/* 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, 1, 8, -1, -6, 0, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([32, 32, -w^4 + 5*w^2 - 4]) primes_array = [ [2, 2, -w],\ [13, 13, -w^2 + 3],\ [17, 17, -w^5 + w^4 + 5*w^3 - 3*w^2 - 5*w + 1],\ [23, 23, w^4 - w^3 - 4*w^2 + 2*w + 1],\ [31, 31, w^5 - w^4 - 6*w^3 + 5*w^2 + 7*w - 5],\ [32, 2, w^5 - 6*w^3 - w^2 + 8*w + 1],\ [43, 43, w^4 - 5*w^2 + 3],\ [47, 47, -w^4 + w^3 + 5*w^2 - 2*w - 5],\ [49, 7, -w^5 + w^4 + 5*w^3 - 3*w^2 - 4*w - 1],\ [53, 53, w^5 - 2*w^4 - 5*w^3 + 9*w^2 + 5*w - 7],\ [59, 59, w^5 - w^4 - 6*w^3 + 4*w^2 + 7*w - 3],\ [71, 71, w^5 - w^4 - 5*w^3 + 4*w^2 + 4*w - 5],\ [79, 79, w^3 - w^2 - 4*w + 1],\ [83, 83, w^5 - 6*w^3 - w^2 + 7*w - 1],\ [83, 83, 2*w^5 - w^4 - 11*w^3 + 2*w^2 + 12*w + 1],\ [89, 89, -w^5 + w^4 + 4*w^3 - 2*w^2 - w - 1],\ [89, 89, -w^5 + w^4 + 5*w^3 - 3*w^2 - 4*w + 3],\ [89, 89, w^5 + w^4 - 6*w^3 - 6*w^2 + 6*w + 3],\ [89, 89, 2*w^4 - w^3 - 9*w^2 + w + 5],\ [101, 101, w^5 - 5*w^3 - w^2 + 5*w - 1],\ [107, 107, w^4 - w^3 - 3*w^2 + 2*w - 1],\ [107, 107, -w^5 + w^4 + 6*w^3 - 3*w^2 - 7*w + 1],\ [107, 107, -w^5 + 4*w^3 + 2*w^2 + 1],\ [113, 113, 2*w^5 - w^4 - 10*w^3 + 2*w^2 + 8*w - 3],\ [121, 11, 2*w^5 - w^4 - 10*w^3 + 2*w^2 + 8*w - 1],\ [125, 5, -w^3 + 4*w - 1],\ [125, 5, -2*w^5 + w^4 + 10*w^3 - 3*w^2 - 9*w + 3],\ [131, 131, w^5 + 2*w^4 - 7*w^3 - 10*w^2 + 9*w + 5],\ [137, 137, w^5 - 2*w^4 - 5*w^3 + 9*w^2 + 6*w - 7],\ [137, 137, w^4 - 3*w^2 - 2*w + 1],\ [139, 139, -w^5 + w^4 + 5*w^3 - 4*w^2 - 5*w + 1],\ [139, 139, w^5 - w^4 - 5*w^3 + 4*w^2 + 3*w - 3],\ [149, 149, -w^5 + 2*w^4 + 4*w^3 - 9*w^2 - w + 7],\ [151, 151, w^4 - 6*w^2 - w + 5],\ [163, 163, -w^5 + w^4 + 4*w^3 - 3*w^2 - w + 3],\ [167, 167, -w^5 + w^4 + 5*w^3 - 2*w^2 - 6*w + 1],\ [169, 13, -2*w^5 + 12*w^3 + w^2 - 13*w + 1],\ [179, 179, -w^5 + 6*w^3 + 2*w^2 - 9*w - 3],\ [179, 179, -w^3 + 5*w - 1],\ [191, 191, w^4 - 3*w^2 - 1],\ [193, 193, -w^5 - 3*w^4 + 6*w^3 + 17*w^2 - 5*w - 15],\ [197, 197, 2*w^5 - 11*w^3 - 2*w^2 + 10*w + 1],\ [197, 197, w^5 - 5*w^3 - 2*w^2 + 3*w - 1],\ [227, 227, w^5 - w^4 - 5*w^3 + 5*w^2 + 4*w - 5],\ [229, 229, -2*w^4 + w^3 + 10*w^2 - w - 9],\ [241, 241, 3*w^5 - 2*w^4 - 16*w^3 + 6*w^2 + 17*w - 3],\ [263, 263, -2*w^5 + 11*w^3 + w^2 - 11*w - 1],\ [269, 269, -w^5 + w^4 + 5*w^3 - 4*w^2 - 7*w + 3],\ [277, 277, 2*w^5 - w^4 - 10*w^3 + w^2 + 7*w + 1],\ [277, 277, 2*w^5 - 2*w^4 - 10*w^3 + 7*w^2 + 11*w - 5],\ [283, 283, -w^4 + w^3 + 4*w^2 - 3],\ [289, 17, -w^5 - w^4 + 7*w^3 + 4*w^2 - 10*w - 1],\ [293, 293, -3*w^5 + 16*w^3 + 3*w^2 - 15*w - 1],\ [293, 293, -2*w^5 - w^4 + 10*w^3 + 7*w^2 - 6*w - 5],\ [307, 307, -2*w^5 + w^4 + 9*w^3 - w^2 - 4*w - 1],\ [311, 311, w^4 - w^3 - 5*w^2 + 4*w + 1],\ [311, 311, -3*w^5 + 17*w^3 + 4*w^2 - 18*w - 7],\ [311, 311, -w^5 - w^4 + 5*w^3 + 6*w^2 - 2*w - 5],\ [311, 311, 2*w^4 - w^3 - 10*w^2 + w + 5],\ [313, 313, w^5 - 5*w^3 - 3*w^2 + 3*w + 3],\ [313, 313, -w^5 - w^4 + 5*w^3 + 7*w^2 - 3*w - 3],\ [317, 317, -w^5 + 2*w^4 + 3*w^3 - 8*w^2 + w + 5],\ [337, 337, 3*w^5 + w^4 - 16*w^3 - 10*w^2 + 14*w + 11],\ [337, 337, -w^5 - 2*w^4 + 6*w^3 + 10*w^2 - 6*w - 7],\ [337, 337, 2*w^5 + w^4 - 11*w^3 - 7*w^2 + 10*w + 3],\ [337, 337, w^4 - w^3 - 6*w^2 + 4*w + 7],\ [347, 347, -2*w^5 + 2*w^4 + 9*w^3 - 7*w^2 - 7*w + 3],\ [347, 347, -w^5 - w^4 + 5*w^3 + 7*w^2 - 2*w - 7],\ [347, 347, -w^5 - w^4 + 6*w^3 + 7*w^2 - 5*w - 7],\ [349, 349, 3*w^5 - w^4 - 17*w^3 + 3*w^2 + 19*w - 3],\ [349, 349, w^4 + w^3 - 6*w^2 - 4*w + 7],\ [359, 359, -2*w^5 + w^4 + 11*w^3 - 3*w^2 - 13*w + 1],\ [373, 373, -2*w^5 + 10*w^3 + 2*w^2 - 9*w - 1],\ [379, 379, -w^5 + 2*w^4 + 4*w^3 - 9*w^2 - w + 9],\ [389, 389, -2*w^5 + w^4 + 11*w^3 - 2*w^2 - 13*w - 1],\ [397, 397, w^5 - 3*w^4 - 5*w^3 + 14*w^2 + 7*w - 11],\ [419, 419, w^5 - 6*w^3 + w^2 + 7*w - 1],\ [419, 419, -w^5 + w^4 + 3*w^3 - 2*w^2 + w - 1],\ [433, 433, -w^5 + w^4 + 4*w^3 - 4*w^2 - w + 5],\ [433, 433, w^5 + w^4 - 7*w^3 - 4*w^2 + 9*w - 1],\ [439, 439, w^4 - 2*w^3 - 4*w^2 + 7*w + 3],\ [443, 443, -w^3 + w^2 + 3*w - 5],\ [443, 443, w^5 - w^4 - 5*w^3 + 2*w^2 + 5*w + 3],\ [449, 449, 2*w^5 - 2*w^4 - 11*w^3 + 8*w^2 + 12*w - 5],\ [457, 457, -2*w^5 - 2*w^4 + 11*w^3 + 13*w^2 - 10*w - 11],\ [461, 461, w^5 + 2*w^4 - 6*w^3 - 10*w^2 + 5*w + 7],\ [479, 479, -2*w^5 + 10*w^3 + w^2 - 6*w + 1],\ [479, 479, w^5 - w^4 - 6*w^3 + 3*w^2 + 9*w - 1],\ [491, 491, w^5 - w^4 - 4*w^3 + w^2 + w + 3],\ [499, 499, -w^3 + 5*w - 3],\ [499, 499, w^5 - 2*w^4 - 5*w^3 + 7*w^2 + 6*w - 1],\ [509, 509, -2*w^5 + 2*w^4 + 10*w^3 - 9*w^2 - 9*w + 11],\ [521, 521, -w^5 + 4*w^3 + 2*w^2 - w - 3],\ [521, 521, 2*w^5 + 2*w^4 - 11*w^3 - 13*w^2 + 9*w + 13],\ [521, 521, -w^4 + 2*w^3 + 4*w^2 - 5*w - 3],\ [523, 523, -w^5 + w^4 + 6*w^3 - 5*w^2 - 8*w + 3],\ [523, 523, w^5 + w^4 - 5*w^3 - 8*w^2 + 3*w + 7],\ [541, 541, w^5 + 3*w^4 - 7*w^3 - 16*w^2 + 8*w + 13],\ [547, 547, w^5 - 2*w^4 - 5*w^3 + 7*w^2 + 7*w - 3],\ [557, 557, 3*w^5 - w^4 - 15*w^3 + w^2 + 11*w - 1],\ [557, 557, 2*w^5 - w^4 - 11*w^3 + 2*w^2 + 11*w + 1],\ [563, 563, w^5 - 3*w^4 - 4*w^3 + 13*w^2 + 3*w - 9],\ [569, 569, -w^5 - w^4 + 5*w^3 + 6*w^2 - 4*w - 1],\ [571, 571, -3*w^5 + 2*w^4 + 15*w^3 - 7*w^2 - 10*w + 3],\ [577, 577, 2*w^5 - 11*w^3 - 2*w^2 + 9*w + 3],\ [587, 587, w^4 - 6*w^2 - 2*w + 7],\ [599, 599, -2*w^5 + 2*w^4 + 10*w^3 - 7*w^2 - 10*w + 3],\ [599, 599, w^5 - 2*w^4 - 4*w^3 + 9*w^2 + 3*w - 7],\ [601, 601, 2*w^2 - 5],\ [607, 607, -w^5 - w^4 + 4*w^3 + 7*w^2 - 5],\ [613, 613, -w^5 + 5*w^3 + w^2 - 3*w + 3],\ [617, 617, -2*w^4 + 11*w^2 + 2*w - 9],\ [619, 619, -2*w^5 + 10*w^3 + w^2 - 7*w + 1],\ [619, 619, -2*w^5 + 4*w^4 + 8*w^3 - 15*w^2 - 5*w + 7],\ [643, 643, -2*w^5 - w^4 + 12*w^3 + 6*w^2 - 12*w - 3],\ [647, 647, 3*w^5 - w^4 - 18*w^3 + w^2 + 23*w + 3],\ [647, 647, 2*w^5 + w^4 - 10*w^3 - 9*w^2 + 9*w + 9],\ [653, 653, -w^4 + 7*w^2 + w - 7],\ [673, 673, -2*w^4 + w^3 + 8*w^2 - w - 5],\ [673, 673, -2*w^5 + 11*w^3 + w^2 - 9*w + 1],\ [677, 677, 4*w^5 - w^4 - 20*w^3 + w^2 + 15*w - 3],\ [683, 683, 3*w^5 - 2*w^4 - 14*w^3 + 6*w^2 + 11*w - 5],\ [683, 683, 2*w^5 - w^4 - 11*w^3 + 3*w^2 + 10*w - 3],\ [683, 683, -w^5 + 5*w^3 + w^2 - 4*w + 3],\ [691, 691, w^5 + w^4 - 6*w^3 - 6*w^2 + 8*w + 3],\ [709, 709, -2*w^5 + 2*w^4 + 11*w^3 - 7*w^2 - 13*w + 3],\ [719, 719, 2*w^5 - 2*w^4 - 10*w^3 + 8*w^2 + 9*w - 9],\ [719, 719, -2*w^5 + w^4 + 11*w^3 - 4*w^2 - 12*w + 3],\ [727, 727, w^5 - w^4 - 6*w^3 + 4*w^2 + 7*w - 1],\ [729, 3, -3],\ [739, 739, 2*w^4 - w^3 - 9*w^2 + 5],\ [739, 739, -w^5 + w^4 + 4*w^3 - 3*w^2 - 3*w - 1],\ [751, 751, -2*w^4 + w^3 + 9*w^2 - 3*w - 3],\ [761, 761, -w^5 - 2*w^4 + 7*w^3 + 12*w^2 - 9*w - 13],\ [773, 773, -w^5 + 5*w^3 + 2*w^2 - 2*w - 3],\ [787, 787, 2*w^5 - 12*w^3 - 2*w^2 + 13*w + 3],\ [787, 787, -2*w^5 + w^4 + 10*w^3 - 3*w^2 - 10*w + 1],\ [797, 797, -w^5 + 2*w^4 + 5*w^3 - 8*w^2 - 7*w + 3],\ [811, 811, -2*w^5 + 12*w^3 + w^2 - 15*w - 1],\ [821, 821, 4*w^5 + w^4 - 22*w^3 - 8*w^2 + 21*w + 3],\ [821, 821, w^5 - 2*w^4 - 3*w^3 + 8*w^2 - 3*w - 3],\ [821, 821, -w^5 + 7*w^3 - 9*w - 1],\ [827, 827, w^5 - 7*w^3 - w^2 + 10*w + 1],\ [829, 829, -2*w^4 + 10*w^2 + w - 7],\ [839, 839, -w^5 + 2*w^4 + 6*w^3 - 10*w^2 - 8*w + 9],\ [853, 853, -w^5 + w^4 + 3*w^3 - 3*w^2 + 2*w + 1],\ [859, 859, -w^5 + 6*w^3 - 9*w + 1],\ [859, 859, -2*w^5 + 3*w^4 + 8*w^3 - 10*w^2 - 6*w + 5],\ [859, 859, w^5 - 2*w^4 - 4*w^3 + 8*w^2 + 2*w - 7],\ [859, 859, 2*w^5 - 2*w^4 - 10*w^3 + 6*w^2 + 10*w - 3],\ [863, 863, 3*w^5 - 4*w^4 - 14*w^3 + 14*w^2 + 13*w - 7],\ [877, 877, w^3 + w^2 - 4*w - 1],\ [881, 881, -w^5 + w^4 + 4*w^3 - 4*w^2 - 2*w + 5],\ [881, 881, -2*w^5 + 10*w^3 + 4*w^2 - 7*w - 5],\ [907, 907, -w^4 + w^3 + 4*w^2 - w + 1],\ [937, 937, w^5 - 7*w^3 + w^2 + 11*w - 3],\ [941, 941, -2*w^5 + w^4 + 11*w^3 - w^2 - 12*w - 1],\ [947, 947, -w^5 + 5*w^3 + 3*w^2 - 5*w - 7],\ [953, 953, 2*w^5 - w^4 - 11*w^3 + 4*w^2 + 10*w - 7],\ [971, 971, w^5 - w^4 - 5*w^3 + 3*w^2 + 7*w + 1],\ [983, 983, -2*w^5 + 9*w^3 + 3*w^2 - 5*w - 3],\ [997, 997, 3*w^5 - 18*w^3 - 2*w^2 + 22*w + 1]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x K = QQ e = 1 hecke_eigenvalues_array = [0, 2, 6, -8, 4, -9, -4, -8, 6, 2, 8, -8, -8, 4, -4, -14, 10, -2, 2, -10, 0, 0, 4, -2, -2, -2, -18, 20, 6, -6, -4, 12, -22, -20, 4, -16, 14, -12, -20, -8, -2, -10, -18, 4, -10, -18, 24, 18, -2, -30, 0, 10, 18, -6, -4, -12, 12, 24, 0, -18, -10, -6, 2, -2, -18, -22, -36, -12, -4, -22, -10, 28, 22, 24, 10, 30, 0, 16, 22, 10, 16, -20, -20, -10, 10, -30, 0, 24, -12, -24, -16, -30, -14, 30, -30, 28, -44, -10, -8, -46, 30, -32, 2, -16, -22, 16, 20, 36, -2, 16, 6, -46, -4, -20, -16, 0, -32, 34, 2, 14, 6, -36, 24, 12, 20, 38, 24, -40, -32, 10, -20, 12, 8, 6, -10, 28, -4, 42, 4, -42, 14, 30, 44, 18, 12, 22, -8, 28, -20, 4, 8, 38, -18, 2, -28, 30, -54, 20, 42, -36, 32, -14] 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 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]