from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(185, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([9,32]))
chi.galois_orbit()
[g,chi] = znchar(Mod(7,185))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(185\) | |
| Conductor: | \(185\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(36\) | 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: | \(\Q(\zeta_{36})\) |
| Fixed field: | 36.0.1134084166835624937413663701523229292665702790961273014545440673828125.1 |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{185}(7,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(1\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{19}{36}\right)\) |
| \(\chi_{185}(12,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(1\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{11}{36}\right)\) |
| \(\chi_{185}(33,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(1\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{13}{36}\right)\) |
| \(\chi_{185}(53,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(1\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{17}{36}\right)\) |
| \(\chi_{185}(83,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(1\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{5}{36}\right)\) |
| \(\chi_{185}(107,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(1\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{31}{36}\right)\) |
| \(\chi_{185}(108,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(1\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{25}{36}\right)\) |
| \(\chi_{185}(118,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(1\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{1}{36}\right)\) |
| \(\chi_{185}(123,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(1\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{29}{36}\right)\) |
| \(\chi_{185}(127,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(1\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{35}{36}\right)\) |
| \(\chi_{185}(157,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(1\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{23}{36}\right)\) |
| \(\chi_{185}(182,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(1\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{7}{36}\right)\) |