from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1989, base_ring=CyclotomicField(24))
M = H._module
chi = DirichletCharacter(H, M([8,4,21]))
chi.galois_orbit()
[g,chi] = znchar(Mod(121,1989))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
Modulus: | \(1989\) | |
Conductor: | \(1989\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(24\) | 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: | \(\Q(\zeta_{24})\) |
Fixed field: | Number field defined by a degree 24 polynomial |
Characters in Galois orbit
Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(14\) | \(16\) | \(19\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\chi_{1989}(121,\cdot)\) | \(1\) | \(1\) | \(-i\) | \(-1\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(i\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(1\) | \(e\left(\frac{1}{12}\right)\) |
\(\chi_{1989}(355,\cdot)\) | \(1\) | \(1\) | \(-i\) | \(-1\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(i\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(1\) | \(e\left(\frac{1}{12}\right)\) |
\(\chi_{1989}(706,\cdot)\) | \(1\) | \(1\) | \(i\) | \(-1\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(-i\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(1\) | \(e\left(\frac{7}{12}\right)\) |
\(\chi_{1989}(790,\cdot)\) | \(1\) | \(1\) | \(i\) | \(-1\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(-i\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(1\) | \(e\left(\frac{11}{12}\right)\) |
\(\chi_{1989}(1141,\cdot)\) | \(1\) | \(1\) | \(-i\) | \(-1\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(i\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(1\) | \(e\left(\frac{5}{12}\right)\) |
\(\chi_{1989}(1375,\cdot)\) | \(1\) | \(1\) | \(-i\) | \(-1\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(i\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(1\) | \(e\left(\frac{5}{12}\right)\) |
\(\chi_{1989}(1726,\cdot)\) | \(1\) | \(1\) | \(i\) | \(-1\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(-i\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(1\) | \(e\left(\frac{11}{12}\right)\) |
\(\chi_{1989}(1759,\cdot)\) | \(1\) | \(1\) | \(i\) | \(-1\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(-i\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(1\) | \(e\left(\frac{7}{12}\right)\) |