from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(666, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,35]))
chi.galois_orbit()
[g,chi] = znchar(Mod(19,666))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(666\) | |
| Conductor: | \(37\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 37.i | 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: | Number field defined by a degree 36 polynomial |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{666}(19,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(-i\) |
| \(\chi_{666}(55,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(i\) |
| \(\chi_{666}(91,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(-i\) |
| \(\chi_{666}(109,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(-i\) |
| \(\chi_{666}(163,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(i\) |
| \(\chi_{666}(217,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(i\) |
| \(\chi_{666}(235,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(-i\) |
| \(\chi_{666}(505,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(i\) |
| \(\chi_{666}(523,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(-i\) |
| \(\chi_{666}(577,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(-i\) |
| \(\chi_{666}(631,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(i\) |
| \(\chi_{666}(649,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(i\) |