sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(700, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([0,3,5]))
pari:[g,chi] = znchar(Mod(129,700))
χ700(89,⋅)
χ700(129,⋅)
χ700(229,⋅)
χ700(269,⋅)
χ700(369,⋅)
χ700(409,⋅)
χ700(509,⋅)
χ700(689,⋅)
sage:chi.galois_orbit()
pari:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
(351,477,101) → (1,e(101),e(61))
a |
−1 | 1 | 3 | 9 | 11 | 13 | 17 | 19 | 23 | 27 | 29 | 31 |
χ700(129,a) |
−1 | 1 | e(1513) | e(1511) | e(154) | e(52) | e(157) | e(3019) | e(3013) | e(53) | e(51) | e(3029) |
sage:chi.jacobi_sum(n)
sage:chi.gauss_sum(a)
pari:znchargauss(g,chi,a)
sage:chi.jacobi_sum(n)
sage:chi.kloosterman_sum(a,b)