from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6422, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([12,13]))
pari: [g,chi] = znchar(Mod(911,6422))
Basic properties
| Modulus: | \(6422\) | |
| Conductor: | \(3211\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(26\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{3211}(911,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6422.bs
\(\chi_{6422}(417,\cdot)\) \(\chi_{6422}(911,\cdot)\) \(\chi_{6422}(1405,\cdot)\) \(\chi_{6422}(1899,\cdot)\) \(\chi_{6422}(2393,\cdot)\) \(\chi_{6422}(2887,\cdot)\) \(\chi_{6422}(3875,\cdot)\) \(\chi_{6422}(4369,\cdot)\) \(\chi_{6422}(4863,\cdot)\) \(\chi_{6422}(5357,\cdot)\) \(\chi_{6422}(5851,\cdot)\) \(\chi_{6422}(6345,\cdot)\)
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
\((4903,4733)\) → \((e\left(\frac{6}{13}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(15\) | \(17\) | \(21\) | \(23\) | \(25\) |
| \( \chi_{ 6422 }(911, a) \) | \(-1\) | \(1\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(1\) | \(e\left(\frac{4}{13}\right)\) |
sage: chi.jacobi_sum(n)