sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(135, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([14,9]))
pari:[g,chi] = znchar(Mod(47,135))
Modulus: | 135 | |
Conductor: | 135 |
sage:chi.conductor()
pari:znconreyconductor(g,chi)
|
Order: | 36 |
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)
|
χ135(2,⋅)
χ135(23,⋅)
χ135(32,⋅)
χ135(38,⋅)
χ135(47,⋅)
χ135(68,⋅)
χ135(77,⋅)
χ135(83,⋅)
χ135(92,⋅)
χ135(113,⋅)
χ135(122,⋅)
χ135(128,⋅)
sage:chi.galois_orbit()
pari:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
(56,82) → (e(187),i)
a |
−1 | 1 | 2 | 4 | 7 | 8 | 11 | 13 | 14 | 16 | 17 | 19 |
χ135(47,a) |
1 | 1 | e(3623) | e(185) | e(3617) | e(1211) | e(181) | e(3631) | e(91) | e(95) | e(121) | e(61) |
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)