from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8470, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,55,36]))
chi.galois_orbit()
[g,chi] = znchar(Mod(89,8470))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(8470\) | |
| Conductor: | \(4235\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 4235.ck | 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 | \(-1\) | \(1\) | \(3\) | \(9\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) | \(37\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{8470}(89,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(1\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{5}{66}\right)\) |
| \(\chi_{8470}(199,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(1\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{47}{66}\right)\) | \(e\left(\frac{25}{66}\right)\) |
| \(\chi_{8470}(859,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(1\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{35}{66}\right)\) |
| \(\chi_{8470}(1629,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{47}{66}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(1\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{65}{66}\right)\) |
| \(\chi_{8470}(1739,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{61}{66}\right)\) | \(1\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{19}{66}\right)\) |
| \(\chi_{8470}(2399,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(1\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{29}{66}\right)\) |
| \(\chi_{8470}(2509,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(1\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{49}{66}\right)\) |
| \(\chi_{8470}(3169,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{47}{66}\right)\) | \(1\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{59}{66}\right)\) |
| \(\chi_{8470}(3279,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(1\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{13}{66}\right)\) |
| \(\chi_{8470}(3939,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(1\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{23}{66}\right)\) |
| \(\chi_{8470}(4049,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{13}{66}\right)\) | \(1\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{43}{66}\right)\) |
| \(\chi_{8470}(4709,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(1\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{53}{66}\right)\) |
| \(\chi_{8470}(4819,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(1\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{7}{66}\right)\) |
| \(\chi_{8470}(5479,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(1\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{17}{66}\right)\) |
| \(\chi_{8470}(5589,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(1\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{37}{66}\right)\) |
| \(\chi_{8470}(6249,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(1\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{47}{66}\right)\) |
| \(\chi_{8470}(6359,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(1\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{1}{66}\right)\) |
| \(\chi_{8470}(7129,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{37}{66}\right)\) | \(1\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{31}{66}\right)\) |
| \(\chi_{8470}(7789,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{17}{66}\right)\) | \(1\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{41}{66}\right)\) |
| \(\chi_{8470}(7899,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{43}{66}\right)\) | \(1\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{61}{66}\right)\) |