from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4056, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([0,13,13,20]))
chi.galois_orbit()
[g,chi] = znchar(Mod(53,4056))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(4056\) | |
| Conductor: | \(4056\) | 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: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Related number fields
| Field of values: | \(\Q(\zeta_{13})\) |
| Fixed field: | Number field defined by a degree 26 polynomial |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{4056}(53,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(-1\) | \(-1\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{3}{13}\right)\) |
| \(\chi_{4056}(365,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(-1\) | \(-1\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{8}{13}\right)\) |
| \(\chi_{4056}(989,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(-1\) | \(-1\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{5}{13}\right)\) |
| \(\chi_{4056}(1301,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(-1\) | \(-1\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{10}{13}\right)\) |
| \(\chi_{4056}(1613,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(-1\) | \(-1\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{2}{13}\right)\) |
| \(\chi_{4056}(1925,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(-1\) | \(-1\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{7}{13}\right)\) |
| \(\chi_{4056}(2237,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(-1\) | \(-1\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{12}{13}\right)\) |
| \(\chi_{4056}(2549,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(-1\) | \(-1\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{4}{13}\right)\) |
| \(\chi_{4056}(2861,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(-1\) | \(-1\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{9}{13}\right)\) |
| \(\chi_{4056}(3173,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(-1\) | \(-1\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{1}{13}\right)\) |
| \(\chi_{4056}(3485,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(-1\) | \(-1\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{6}{13}\right)\) |
| \(\chi_{4056}(3797,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(-1\) | \(-1\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{11}{13}\right)\) |