Properties

Label 3360.853
Modulus $3360$
Conductor $1120$
Order $8$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3360, base_ring=CyclotomicField(8))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,5,0,6,4]))
 
pari: [g,chi] = znchar(Mod(853,3360))
 

Basic properties

Modulus: \(3360\)
Conductor: \(1120\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(8\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1120}(853,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3360.ez

\(\chi_{3360}(853,\cdot)\) \(\chi_{3360}(1357,\cdot)\) \(\chi_{3360}(2533,\cdot)\) \(\chi_{3360}(3037,\cdot)\)

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: \(\Q(\zeta_{8})\)
Fixed field: 8.8.80564191232000000.5

Values on generators

\((1471,421,1121,2017,1921)\) → \((1,e\left(\frac{5}{8}\right),1,-i,-1)\)

First values

\(a\) \(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 3360 }(853, a) \) \(1\)\(1\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{1}{8}\right)\)\(-i\)\(e\left(\frac{3}{8}\right)\)\(1\)\(e\left(\frac{3}{8}\right)\)\(-1\)\(e\left(\frac{3}{8}\right)\)\(i\)\(e\left(\frac{3}{8}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3360 }(853,a) \;\) at \(\;a = \) e.g. 2