from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9660, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,33,0,55,3]))
chi.galois_orbit()
[g,chi] = znchar(Mod(971,9660))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(9660\) | |
| Conductor: | \(1932\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 1932.bz | 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: | Number field defined by a degree 66 polynomial |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{9660}(971,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{1}{6}\right)\) |
| \(\chi_{9660}(1391,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{1}{6}\right)\) |
| \(\chi_{9660}(1571,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{47}{66}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{5}{6}\right)\) |
| \(\chi_{9660}(1811,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{1}{6}\right)\) |
| \(\chi_{9660}(2411,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{5}{6}\right)\) |
| \(\chi_{9660}(3671,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{5}{6}\right)\) |
| \(\chi_{9660}(4091,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{66}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{5}{6}\right)\) |
| \(\chi_{9660}(4331,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{1}{6}\right)\) |
| \(\chi_{9660}(5171,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{1}{6}\right)\) |
| \(\chi_{9660}(5351,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{5}{6}\right)\) |
| \(\chi_{9660}(5771,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{5}{6}\right)\) |
| \(\chi_{9660}(6431,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{47}{66}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{1}{6}\right)\) |
| \(\chi_{9660}(6611,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{5}{6}\right)\) |
| \(\chi_{9660}(6851,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{1}{6}\right)\) |
| \(\chi_{9660}(7871,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{5}{6}\right)\) |
| \(\chi_{9660}(8111,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{1}{6}\right)\) |
| \(\chi_{9660}(8291,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{5}{6}\right)\) |
| \(\chi_{9660}(8531,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{1}{6}\right)\) |
| \(\chi_{9660}(8711,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{5}{6}\right)\) |
| \(\chi_{9660}(9371,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{47}{66}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{1}{6}\right)\) |