from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(322, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([11,48]))
chi.galois_orbit()
[g,chi] = znchar(Mod(3,322))
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: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 161.n | 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: | \(\Q(\zeta_{33})\) |
| Fixed field: | Number field defined by a degree 66 polynomial |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(25\) | \(27\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{322}(3,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{9}{22}\right)\) |
| \(\chi_{322}(31,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{13}{22}\right)\) |
| \(\chi_{322}(59,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{1}{22}\right)\) |
| \(\chi_{322}(73,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{5}{22}\right)\) |
| \(\chi_{322}(75,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{17}{22}\right)\) |
| \(\chi_{322}(87,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{15}{22}\right)\) |
| \(\chi_{322}(101,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{7}{22}\right)\) |
| \(\chi_{322}(117,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{19}{22}\right)\) |
| \(\chi_{322}(131,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{21}{22}\right)\) |
| \(\chi_{322}(173,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{3}{22}\right)\) |
| \(\chi_{322}(187,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{9}{22}\right)\) |
| \(\chi_{322}(213,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{17}{22}\right)\) |
| \(\chi_{322}(215,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{13}{22}\right)\) |
| \(\chi_{322}(243,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{47}{66}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{1}{22}\right)\) |
| \(\chi_{322}(255,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{19}{22}\right)\) |
| \(\chi_{322}(257,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{5}{22}\right)\) |
| \(\chi_{322}(269,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{47}{66}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{21}{22}\right)\) |
| \(\chi_{322}(271,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{47}{66}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{15}{22}\right)\) |
| \(\chi_{322}(285,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{7}{22}\right)\) |
| \(\chi_{322}(311,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{66}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{3}{22}\right)\) |