from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8112, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([13,0,13,7]))
chi.galois_orbit()
[g,chi] = znchar(Mod(623,8112))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(8112\) | |
| Conductor: | \(2028\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(26\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 2028.z | 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: | 26.26.409808027834211389172943903711535494765866720064678008376474736787456.1 |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{8112}(623,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(1\) | \(1\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{3}{13}\right)\) |
| \(\chi_{8112}(1247,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(1\) | \(1\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{6}{13}\right)\) |
| \(\chi_{8112}(1871,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(1\) | \(1\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{9}{13}\right)\) |
| \(\chi_{8112}(2495,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(1\) | \(1\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{12}{13}\right)\) |
| \(\chi_{8112}(3119,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(1\) | \(1\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{2}{13}\right)\) |
| \(\chi_{8112}(3743,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(1\) | \(1\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{5}{13}\right)\) |
| \(\chi_{8112}(4367,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(1\) | \(1\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{8}{13}\right)\) |
| \(\chi_{8112}(4991,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(1\) | \(1\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{11}{13}\right)\) |
| \(\chi_{8112}(5615,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(1\) | \(1\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{1}{13}\right)\) |
| \(\chi_{8112}(6239,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(1\) | \(1\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{4}{13}\right)\) |
| \(\chi_{8112}(6863,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(1\) | \(1\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{7}{13}\right)\) |
| \(\chi_{8112}(7487,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(1\) | \(1\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{10}{13}\right)\) |