from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1058, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([8]))
chi.galois_orbit()
[g,chi] = znchar(Mod(177,1058))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1058\) | |
| Conductor: | \(23\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(11\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 23.c | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Related number fields
| Field of values: | \(\Q(\zeta_{11})\) |
| Fixed field: | \(\Q(\zeta_{23})^+\) |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1058}(177,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{8}{11}\right)\) |
| \(\chi_{1058}(255,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{2}{11}\right)\) |
| \(\chi_{1058}(399,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{6}{11}\right)\) |
| \(\chi_{1058}(487,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) |
| \(\chi_{1058}(501,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{1}{11}\right)\) |
| \(\chi_{1058}(647,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{5}{11}\right)\) |
| \(\chi_{1058}(699,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{10}{11}\right)\) |
| \(\chi_{1058}(795,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{3}{11}\right)\) |
| \(\chi_{1058}(863,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{9}{11}\right)\) |
| \(\chi_{1058}(995,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{7}{11}\right)\) |