from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1001, base_ring=CyclotomicField(4))
M = H._module
chi = DirichletCharacter(H, M([0,0,1]))
chi.galois_orbit()
[g,chi] = znchar(Mod(463,1001))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1001\) | |
| Conductor: | \(13\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(4\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 13.d | 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: | 4.0.2197.1 |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(12\) | \(15\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1001}(463,\cdot)\) | \(-1\) | \(1\) | \(i\) | \(1\) | \(-1\) | \(i\) | \(i\) | \(-i\) | \(1\) | \(-1\) | \(-1\) | \(i\) |
| \(\chi_{1001}(694,\cdot)\) | \(-1\) | \(1\) | \(-i\) | \(1\) | \(-1\) | \(-i\) | \(-i\) | \(i\) | \(1\) | \(-1\) | \(-1\) | \(-i\) |