sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(475, base_ring=CyclotomicField(90))
M = H._module
chi = DirichletCharacter(H, M([81,80]))
pari:[g,chi] = znchar(Mod(119,475))
Modulus: | 475 | |
Conductor: | 475 |
sage:chi.conductor()
pari:znconreyconductor(g,chi)
|
Order: | 90 |
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)
|
χ475(4,⋅)
χ475(9,⋅)
χ475(44,⋅)
χ475(54,⋅)
χ475(104,⋅)
χ475(119,⋅)
χ475(139,⋅)
χ475(169,⋅)
χ475(194,⋅)
χ475(214,⋅)
χ475(234,⋅)
χ475(244,⋅)
χ475(264,⋅)
χ475(289,⋅)
χ475(294,⋅)
χ475(309,⋅)
χ475(329,⋅)
χ475(339,⋅)
χ475(359,⋅)
χ475(384,⋅)
χ475(389,⋅)
χ475(404,⋅)
χ475(434,⋅)
χ475(454,⋅)
sage:chi.galois_orbit()
pari:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
(77,401) → (e(109),e(98))
a |
−1 | 1 | 2 | 3 | 4 | 6 | 7 | 8 | 9 | 11 | 12 | 13 |
χ475(119,a) |
1 | 1 | e(9071) | e(9077) | e(4526) | e(4529) | e(65) | e(3011) | e(4532) | e(151) | e(3013) | e(9049) |
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)