from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1368, base_ring=CyclotomicField(6))
M = H._module
chi = DirichletCharacter(H, M([3,3,1,1]))
pari: [g,chi] = znchar(Mod(1091,1368))
Basic properties
| Modulus: | \(1368\) | |
| Conductor: | \(1368\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(6\) | 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 1368.cz
\(\chi_{1368}(563,\cdot)\) \(\chi_{1368}(1091,\cdot)\)
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: | \(\mathbb{Q}(\zeta_3)\) |
| Fixed field: | 6.0.24953372987904.2 |
Values on generators
\((343,685,1217,1009)\) → \((-1,-1,e\left(\frac{1}{6}\right),e\left(\frac{1}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 1368 }(1091, a) \) | \(-1\) | \(1\) | \(1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(1\) | \(-1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) |
sage: chi.jacobi_sum(n)