sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(440, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([10,0,15,14]))
pari:[g,chi] = znchar(Mod(183,440))
χ440(7,⋅)
χ440(63,⋅)
χ440(127,⋅)
χ440(167,⋅)
χ440(183,⋅)
χ440(303,⋅)
χ440(327,⋅)
χ440(343,⋅)
sage:chi.galois_orbit()
pari:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
(111,221,177,321) → (−1,1,−i,e(107))
a |
−1 | 1 | 3 | 7 | 9 | 13 | 17 | 19 | 21 | 23 | 27 | 29 |
χ440(183,a) |
−1 | 1 | e(207) | e(203) | e(107) | e(2019) | e(201) | e(101) | −1 | −i | e(201) | e(52) |
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)