from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(320, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([8,7,8]))
chi.galois_orbit()
[g,chi] = znchar(Mod(19,320))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(320\) | |
| Conductor: | \(320\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(16\) | 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: | \(\Q(\zeta_{16})\) |
| Fixed field: | 16.0.236118324143482260684800000000.1 |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{320}(19,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(-i\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{15}{16}\right)\) |
| \(\chi_{320}(59,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(i\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{9}{16}\right)\) |
| \(\chi_{320}(99,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(-i\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{3}{16}\right)\) |
| \(\chi_{320}(139,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(i\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{13}{16}\right)\) |
| \(\chi_{320}(179,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(-i\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{7}{16}\right)\) |
| \(\chi_{320}(219,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(i\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{1}{16}\right)\) |
| \(\chi_{320}(259,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(-i\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{11}{16}\right)\) |
| \(\chi_{320}(299,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(i\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{5}{16}\right)\) |