sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(392, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,21,41]))
pari:[g,chi] = znchar(Mod(229,392))
Modulus: | 392 | |
Conductor: | 392 |
sage:chi.conductor()
pari:znconreyconductor(g,chi)
|
Order: | 42 |
sage:chi.multiplicative_order()
pari:charorder(g,chi)
|
Real: | no |
Primitive: | yes |
sage:chi.is_primitive()
pari:#znconreyconductor(g,chi)==1
|
Minimal: | yes |
Parity: | odd |
sage:chi.is_odd()
pari:zncharisodd(g,chi)
|
χ392(5,⋅)
χ392(45,⋅)
χ392(61,⋅)
χ392(101,⋅)
χ392(157,⋅)
χ392(173,⋅)
χ392(213,⋅)
χ392(229,⋅)
χ392(269,⋅)
χ392(285,⋅)
χ392(341,⋅)
χ392(381,⋅)
sage:chi.galois_orbit()
pari:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
(295,197,297) → (1,−1,e(4241))
a |
−1 | 1 | 3 | 5 | 9 | 11 | 13 | 15 | 17 | 19 | 23 | 25 |
χ392(229,a) |
−1 | 1 | e(2110) | e(2117) | e(2120) | e(4223) | e(75) | e(72) | e(4217) | e(32) | e(212) | e(2113) |
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)