from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(357, base_ring=CyclotomicField(8))
M = H._module
chi = DirichletCharacter(H, M([4,0,5]))
chi.galois_orbit()
[g,chi] = znchar(Mod(8,357))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(357\) | |
| Conductor: | \(51\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(8\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 51.g | 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: | \(\Q(\zeta_{8})\) |
| Fixed field: | 8.0.33237432513.1 |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(19\) | \(20\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{357}(8,\cdot)\) | \(-1\) | \(1\) | \(i\) | \(-1\) | \(e\left(\frac{5}{8}\right)\) | \(-i\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(-1\) | \(1\) | \(-i\) | \(e\left(\frac{1}{8}\right)\) |
| \(\chi_{357}(134,\cdot)\) | \(-1\) | \(1\) | \(-i\) | \(-1\) | \(e\left(\frac{3}{8}\right)\) | \(i\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(-1\) | \(1\) | \(i\) | \(e\left(\frac{7}{8}\right)\) |
| \(\chi_{357}(155,\cdot)\) | \(-1\) | \(1\) | \(-i\) | \(-1\) | \(e\left(\frac{7}{8}\right)\) | \(i\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(-1\) | \(1\) | \(i\) | \(e\left(\frac{3}{8}\right)\) |
| \(\chi_{357}(281,\cdot)\) | \(-1\) | \(1\) | \(i\) | \(-1\) | \(e\left(\frac{1}{8}\right)\) | \(-i\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(-1\) | \(1\) | \(-i\) | \(e\left(\frac{5}{8}\right)\) |