sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(323, base_ring=CyclotomicField(24))
M = H._module
chi = DirichletCharacter(H, M([15,4]))
pari:[g,chi] = znchar(Mod(8,323))
Modulus: | 323 | |
Conductor: | 323 |
sage:chi.conductor()
pari:znconreyconductor(g,chi)
|
Order: | 24 |
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)
|
χ323(8,⋅)
χ323(145,⋅)
χ323(179,⋅)
χ323(202,⋅)
χ323(236,⋅)
χ323(240,⋅)
χ323(274,⋅)
χ323(297,⋅)
sage:chi.galois_orbit()
pari:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
(20,154) → (e(85),e(61))
a |
−1 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
χ323(8,a) |
−1 | 1 | e(1211) | e(2419) | e(65) | e(2419) | e(2417) | e(87) | −i | e(127) | e(2417) | e(83) |
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)