Properties

Label 6864.1585
Modulus $6864$
Conductor $13$
Order $2$
Real yes
Primitive no
Minimal no
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6864, base_ring=CyclotomicField(2))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,0,0,0,1]))
 
pari: [g,chi] = znchar(Mod(1585,6864))
 

Basic properties

Modulus: \(6864\)
Conductor: \(13\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(2\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: yes
Primitive: no, induced from \(\chi_{13}(12,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6864.g

\(\chi_{6864}(1585,\cdot)\)

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: \(\Q\)
Fixed field: \(\Q(\sqrt{13}) \)

Values on generators

\((2575,1717,4577,4369,2641)\) → \((1,1,1,1,-1)\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)\(37\)
\( \chi_{ 6864 }(1585, a) \) \(1\)\(1\)\(-1\)\(-1\)\(1\)\(-1\)\(1\)\(1\)\(1\)\(-1\)\(1\)\(-1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6864 }(1585,a) \;\) at \(\;a = \) e.g. 2