from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1045, base_ring=CyclotomicField(4))
M = H._module
chi = DirichletCharacter(H, M([3,2,2]))
chi.galois_orbit()
[g,chi] = znchar(Mod(208,1045))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1045\) | |
| Conductor: | \(1045\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(4\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | 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: | 4.0.5460125.4 |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(13\) | \(14\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1045}(208,\cdot)\) | \(-1\) | \(1\) | \(-i\) | \(-i\) | \(-1\) | \(-1\) | \(i\) | \(i\) | \(-1\) | \(i\) | \(i\) | \(1\) |
| \(\chi_{1045}(417,\cdot)\) | \(-1\) | \(1\) | \(i\) | \(i\) | \(-1\) | \(-1\) | \(-i\) | \(-i\) | \(-1\) | \(-i\) | \(-i\) | \(1\) |