sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(99, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([5,24]))
pari:[g,chi] = znchar(Mod(47,99))
Modulus: | 99 | |
Conductor: | 99 |
sage:chi.conductor()
pari:znconreyconductor(g,chi)
|
Order: | 30 |
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)
|
χ99(5,⋅)
χ99(14,⋅)
χ99(20,⋅)
χ99(38,⋅)
χ99(47,⋅)
χ99(59,⋅)
χ99(86,⋅)
χ99(92,⋅)
sage:chi.galois_orbit()
pari:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
(56,46) → (e(61),e(54))
a |
−1 | 1 | 2 | 4 | 5 | 7 | 8 | 10 | 13 | 14 | 16 | 17 |
χ99(47,a) |
−1 | 1 | e(3029) | e(1514) | e(301) | e(154) | e(109) | 1 | e(152) | e(307) | e(1513) | e(107) |
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)