sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(209, base_ring=CyclotomicField(90))
M = H._module
chi = DirichletCharacter(H, M([18,50]))
pari:[g,chi] = znchar(Mod(169,209))
Modulus: | 209 | |
Conductor: | 209 |
sage:chi.conductor()
pari:znconreyconductor(g,chi)
|
Order: | 45 |
sage:chi.multiplicative_order()
pari:charorder(g,chi)
|
Real: | no |
Primitive: | yes |
sage:chi.is_primitive()
pari:#znconreyconductor(g,chi)==1
|
Minimal: | yes |
Parity: | even |
sage:chi.is_odd()
pari:zncharisodd(g,chi)
|
χ209(4,⋅)
χ209(5,⋅)
χ209(9,⋅)
χ209(16,⋅)
χ209(25,⋅)
χ209(36,⋅)
χ209(42,⋅)
χ209(47,⋅)
χ209(80,⋅)
χ209(81,⋅)
χ209(82,⋅)
χ209(92,⋅)
χ209(93,⋅)
χ209(104,⋅)
χ209(119,⋅)
χ209(130,⋅)
χ209(137,⋅)
χ209(157,⋅)
χ209(158,⋅)
χ209(168,⋅)
χ209(169,⋅)
χ209(180,⋅)
χ209(196,⋅)
χ209(207,⋅)
sage:chi.galois_orbit()
pari:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
(134,78) → (e(51),e(95))
a |
−1 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 12 |
χ209(169,a) |
1 | 1 | e(4534) | e(4537) | e(4523) | e(4531) | e(4526) | e(1511) | e(154) | e(4529) | e(94) | e(31) |
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)