/* 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([1, -2, -5, 0, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([43, 43, -w - 3]) primes_array = [ [4, 2, -w^3 + 4*w + 1],\ [5, 5, w - 1],\ [7, 7, -w^2 + w + 2],\ [13, 13, -w^3 + w^2 + 4*w],\ [13, 13, -w^2 + 3],\ [17, 17, -w^2 + 2],\ [19, 19, -w^3 + 5*w],\ [31, 31, -w^2 + 2*w + 3],\ [37, 37, -2*w^3 + w^2 + 8*w - 1],\ [43, 43, -w - 3],\ [53, 53, -w^3 + 2*w^2 + 3*w - 2],\ [53, 53, w^3 - 6*w - 2],\ [59, 59, 2*w^3 - w^2 - 10*w - 2],\ [61, 61, 2*w^3 - w^2 - 10*w],\ [61, 61, 2*w^3 - w^2 - 8*w],\ [71, 71, 2*w^3 - 9*w - 2],\ [73, 73, -w^3 - w^2 + 6*w + 3],\ [81, 3, -3],\ [83, 83, -w^3 + 2*w^2 + 3*w - 3],\ [101, 101, 2*w^3 - 8*w - 3],\ [125, 5, w^3 + w^2 - 4*w - 6],\ [127, 127, 2*w^3 - w^2 - 7*w - 2],\ [127, 127, w^3 - 7*w - 7],\ [131, 131, w^3 - 3*w - 4],\ [131, 131, 3*w^3 - 13*w - 7],\ [137, 137, w^2 + w - 4],\ [137, 137, -w^3 + 2*w^2 + 4*w - 4],\ [149, 149, 3*w + 5],\ [157, 157, -2*w^3 - w^2 + 7*w + 2],\ [157, 157, -2*w^3 + 2*w^2 + 9*w - 1],\ [163, 163, -2*w^3 + w^2 + 7*w],\ [163, 163, -3*w^3 + w^2 + 15*w + 3],\ [163, 163, 2*w^3 - 7*w - 1],\ [163, 163, 2*w^3 - 11*w - 5],\ [167, 167, -2*w^3 + w^2 + 9*w - 4],\ [167, 167, w^3 - w^2 - 7*w + 1],\ [167, 167, w^3 - w^2 - 7*w - 1],\ [167, 167, w^2 - 2*w - 5],\ [169, 13, w^3 + w^2 - 6*w - 2],\ [173, 173, -3*w^3 + 13*w + 3],\ [181, 181, 2*w^2 + w - 4],\ [191, 191, 2*w^3 - 10*w - 1],\ [191, 191, -2*w^3 + 2*w^2 + 6*w - 3],\ [193, 193, 3*w^3 - 13*w - 6],\ [197, 197, -w^3 + w^2 + 2*w - 3],\ [197, 197, w^3 + w^2 - 7*w - 7],\ [199, 199, -w^2 + 2*w + 6],\ [223, 223, -2*w^3 + w^2 + 11*w + 2],\ [227, 227, -3*w^3 + 2*w^2 + 15*w - 1],\ [229, 229, -w^3 + w^2 + 6*w - 5],\ [233, 233, 2*w^3 - 11*w - 4],\ [239, 239, -2*w^3 + 11*w + 3],\ [251, 251, 2*w^3 - w^2 - 10*w - 4],\ [257, 257, -2*w^3 + w^2 + 6*w + 1],\ [257, 257, w^3 - 7*w - 2],\ [263, 263, 2*w^3 - 9*w],\ [269, 269, w^3 - 4*w - 6],\ [269, 269, -2*w^3 + 8*w - 3],\ [271, 271, 2*w^2 - w - 5],\ [277, 277, -3*w^3 + 2*w^2 + 11*w - 2],\ [277, 277, 3*w^3 - w^2 - 15*w - 5],\ [281, 281, 2*w^2 - 5],\ [283, 283, -3*w^3 + 2*w^2 + 12*w + 2],\ [283, 283, -2*w^3 + w^2 + 6*w + 6],\ [293, 293, -2*w^3 - w^2 + 12*w + 7],\ [293, 293, -2*w^3 + 9*w - 1],\ [311, 311, -3*w^3 + w^2 + 12*w + 4],\ [313, 313, -w^3 + w^2 + 2*w - 4],\ [313, 313, -4*w^3 + 2*w^2 + 16*w - 3],\ [331, 331, 3*w^3 - w^2 - 16*w - 5],\ [331, 331, 2*w^3 - 2*w^2 - 8*w - 1],\ [343, 7, 2*w^3 - 3*w^2 - 9*w + 6],\ [359, 359, -2*w^3 + 10*w - 1],\ [367, 367, -w^3 + 2*w^2 + w - 4],\ [367, 367, 3*w^3 - 11*w - 1],\ [373, 373, -w^3 + w^2 + 2*w - 5],\ [373, 373, 2*w^3 - 2*w^2 - 11*w - 1],\ [389, 389, -2*w^3 + 3*w^2 + 8*w - 6],\ [409, 409, 3*w^3 - 2*w^2 - 13*w - 1],\ [419, 419, 3*w^3 - w^2 - 12*w - 1],\ [419, 419, w^3 - w^2 - 4*w - 4],\ [439, 439, w^2 + 2*w - 4],\ [439, 439, 3*w^3 + w^2 - 11*w - 5],\ [443, 443, 2*w^3 + 2*w^2 - 7*w - 8],\ [449, 449, 4*w^3 - w^2 - 16*w - 8],\ [457, 457, 2*w^3 + w^2 - 11*w - 4],\ [461, 461, -5*w^3 + 2*w^2 + 24*w + 2],\ [463, 463, -w^3 + 3*w^2 + 6*w - 5],\ [467, 467, w^3 - 3*w^2 - 6*w + 4],\ [467, 467, 3*w^3 - w^2 - 12*w - 2],\ [479, 479, -w^3 + w^2 + 4*w - 6],\ [479, 479, -w^3 + 2*w + 8],\ [491, 491, -w^3 + 2*w^2 + 2*w - 6],\ [491, 491, w - 5],\ [499, 499, 2*w^3 + w^2 - 10*w - 4],\ [509, 509, -4*w^3 + w^2 + 20*w + 5],\ [521, 521, w^3 - w + 6],\ [523, 523, -4*w^3 + 18*w + 5],\ [523, 523, -2*w^3 - 2*w^2 + 9*w + 12],\ [557, 557, w^2 - 3*w - 6],\ [571, 571, 3*w^3 - 11*w - 8],\ [577, 577, 2*w^3 + 2*w^2 - 11*w - 7],\ [577, 577, -3*w^3 + 4*w^2 + 11*w - 6],\ [587, 587, -w^3 - w^2 + 7*w + 11],\ [593, 593, w^3 + w^2 - 8*w - 8],\ [593, 593, -2*w^3 + 4*w^2 + 6*w - 9],\ [599, 599, 4*w^3 - 17*w - 9],\ [599, 599, w^3 + w^2 - 6*w - 10],\ [601, 601, 3*w^2 - 10],\ [607, 607, 4*w^3 - w^2 - 18*w - 8],\ [619, 619, w^2 - w - 8],\ [641, 641, -w^3 + 3*w^2 + 2*w - 10],\ [643, 643, 3*w^3 - w^2 - 16*w + 1],\ [653, 653, -2*w^3 + 2*w^2 + 9*w + 3],\ [677, 677, w^3 + w^2 - 8*w - 11],\ [677, 677, 2*w^3 - w^2 - 12*w - 3],\ [683, 683, 3*w^3 - 2*w^2 - 13*w - 2],\ [683, 683, 3*w^3 + 2*w^2 - 16*w - 10],\ [691, 691, -2*w^3 - 2*w^2 + 12*w + 15],\ [691, 691, 3*w^3 - 2*w^2 - 15*w - 2],\ [691, 691, w^3 - 2*w^2 - 6*w + 6],\ [691, 691, -4*w^3 + 3*w^2 + 15*w - 2],\ [719, 719, 2*w^3 - 7*w - 8],\ [719, 719, w^2 + 2*w - 5],\ [727, 727, -w^3 + 3*w^2 + 4*w - 8],\ [727, 727, 2*w^3 - 9*w - 9],\ [733, 733, w^3 - 8*w - 2],\ [739, 739, -w^3 - 2*w^2 + 4*w + 10],\ [743, 743, w^3 - w^2 - 2*w + 8],\ [743, 743, 4*w^3 - w^2 - 17*w - 4],\ [751, 751, -2*w^3 + 2*w^2 + 6*w - 5],\ [751, 751, w^3 - 5*w - 7],\ [769, 769, 3*w^3 - 2*w^2 - 10*w],\ [787, 787, -2*w^3 + 3*w^2 + 8*w],\ [787, 787, 3*w^3 - w^2 - 10*w - 4],\ [809, 809, 3*w^3 + w^2 - 16*w - 6],\ [811, 811, 3*w^3 - 14*w - 2],\ [811, 811, -w^3 + 3*w^2 + 5*w - 5],\ [811, 811, w^3 - 2*w - 6],\ [811, 811, 3*w^3 - w^2 - 11*w - 3],\ [823, 823, w^3 + 2*w^2 - 3*w - 8],\ [827, 827, -w^3 + 3*w^2 + 3*w - 7],\ [829, 829, w^3 + w^2 - 3*w - 7],\ [853, 853, 3*w^3 + w^2 - 14*w - 6],\ [857, 857, 3*w^3 + w^2 - 13*w - 5],\ [857, 857, 3*w^3 - 3*w^2 - 11*w - 1],\ [859, 859, 4*w^3 - 2*w^2 - 18*w - 3],\ [859, 859, 3*w^3 - 11*w - 3],\ [863, 863, 3*w^3 + w^2 - 17*w - 9],\ [877, 877, -w^3 - 2*w^2 + 9*w + 1],\ [877, 877, 5*w^3 + w^2 - 23*w - 11],\ [911, 911, w^2 - 4*w - 6],\ [911, 911, -2*w^3 + 2*w^2 + 10*w + 3],\ [919, 919, -4*w^3 + 3*w^2 + 18*w - 1],\ [929, 929, 2*w^3 - 13*w - 12],\ [929, 929, -w^3 + 2*w^2 + 7*w - 5],\ [953, 953, -3*w^3 + 3*w^2 + 13*w - 1],\ [967, 967, -3*w^3 + 4*w^2 + 13*w - 7],\ [971, 971, 2*w^2 - 3*w - 12],\ [971, 971, 3*w^3 - w^2 - 16*w - 3]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^4 - 12*x^2 + 3 K. = NumberField(heckePol) hecke_eigenvalues_array = [e, e, -e, 1/2*e^2 - 1/2, -e^3 + 11*e, -1/2*e^3 + 9/2*e, -1/2*e^3 + 11/2*e, 7, -e, -1, e^3 - 14*e, 6, -e^2 + 9, -1/2*e^2 + 11/2, -2*e^3 + 24*e, -2*e^3 + 22*e, 5/2*e^3 - 61/2*e, 7, 3/2*e^3 - 31/2*e, 3*e^3 - 31*e, -3*e^3 + 31*e, -1/2*e^3 + 11/2*e, e^2 + 5, e^3 - 10*e, e^3 - 9*e, -e^3 + 11*e, e^3 - 9*e, e^2 - 3, 5/2*e^3 - 59/2*e, 8, e^3 - 17*e, 2, 7/2*e^3 - 77/2*e, -3/2*e^2 + 29/2, 1/2*e^2 - 21/2, 3/2*e^2 - 21/2, 3*e^2 - 18, -3*e^2 + 21, -4*e^3 + 44*e, 4*e^3 - 49*e, -7/2*e^3 + 95/2*e, -7/2*e^3 + 91/2*e, -e^2 + 27, -e^3 + 6*e, 9, -4*e^2 + 27, e^2 - 7, 10, e^2 - 18, -e^3 + 11*e, -4*e^2 + 18, -3*e^2 + 6, -2*e^2 + 24, 1/2*e^3 - 23/2*e, 2*e^2 - 12, 3/2*e^3 - 23/2*e, -3/2*e^2 - 15/2, -3/2*e^2 + 9/2, e^3 - 18*e, -7/2*e^3 + 81/2*e, e^2 - 14, 3*e^3 - 43*e, -1/2*e^2 + 47/2, -4*e^3 + 45*e, -e^2 + 21, -1/2*e^3 + 15/2*e, 4*e^3 - 44*e, -2*e^2 + 26, 2*e^3 - 26*e, -31, -3/2*e^2 + 37/2, 11/2*e^3 - 113/2*e, e^3 - 13*e, 3/2*e^2 + 17/2, -11/2*e^3 + 137/2*e, -e^2 + 1, -2*e^2 + 1, 3*e^3 - 37*e, -3*e^2 + 32, 4*e^3 - 58*e, 9/2*e^2 - 69/2, 2*e^3 - 22*e, -e^3 + 17*e, -e^3 + 10*e, 2*e^3 - 14*e, 5*e^3 - 53*e, 6, 3/2*e^3 - 39/2*e, 3*e^3 - 47*e, -7*e^3 + 75*e, -3*e^2 - 3, -5/2*e^3 + 41/2*e, -21, -e^2 - 18, 2*e^3 - 15*e, -11/2*e^2 + 51/2, -2*e^2, -2*e^3 + 26*e, -2*e^3 + 36*e, -30, 3*e^3 - 26*e, 4*e^3 - 44*e, -4*e^3 + 58*e, -4*e^2 + 27, 5/2*e^2 - 39/2, 7*e^3 - 71*e, 5*e^3 - 51*e, -e^2 + 27, e^3 - 15*e, -34, 11/2*e^2 - 85/2, -1/2*e^2 - 45/2, 3*e^2 - 22, -13/2*e^2 + 93/2, 6*e^2 - 42, -4*e^2 + 18, -e^2 - 18, -7*e^3 + 91*e, -3*e^2 + 49, -4*e^2 + 28, e^3 - 15*e, 4*e, 6*e^3 - 65*e, -e^3 + 13*e, -e^3 + 25*e, 34, -3*e^2 - 7, 2*e^3 - 30*e, 1/2*e^2 - 69/2, 5*e^3 - 66*e, 3*e^2 - 25, -1/2*e^2 - 65/2, 3*e^3 - 43*e, -15/2*e^2 + 83/2, -2*e^3 + 24*e, 2*e^3 - 32*e, -8*e^3 + 78*e, -3*e^3 + 38*e, 5*e^3 - 63*e, -3*e^3 + 28*e, 3*e^3 - 37*e, 6*e^3 - 60*e, -2*e^3 + 27*e, -15/2*e^3 + 209/2*e, -15*e, -5*e^2 + 3, 7*e^2 - 29, 7*e^3 - 73*e, -6*e^2 + 15, 6*e^3 - 62*e, -5/2*e^3 + 53/2*e, 7/2*e^2 - 81/2, -1/2*e^2 + 39/2, 44, e^2 - 21, -9*e^3 + 107*e, -2*e^2 + 24, 6*e^3 - 60*e, -3*e^2 + 57, -4*e^2 + 42] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([43, 43, -w - 3])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]