sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(812, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,28,36]))
pari:[g,chi] = znchar(Mod(571,812))
Modulus: | 812 | |
Conductor: | 812 |
sage:chi.conductor()
pari:znconreyconductor(g,chi)
|
Order: | 42 |
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)
|
χ812(23,⋅)
χ812(107,⋅)
χ812(123,⋅)
χ812(219,⋅)
χ812(431,⋅)
χ812(459,⋅)
χ812(471,⋅)
χ812(487,⋅)
χ812(571,⋅)
χ812(683,⋅)
χ812(779,⋅)
χ812(807,⋅)
sage:chi.galois_orbit()
pari:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
(407,465,785) → (−1,e(32),e(76))
a |
−1 | 1 | 3 | 5 | 9 | 11 | 13 | 15 | 17 | 19 | 23 | 25 |
χ812(571,a) |
−1 | 1 | e(4219) | e(214) | e(2119) | e(4225) | e(73) | e(149) | e(32) | e(4223) | e(4241) | e(218) |
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)