from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3, base_ring=CyclotomicField(2))
M = H._module
chi = DirichletCharacter(H, M([1]))
pari: [g,chi] = znchar(Mod(2,3))
Kronecker symbol representation
sage: kronecker_character(-3)
pari: znchartokronecker(g,chi)
\(\displaystyle\left(\frac{-3}{\bullet}\right)\)
Basic properties
Modulus: | \(3\) | |
Conductor: | \(3\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(2\) | 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 3.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: | \(\Q\) |
Fixed field: | \(\Q(\sqrt{-3}) \) |
Values on generators
\(2\) → \(-1\)
Values
\(a\) | \(-1\) | \(1\) |
\( \chi_{ 3 }(2, a) \) | \(-1\) | \(1\) |
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)
Additional information
This is the first Dirichlet character taking a value other than $1$. It is also the first Dirichlet character with odd parity.