from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1386, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([0,10,6]))
chi.galois_orbit()
[g,chi] = znchar(Mod(37,1386))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1386\) | |
| Conductor: | \(77\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(15\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 77.m | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Related number fields
| Field of values: | \(\Q(\zeta_{15})\) |
| Fixed field: | 15.15.886528337182930278529.1 |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(5\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1386}(37,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{3}{5}\right)\) |
| \(\chi_{1386}(163,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{4}{5}\right)\) |
| \(\chi_{1386}(235,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{3}{5}\right)\) |
| \(\chi_{1386}(289,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{2}{5}\right)\) |
| \(\chi_{1386}(361,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{4}{5}\right)\) |
| \(\chi_{1386}(487,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{2}{5}\right)\) |
| \(\chi_{1386}(1171,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{1}{5}\right)\) |
| \(\chi_{1386}(1369,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{1}{5}\right)\) |