from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1836, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([8,8,5]))
chi.galois_orbit()
[g,chi] = znchar(Mod(107,1836))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1836\) | |
| Conductor: | \(204\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(16\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 204.t | 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: | 16.0.1230789518357685955321724928.1 |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1836}(107,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(i\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(1\) |
| \(\chi_{1836}(215,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(-i\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(1\) |
| \(\chi_{1836}(431,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(-i\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(1\) |
| \(\chi_{1836}(539,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(i\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(1\) |
| \(\chi_{1836}(755,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(-i\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(1\) |
| \(\chi_{1836}(1187,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(i\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(1\) |
| \(\chi_{1836}(1295,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(i\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(1\) |
| \(\chi_{1836}(1727,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(-i\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(1\) |