sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(319, base_ring=CyclotomicField(140))
M = H._module
chi = DirichletCharacter(H, M([112,55]))
pari:[g,chi] = znchar(Mod(47,319))
Modulus: | 319 | |
Conductor: | 319 |
sage:chi.conductor()
pari:znconreyconductor(g,chi)
|
Order: | 140 |
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)
|
χ319(3,⋅)
χ319(14,⋅)
χ319(15,⋅)
χ319(26,⋅)
χ319(27,⋅)
χ319(31,⋅)
χ319(37,⋅)
χ319(47,⋅)
χ319(48,⋅)
χ319(60,⋅)
χ319(69,⋅)
χ319(97,⋅)
χ319(102,⋅)
χ319(108,⋅)
χ319(113,⋅)
χ319(114,⋅)
χ319(119,⋅)
χ319(124,⋅)
χ319(126,⋅)
χ319(130,⋅)
χ319(135,⋅)
χ319(137,⋅)
χ319(147,⋅)
χ319(148,⋅)
χ319(159,⋅)
χ319(163,⋅)
χ319(185,⋅)
χ319(192,⋅)
χ319(201,⋅)
χ319(213,⋅)
...
sage:chi.galois_orbit()
pari:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
(233,89) → (e(54),e(2811))
a |
−1 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 12 |
χ319(47,a) |
−1 | 1 | e(14027) | e(14051) | e(7027) | e(7059) | e(7039) | e(3511) | e(14081) | e(7051) | e(281) | −i |
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)