from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1859, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([13,20]))
pari: [g,chi] = znchar(Mod(560,1859))
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.v
\(\chi_{1859}(131,\cdot)\) \(\chi_{1859}(274,\cdot)\) \(\chi_{1859}(417,\cdot)\) \(\chi_{1859}(560,\cdot)\) \(\chi_{1859}(703,\cdot)\) \(\chi_{1859}(989,\cdot)\) \(\chi_{1859}(1132,\cdot)\) \(\chi_{1859}(1275,\cdot)\) \(\chi_{1859}(1418,\cdot)\) \(\chi_{1859}(1561,\cdot)\) \(\chi_{1859}(1704,\cdot)\) \(\chi_{1859}(1847,\cdot)\)
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: | Number field defined by a degree 26 polynomial |
Values on generators
\((508,1354)\) → \((-1,e\left(\frac{10}{13}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \( \chi_{ 1859 }(560, a) \) | \(-1\) | \(1\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{12}{13}\right)\) |
sage: chi.jacobi_sum(n)