from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(169, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([21]))
chi.galois_orbit()
[g,chi] = znchar(Mod(12,169))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(169\) | |
| Conductor: | \(169\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(26\) | 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: | \(\Q(\zeta_{13})\) |
| Fixed field: | 26.26.3830224792147131369362629348887201408953937846517364173.1 |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{169}(12,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{5}{26}\right)\) |
| \(\chi_{169}(25,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{23}{26}\right)\) |
| \(\chi_{169}(38,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{15}{26}\right)\) |
| \(\chi_{169}(51,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{7}{26}\right)\) |
| \(\chi_{169}(64,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{25}{26}\right)\) |
| \(\chi_{169}(77,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{17}{26}\right)\) |
| \(\chi_{169}(90,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{9}{26}\right)\) |
| \(\chi_{169}(103,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{1}{26}\right)\) |
| \(\chi_{169}(116,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{19}{26}\right)\) |
| \(\chi_{169}(129,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{11}{26}\right)\) |
| \(\chi_{169}(142,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{3}{26}\right)\) |
| \(\chi_{169}(155,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{21}{26}\right)\) |