sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(475, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([51,10]))
pari:[g,chi] = znchar(Mod(122,475))
Modulus: | 475 | |
Conductor: | 475 |
sage:chi.conductor()
pari:znconreyconductor(g,chi)
|
Order: | 60 |
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)
|
χ475(8,⋅)
χ475(12,⋅)
χ475(27,⋅)
χ475(88,⋅)
χ475(103,⋅)
χ475(122,⋅)
χ475(183,⋅)
χ475(198,⋅)
χ475(202,⋅)
χ475(217,⋅)
χ475(278,⋅)
χ475(297,⋅)
χ475(312,⋅)
χ475(373,⋅)
χ475(388,⋅)
χ475(392,⋅)
sage:chi.galois_orbit()
pari:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
(77,401) → (e(2017),e(61))
a |
−1 | 1 | 2 | 3 | 4 | 6 | 7 | 8 | 9 | 11 | 12 | 13 |
χ475(122,a) |
1 | 1 | e(601) | e(607) | e(301) | e(152) | i | e(201) | e(307) | e(53) | e(203) | e(6059) |
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)