [919, 919, -7*w^4 + 4*w^3 + 42*w^2 - 21],\ [937, 937, 3*w^4 - 2*w^3 - 19*w^2 + 4*w + 13],\ [947, 947, 6*w^4 - 3*w^3 - 36*w^2 - 2*w + 19],\ [953, 953, w^4 - 2*w^3 - 3*w^2 + 6*w - 5],\ [953, 953, -9*w^4 + 3*w^3 + 56*w^2 + 10*w - 31],\ [977, 977, -5*w^4 + w^3 + 33*w^2 + 7*w - 23],\ [977, 977, -5*w^4 + 2*w^3 + 31*w^2 + 2*w - 13],\ [997, 997, 6*w^4 - w^3 - 38*w^2 - 13*w + 21],\ [997, 997, 4*w^4 - 2*w^3 - 24*w^2 - 4*w + 13]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x K = QQ e = 1 hecke_eigenvalues_array = [1, 2, -1, -4, -4, -6, 8, -4, 0, 6, 2, 2, 2, 8, 14, 0, 14, 2, -6, -10, 2, 8, 8, 0, -12, 12, 8, 6, 20, -10, -10, -10, 8, -6, 0, 12, 8, 0, 18, -4, -4, 8, 8, 8, 12, 12, 8, -10, 18, -18, 6, -8, 0, -6, 0, 14, -10, -28, 26, -34, 24, -30, -16, 24, 6, 12, 14, 24, -12, 2, 0, -16, -10, 26, -22, -22, -34, 20, 6, 26, 2, -4, 2, 8, -24, -24, 12, -24, -6, 30, 30, 8, 30, 44, -28, 0, -36, 0, -30, 18, 18, 38, 2, -16, -18, -12, 12, -42, 18, -30, 2, -46, 14, 8, -34, -6, 30, -18, 48, -10, 36, -36, -36, 20, -46, -10, 24, -10, 26, 6, -30, -28, 8, -18, 48, -18, 20, 20, 26, 44, 56, 30, -4, 18, -18, 50, 24, 12, 12, -22, 32, -40, 12, -54, -36, 6, -30, 38, 2] 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([8, 2, w^4 - 7*w^2 - 3*w + 5])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]