from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(322, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([22,30]))
chi.galois_orbit()
[g,chi] = znchar(Mod(9,322))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(322\) | |
| Conductor: | \(161\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(33\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 161.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: | \(\Q(\zeta_{33})\) |
| Fixed field: | 33.33.277966181338944111003326058293667039541136678070715028736001.1 |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(25\) | \(27\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{322}(9,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{9}{11}\right)\) |
| \(\chi_{322}(25,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{4}{11}\right)\) |
| \(\chi_{322}(39,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{5}{11}\right)\) |
| \(\chi_{322}(81,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{7}{11}\right)\) |
| \(\chi_{322}(95,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{10}{11}\right)\) |
| \(\chi_{322}(121,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{3}{11}\right)\) |
| \(\chi_{322}(123,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{1}{11}\right)\) |
| \(\chi_{322}(151,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{6}{11}\right)\) |
| \(\chi_{322}(163,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{4}{11}\right)\) |
| \(\chi_{322}(165,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{8}{11}\right)\) |
| \(\chi_{322}(177,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{5}{11}\right)\) |
| \(\chi_{322}(179,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{2}{11}\right)\) |
| \(\chi_{322}(193,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{9}{11}\right)\) |
| \(\chi_{322}(219,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{7}{11}\right)\) |
| \(\chi_{322}(233,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{10}{11}\right)\) |
| \(\chi_{322}(261,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{1}{11}\right)\) |
| \(\chi_{322}(289,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{6}{11}\right)\) |
| \(\chi_{322}(303,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{8}{11}\right)\) |
| \(\chi_{322}(305,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{3}{11}\right)\) |
| \(\chi_{322}(317,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{2}{11}\right)\) |