from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(357, base_ring=CyclotomicField(4))
M = H._module
chi = DirichletCharacter(H, M([0,2,1]))
chi.galois_orbit()
[g,chi] = znchar(Mod(13,357))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(357\) | |
| Conductor: | \(119\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(4\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 119.f | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Related number fields
| Field of values: | \(\mathbb{Q}(i)\) |
| Fixed field: | 4.0.240737.1 |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(19\) | \(20\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{357}(13,\cdot)\) | \(-1\) | \(1\) | \(-1\) | \(1\) | \(-i\) | \(-1\) | \(i\) | \(-i\) | \(-1\) | \(1\) | \(1\) | \(-i\) |
| \(\chi_{357}(55,\cdot)\) | \(-1\) | \(1\) | \(-1\) | \(1\) | \(i\) | \(-1\) | \(-i\) | \(i\) | \(-1\) | \(1\) | \(1\) | \(i\) |