sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(720, base_ring=CyclotomicField(4))
M = H._module
chi = DirichletCharacter(H, M([0,1,0,0]))
pari:[g,chi] = znchar(Mod(181,720))
χ720(181,⋅)
χ720(541,⋅)
sage:chi.galois_orbit()
pari:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
(271,181,641,577) → (1,i,1,1)
a |
−1 | 1 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 |
χ720(181,a) |
1 | 1 | −1 | i | −i | 1 | −i | −1 | −i | 1 | i | −1 |
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)