from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(160, base_ring=CyclotomicField(8))
M = H._module
chi = DirichletCharacter(H, M([0,5,6]))
chi.galois_orbit()
[g,chi] = znchar(Mod(53,160))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(160\) | |
| Conductor: | \(160\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(8\) | 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: | \(\Q(\zeta_{8})\) |
| Fixed field: | 8.0.33554432000000.2 |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{160}(53,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{8}\right)\) | \(1\) | \(i\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(i\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(1\) | \(e\left(\frac{3}{8}\right)\) |
| \(\chi_{160}(77,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{8}\right)\) | \(1\) | \(-i\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(-i\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(1\) | \(e\left(\frac{1}{8}\right)\) |
| \(\chi_{160}(133,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{8}\right)\) | \(1\) | \(i\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(i\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(1\) | \(e\left(\frac{7}{8}\right)\) |
| \(\chi_{160}(157,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{8}\right)\) | \(1\) | \(-i\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(-i\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(1\) | \(e\left(\frac{5}{8}\right)\) |