from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1098, base_ring=CyclotomicField(12))
M = H._module
chi = DirichletCharacter(H, M([6,11]))
chi.galois_orbit()
[g,chi] = znchar(Mod(143,1098))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
Modulus: | \(1098\) | |
Conductor: | \(183\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(12\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from 183.o | 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: | \(\Q(\zeta_{12})\) |
Fixed field: | 12.12.31721645938736726383869.1 |
Characters in Galois orbit
Character | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\chi_{1098}(143,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(i\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(-i\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{12}\right)\) |
\(\chi_{1098}(215,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(-i\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(i\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{11}{12}\right)\) |
\(\chi_{1098}(395,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(i\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(-i\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{5}{12}\right)\) |
\(\chi_{1098}(467,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(-i\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(i\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{7}{12}\right)\) |