from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3762, base_ring=CyclotomicField(18))
M = H._module
chi = DirichletCharacter(H, M([6,0,10]))
chi.galois_orbit()
[g,chi] = znchar(Mod(1651,3762))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(3762\) | |
| Conductor: | \(171\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(9\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 171.v | 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: | \(\Q(\zeta_{9})\) |
| Fixed field: | 9.9.9025761726072081.1 |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(5\) | \(7\) | \(13\) | \(17\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) | \(37\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{3762}(1651,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(1\) | \(e\left(\frac{2}{9}\right)\) | \(1\) |
| \(\chi_{3762}(2113,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(1\) | \(e\left(\frac{4}{9}\right)\) | \(1\) |
| \(\chi_{3762}(2707,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(1\) | \(e\left(\frac{7}{9}\right)\) | \(1\) |
| \(\chi_{3762}(3037,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(1\) | \(e\left(\frac{8}{9}\right)\) | \(1\) |
| \(\chi_{3762}(3103,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(1\) | \(e\left(\frac{1}{9}\right)\) | \(1\) |
| \(\chi_{3762}(3235,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(1\) | \(e\left(\frac{5}{9}\right)\) | \(1\) |