from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(19, base_ring=CyclotomicField(2))
M = H._module
chi = DirichletCharacter(H, M([1]))
pari: [g,chi] = znchar(Mod(18,19))
Kronecker symbol representation
sage: kronecker_character(-19)
pari: znchartokronecker(g,chi)
Basic properties
Modulus: | ||
Conductor: | sage: chi.conductor()
pari: znconreyconductor(g,chi)
| |
Order: | sage: chi.multiplicative_order()
pari: charorder(g,chi)
| |
Real: | yes | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 19.b
sage: chi.galois_orbit()
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Related number fields
Field of values: | |
Fixed field: |
Values on generators
→
Values
sage: chi.jacobi_sum(n)
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
Jacobi sum
sage: chi.jacobi_sum(n)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)