from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1197, base_ring=CyclotomicField(18))
M = H._module
chi = DirichletCharacter(H, M([15,6,14]))
pari: [g,chi] = znchar(Mod(842,1197))
Basic properties
| Modulus: | \(1197\) | |
| Conductor: | \(1197\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(18\) | 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)
|
Galois orbit 1197.fv
\(\chi_{1197}(23,\cdot)\) \(\chi_{1197}(74,\cdot)\) \(\chi_{1197}(263,\cdot)\) \(\chi_{1197}(275,\cdot)\) \(\chi_{1197}(842,\cdot)\) \(\chi_{1197}(1145,\cdot)\)
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: | 18.0.30444438798948941043166903413701158032323547.4 |
Values on generators
\((533,514,1009)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{1}{3}\right),e\left(\frac{7}{9}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(20\) |
| \( \chi_{ 1197 }(842, a) \) | \(-1\) | \(1\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(-1\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{5}{6}\right)\) |
sage: chi.jacobi_sum(n)