from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1859, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([13,23]))
pari: [g,chi] = znchar(Mod(142,1859))
Basic properties
| Modulus: | \(1859\) | |
| Conductor: | \(1859\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(26\) | 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 1859.u
\(\chi_{1859}(142,\cdot)\) \(\chi_{1859}(285,\cdot)\) \(\chi_{1859}(428,\cdot)\) \(\chi_{1859}(571,\cdot)\) \(\chi_{1859}(714,\cdot)\) \(\chi_{1859}(857,\cdot)\) \(\chi_{1859}(1000,\cdot)\) \(\chi_{1859}(1143,\cdot)\) \(\chi_{1859}(1286,\cdot)\) \(\chi_{1859}(1429,\cdot)\) \(\chi_{1859}(1572,\cdot)\) \(\chi_{1859}(1715,\cdot)\)
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: | \(\Q(\zeta_{13})\) |
| Fixed field: | 26.0.132229747945843362449200441698112979541994803533368394530687918784063.1 |
Values on generators
\((508,1354)\) → \((-1,e\left(\frac{23}{26}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \( \chi_{ 1859 }(142, a) \) | \(-1\) | \(1\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{6}{13}\right)\) |
sage: chi.jacobi_sum(n)