from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3024, base_ring=CyclotomicField(18))
M = H._module
chi = DirichletCharacter(H, M([0,9,10,15]))
pari: [g,chi] = znchar(Mod(1321,3024))
Basic properties
| Modulus: | \(3024\) | |
| Conductor: | \(1512\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(18\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1512}(565,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3024.fq
\(\chi_{3024}(313,\cdot)\) \(\chi_{3024}(409,\cdot)\) \(\chi_{3024}(1321,\cdot)\) \(\chi_{3024}(1417,\cdot)\) \(\chi_{3024}(2329,\cdot)\) \(\chi_{3024}(2425,\cdot)\)
sage: chi.galois_orbit()
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Related number fields
| Field of values: | \(\Q(\zeta_{9})\) |
| Fixed field: | 18.0.627502832899903433922253495745464472961024.1 |
Values on generators
\((1135,757,785,2593)\) → \((1,-1,e\left(\frac{5}{9}\right),e\left(\frac{5}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 3024 }(1321, a) \) | \(-1\) | \(1\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(-1\) |
sage: chi.jacobi_sum(n)