sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(85, base_ring=CyclotomicField(8))
M = H._module
chi = DirichletCharacter(H, M([4,7]))
pari:[g,chi] = znchar(Mod(19,85))
Modulus: | 85 | |
Conductor: | 85 |
sage:chi.conductor()
pari:znconreyconductor(g,chi)
|
Order: | 8 |
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)
|
χ85(9,⋅)
χ85(19,⋅)
χ85(49,⋅)
χ85(59,⋅)
sage:chi.galois_orbit()
pari:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
(52,71) → (−1,e(87))
a |
−1 | 1 | 2 | 3 | 4 | 6 | 7 | 8 | 9 | 11 | 12 | 13 |
χ85(19,a) |
1 | 1 | −i | e(83) | −1 | e(81) | e(81) | i | −i | e(81) | e(87) | 1 |
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)