from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5, base_ring=CyclotomicField(2))
M = H._module
chi = DirichletCharacter(H, M([1]))
chi.galois_orbit()
[g,chi] = znchar(Mod(4,5))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Kronecker symbol representation
sage: kronecker_character(5)
pari: znchartokronecker(g,chi)
\(\displaystyle\left(\frac{5}{\bullet}\right)\)
Basic properties
| Modulus: | \(5\) | |
| Conductor: | \(5\) | 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: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Related number fields
| Field of values: | \(\Q\) |
| Fixed field: | \(\Q(\sqrt{5}) \) |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) |
|---|---|---|---|---|
| \(\chi_{5}(4,\cdot)\) | \(1\) | \(1\) | \(-1\) | \(-1\) |