from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2347, base_ring=CyclotomicField(34))
M = H._module
chi = DirichletCharacter(H, M([13]))
pari: [g,chi] = znchar(Mod(28,2347))
Basic properties
| Modulus: | \(2347\) | |
| Conductor: | \(2347\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(34\) | 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 2347.g
\(\chi_{2347}(28,\cdot)\) \(\chi_{2347}(258,\cdot)\) \(\chi_{2347}(274,\cdot)\) \(\chi_{2347}(430,\cdot)\) \(\chi_{2347}(513,\cdot)\) \(\chi_{2347}(829,\cdot)\) \(\chi_{2347}(855,\cdot)\) \(\chi_{2347}(1239,\cdot)\) \(\chi_{2347}(1425,\cdot)\) \(\chi_{2347}(1499,\cdot)\) \(\chi_{2347}(1563,\cdot)\) \(\chi_{2347}(1716,\cdot)\) \(\chi_{2347}(1877,\cdot)\) \(\chi_{2347}(2042,\cdot)\) \(\chi_{2347}(2065,\cdot)\) \(\chi_{2347}(2164,\cdot)\)
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: | \(\Q(\zeta_{17})\) |
| Fixed field: | Number field defined by a degree 34 polynomial |
Values on generators
\(3\) → \(e\left(\frac{13}{34}\right)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 2347 }(28, a) \) | \(-1\) | \(1\) | \(e\left(\frac{33}{34}\right)\) | \(e\left(\frac{13}{34}\right)\) | \(e\left(\frac{16}{17}\right)\) | \(e\left(\frac{11}{34}\right)\) | \(e\left(\frac{6}{17}\right)\) | \(e\left(\frac{1}{34}\right)\) | \(e\left(\frac{31}{34}\right)\) | \(e\left(\frac{13}{17}\right)\) | \(e\left(\frac{5}{17}\right)\) | \(e\left(\frac{13}{34}\right)\) |
sage: chi.jacobi_sum(n)