from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6422, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([8,0]))
pari: [g,chi] = znchar(Mod(495,6422))
Basic properties
Modulus: | \(6422\) | |
Conductor: | \(169\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(13\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{169}(157,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6422.bh
\(\chi_{6422}(495,\cdot)\) \(\chi_{6422}(989,\cdot)\) \(\chi_{6422}(1483,\cdot)\) \(\chi_{6422}(1977,\cdot)\) \(\chi_{6422}(2471,\cdot)\) \(\chi_{6422}(2965,\cdot)\) \(\chi_{6422}(3459,\cdot)\) \(\chi_{6422}(3953,\cdot)\) \(\chi_{6422}(4447,\cdot)\) \(\chi_{6422}(4941,\cdot)\) \(\chi_{6422}(5435,\cdot)\) \(\chi_{6422}(5929,\cdot)\)
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: | 13.13.542800770374370512771595361.1 |
Values on generators
\((4903,4733)\) → \((e\left(\frac{4}{13}\right),1)\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(15\) | \(17\) | \(21\) | \(23\) | \(25\) |
\( \chi_{ 6422 }(495, a) \) | \(1\) | \(1\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(1\) | \(e\left(\frac{7}{13}\right)\) |
sage: chi.jacobi_sum(n)