sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(253, base_ring=CyclotomicField(110))
M = H._module
chi = DirichletCharacter(H, M([22,80]))
pari:[g,chi] = znchar(Mod(26,253))
Modulus: | 253 | |
Conductor: | 253 |
sage:chi.conductor()
pari:znconreyconductor(g,chi)
|
Order: | 55 |
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)
|
χ253(3,⋅)
χ253(4,⋅)
χ253(9,⋅)
χ253(16,⋅)
χ253(25,⋅)
χ253(26,⋅)
χ253(27,⋅)
χ253(31,⋅)
χ253(36,⋅)
χ253(48,⋅)
χ253(49,⋅)
χ253(58,⋅)
χ253(59,⋅)
χ253(64,⋅)
χ253(71,⋅)
χ253(75,⋅)
χ253(81,⋅)
χ253(82,⋅)
χ253(104,⋅)
χ253(108,⋅)
χ253(119,⋅)
χ253(124,⋅)
χ253(141,⋅)
χ253(146,⋅)
χ253(147,⋅)
χ253(163,⋅)
χ253(169,⋅)
χ253(170,⋅)
χ253(174,⋅)
χ253(179,⋅)
...
sage:chi.galois_orbit()
pari:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
(24,166) → (e(51),e(118))
a |
−1 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 12 |
χ253(26,a) |
1 | 1 | e(5536) | e(5513) | e(5517) | e(5529) | e(5549) | e(5512) | e(5553) | e(5526) | e(112) | e(116) |
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)