from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(683, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([7]))
chi.galois_orbit()
[g,chi] = znchar(Mod(2,683))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(683\) | |
| Conductor: | \(683\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(22\) | 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: | \(\Q(\zeta_{11})\) |
| Fixed field: | Number field defined by a degree 22 polynomial |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{683}(2,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{22}\right)\) | \(1\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(1\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{1}{22}\right)\) |
| \(\chi_{683}(8,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{22}\right)\) | \(1\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(1\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{3}{22}\right)\) |
| \(\chi_{683}(32,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{22}\right)\) | \(1\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(1\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{5}{22}\right)\) |
| \(\chi_{683}(128,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{22}\right)\) | \(1\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(1\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{7}{22}\right)\) |
| \(\chi_{683}(342,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{21}{22}\right)\) | \(1\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(1\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{21}{22}\right)\) |
| \(\chi_{683}(427,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{22}\right)\) | \(1\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(1\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{19}{22}\right)\) |
| \(\chi_{683}(512,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{22}\right)\) | \(1\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(1\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{9}{22}\right)\) |
| \(\chi_{683}(619,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{22}\right)\) | \(1\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(1\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{17}{22}\right)\) |
| \(\chi_{683}(667,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{15}{22}\right)\) | \(1\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(1\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{15}{22}\right)\) |
| \(\chi_{683}(679,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{22}\right)\) | \(1\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(1\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{13}{22}\right)\) |