from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2159, base_ring=CyclotomicField(6))
M = H._module
chi = DirichletCharacter(H, M([0,2]))
pari: [g,chi] = znchar(Mod(1123,2159))
Basic properties
| Modulus: | \(2159\) | |
| Conductor: | \(127\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(3\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{127}(107,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2159.e
\(\chi_{2159}(273,\cdot)\) \(\chi_{2159}(1123,\cdot)\)
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: | 3.3.16129.1 |
Values on generators
\((1652,511)\) → \((1,e\left(\frac{1}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 2159 }(1123, a) \) | \(1\) | \(1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(1\) | \(e\left(\frac{2}{3}\right)\) |
sage: chi.jacobi_sum(n)