from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(116, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([14,5]))
chi.galois_orbit()
[g,chi] = znchar(Mod(3,116))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(116\) | |
| Conductor: | \(116\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(28\) | 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: | \(\Q(\zeta_{28})\) |
| Fixed field: | \(\Q(\zeta_{116})^+\) |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{116}(3,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(-i\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{1}{28}\right)\) |
| \(\chi_{116}(11,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(-i\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{5}{28}\right)\) |
| \(\chi_{116}(15,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(i\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{11}{28}\right)\) |
| \(\chi_{116}(19,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(-i\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{13}{28}\right)\) |
| \(\chi_{116}(27,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(i\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{3}{28}\right)\) |
| \(\chi_{116}(31,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(-i\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{17}{28}\right)\) |
| \(\chi_{116}(39,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(i\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{27}{28}\right)\) |
| \(\chi_{116}(43,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(-i\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{25}{28}\right)\) |
| \(\chi_{116}(47,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(i\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{19}{28}\right)\) |
| \(\chi_{116}(55,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(i\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{15}{28}\right)\) |
| \(\chi_{116}(79,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(-i\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{9}{28}\right)\) |
| \(\chi_{116}(95,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(i\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{23}{28}\right)\) |