from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2312, base_ring=CyclotomicField(34))
M = H._module
chi = DirichletCharacter(H, M([0,17,9]))
chi.galois_orbit()
[g,chi] = znchar(Mod(101,2312))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(2312\) | |
| Conductor: | \(2312\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(34\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | 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: | \(\Q(\zeta_{17})\) |
| Fixed field: | 34.34.214639510116448210748631066635120907555924504566182022171002423006964754343412080865644260622336.1 |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(19\) | \(21\) | \(23\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{2312}(101,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{17}\right)\) | \(e\left(\frac{2}{17}\right)\) | \(e\left(\frac{1}{34}\right)\) | \(e\left(\frac{9}{17}\right)\) | \(e\left(\frac{10}{17}\right)\) | \(e\left(\frac{13}{34}\right)\) | \(e\left(\frac{15}{17}\right)\) | \(e\left(\frac{7}{34}\right)\) | \(e\left(\frac{27}{34}\right)\) | \(e\left(\frac{1}{34}\right)\) |
| \(\chi_{2312}(237,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{17}\right)\) | \(e\left(\frac{16}{17}\right)\) | \(e\left(\frac{25}{34}\right)\) | \(e\left(\frac{4}{17}\right)\) | \(e\left(\frac{12}{17}\right)\) | \(e\left(\frac{19}{34}\right)\) | \(e\left(\frac{1}{17}\right)\) | \(e\left(\frac{5}{34}\right)\) | \(e\left(\frac{29}{34}\right)\) | \(e\left(\frac{25}{34}\right)\) |
| \(\chi_{2312}(373,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{17}\right)\) | \(e\left(\frac{13}{17}\right)\) | \(e\left(\frac{15}{34}\right)\) | \(e\left(\frac{16}{17}\right)\) | \(e\left(\frac{14}{17}\right)\) | \(e\left(\frac{25}{34}\right)\) | \(e\left(\frac{4}{17}\right)\) | \(e\left(\frac{3}{34}\right)\) | \(e\left(\frac{31}{34}\right)\) | \(e\left(\frac{15}{34}\right)\) |
| \(\chi_{2312}(509,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{14}{17}\right)\) | \(e\left(\frac{10}{17}\right)\) | \(e\left(\frac{5}{34}\right)\) | \(e\left(\frac{11}{17}\right)\) | \(e\left(\frac{16}{17}\right)\) | \(e\left(\frac{31}{34}\right)\) | \(e\left(\frac{7}{17}\right)\) | \(e\left(\frac{1}{34}\right)\) | \(e\left(\frac{33}{34}\right)\) | \(e\left(\frac{5}{34}\right)\) |
| \(\chi_{2312}(645,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{17}\right)\) | \(e\left(\frac{7}{17}\right)\) | \(e\left(\frac{29}{34}\right)\) | \(e\left(\frac{6}{17}\right)\) | \(e\left(\frac{1}{17}\right)\) | \(e\left(\frac{3}{34}\right)\) | \(e\left(\frac{10}{17}\right)\) | \(e\left(\frac{33}{34}\right)\) | \(e\left(\frac{1}{34}\right)\) | \(e\left(\frac{29}{34}\right)\) |
| \(\chi_{2312}(781,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{17}\right)\) | \(e\left(\frac{4}{17}\right)\) | \(e\left(\frac{19}{34}\right)\) | \(e\left(\frac{1}{17}\right)\) | \(e\left(\frac{3}{17}\right)\) | \(e\left(\frac{9}{34}\right)\) | \(e\left(\frac{13}{17}\right)\) | \(e\left(\frac{31}{34}\right)\) | \(e\left(\frac{3}{34}\right)\) | \(e\left(\frac{19}{34}\right)\) |
| \(\chi_{2312}(917,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{15}{17}\right)\) | \(e\left(\frac{1}{17}\right)\) | \(e\left(\frac{9}{34}\right)\) | \(e\left(\frac{13}{17}\right)\) | \(e\left(\frac{5}{17}\right)\) | \(e\left(\frac{15}{34}\right)\) | \(e\left(\frac{16}{17}\right)\) | \(e\left(\frac{29}{34}\right)\) | \(e\left(\frac{5}{34}\right)\) | \(e\left(\frac{9}{34}\right)\) |
| \(\chi_{2312}(1053,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{17}\right)\) | \(e\left(\frac{15}{17}\right)\) | \(e\left(\frac{33}{34}\right)\) | \(e\left(\frac{8}{17}\right)\) | \(e\left(\frac{7}{17}\right)\) | \(e\left(\frac{21}{34}\right)\) | \(e\left(\frac{2}{17}\right)\) | \(e\left(\frac{27}{34}\right)\) | \(e\left(\frac{7}{34}\right)\) | \(e\left(\frac{33}{34}\right)\) |
| \(\chi_{2312}(1189,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{10}{17}\right)\) | \(e\left(\frac{12}{17}\right)\) | \(e\left(\frac{23}{34}\right)\) | \(e\left(\frac{3}{17}\right)\) | \(e\left(\frac{9}{17}\right)\) | \(e\left(\frac{27}{34}\right)\) | \(e\left(\frac{5}{17}\right)\) | \(e\left(\frac{25}{34}\right)\) | \(e\left(\frac{9}{34}\right)\) | \(e\left(\frac{23}{34}\right)\) |
| \(\chi_{2312}(1325,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{17}\right)\) | \(e\left(\frac{9}{17}\right)\) | \(e\left(\frac{13}{34}\right)\) | \(e\left(\frac{15}{17}\right)\) | \(e\left(\frac{11}{17}\right)\) | \(e\left(\frac{33}{34}\right)\) | \(e\left(\frac{8}{17}\right)\) | \(e\left(\frac{23}{34}\right)\) | \(e\left(\frac{11}{34}\right)\) | \(e\left(\frac{13}{34}\right)\) |
| \(\chi_{2312}(1461,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{17}\right)\) | \(e\left(\frac{6}{17}\right)\) | \(e\left(\frac{3}{34}\right)\) | \(e\left(\frac{10}{17}\right)\) | \(e\left(\frac{13}{17}\right)\) | \(e\left(\frac{5}{34}\right)\) | \(e\left(\frac{11}{17}\right)\) | \(e\left(\frac{21}{34}\right)\) | \(e\left(\frac{13}{34}\right)\) | \(e\left(\frac{3}{34}\right)\) |
| \(\chi_{2312}(1597,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{17}\right)\) | \(e\left(\frac{3}{17}\right)\) | \(e\left(\frac{27}{34}\right)\) | \(e\left(\frac{5}{17}\right)\) | \(e\left(\frac{15}{17}\right)\) | \(e\left(\frac{11}{34}\right)\) | \(e\left(\frac{14}{17}\right)\) | \(e\left(\frac{19}{34}\right)\) | \(e\left(\frac{15}{34}\right)\) | \(e\left(\frac{27}{34}\right)\) |
| \(\chi_{2312}(1869,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{6}{17}\right)\) | \(e\left(\frac{14}{17}\right)\) | \(e\left(\frac{7}{34}\right)\) | \(e\left(\frac{12}{17}\right)\) | \(e\left(\frac{2}{17}\right)\) | \(e\left(\frac{23}{34}\right)\) | \(e\left(\frac{3}{17}\right)\) | \(e\left(\frac{15}{34}\right)\) | \(e\left(\frac{19}{34}\right)\) | \(e\left(\frac{7}{34}\right)\) |
| \(\chi_{2312}(2005,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{12}{17}\right)\) | \(e\left(\frac{11}{17}\right)\) | \(e\left(\frac{31}{34}\right)\) | \(e\left(\frac{7}{17}\right)\) | \(e\left(\frac{4}{17}\right)\) | \(e\left(\frac{29}{34}\right)\) | \(e\left(\frac{6}{17}\right)\) | \(e\left(\frac{13}{34}\right)\) | \(e\left(\frac{21}{34}\right)\) | \(e\left(\frac{31}{34}\right)\) |
| \(\chi_{2312}(2141,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{17}\right)\) | \(e\left(\frac{8}{17}\right)\) | \(e\left(\frac{21}{34}\right)\) | \(e\left(\frac{2}{17}\right)\) | \(e\left(\frac{6}{17}\right)\) | \(e\left(\frac{1}{34}\right)\) | \(e\left(\frac{9}{17}\right)\) | \(e\left(\frac{11}{34}\right)\) | \(e\left(\frac{23}{34}\right)\) | \(e\left(\frac{21}{34}\right)\) |
| \(\chi_{2312}(2277,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{17}\right)\) | \(e\left(\frac{5}{17}\right)\) | \(e\left(\frac{11}{34}\right)\) | \(e\left(\frac{14}{17}\right)\) | \(e\left(\frac{8}{17}\right)\) | \(e\left(\frac{7}{34}\right)\) | \(e\left(\frac{12}{17}\right)\) | \(e\left(\frac{9}{34}\right)\) | \(e\left(\frac{25}{34}\right)\) | \(e\left(\frac{11}{34}\right)\) |