sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(143, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([36,55]))
pari:[g,chi] = znchar(Mod(20,143))
Modulus: | 143 | |
Conductor: | 143 |
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: | odd |
sage:chi.is_odd()
pari:zncharisodd(g,chi)
|
χ143(15,⋅)
χ143(20,⋅)
χ143(37,⋅)
χ143(58,⋅)
χ143(59,⋅)
χ143(71,⋅)
χ143(80,⋅)
χ143(93,⋅)
χ143(97,⋅)
χ143(102,⋅)
χ143(115,⋅)
χ143(119,⋅)
χ143(124,⋅)
χ143(136,⋅)
χ143(137,⋅)
χ143(141,⋅)
sage:chi.galois_orbit()
pari:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
(79,67) → (e(53),e(1211))
a |
−1 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 12 |
χ143(20,a) |
−1 | 1 | e(6031) | e(157) | e(301) | e(2013) | e(6059) | e(6017) | e(2011) | e(1514) | e(61) | −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)