from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2873, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([6,13]))
pari: [g,chi] = znchar(Mod(118,2873))
Basic properties
| Modulus: | \(2873\) | |
| Conductor: | \(2873\) | 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: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2873.bh
\(\chi_{2873}(118,\cdot)\) \(\chi_{2873}(560,\cdot)\) \(\chi_{2873}(781,\cdot)\) \(\chi_{2873}(1002,\cdot)\) \(\chi_{2873}(1223,\cdot)\) \(\chi_{2873}(1444,\cdot)\) \(\chi_{2873}(1665,\cdot)\) \(\chi_{2873}(1886,\cdot)\) \(\chi_{2873}(2107,\cdot)\) \(\chi_{2873}(2328,\cdot)\) \(\chi_{2873}(2549,\cdot)\) \(\chi_{2873}(2770,\cdot)\)
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
\((171,2536)\) → \((e\left(\frac{3}{13}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 2873 }(118, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{7}{26}\right)\) |
sage: chi.jacobi_sum(n)