Properties

Label 2898.1103
Modulus $2898$
Conductor $1449$
Order $6$
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(2898, base_ring=CyclotomicField(6))
 
M = H._module
 
chi = DirichletCharacter(H, M([5,4,3]))
 
pari: [g,chi] = znchar(Mod(1103,2898))
 

Basic properties

Modulus: \(2898\)
Conductor: \(1449\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(6\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1449}(1103,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2898.bn

\(\chi_{2898}(1103,\cdot)\) \(\chi_{2898}(2207,\cdot)\)

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: \(\Q(\sqrt{-3}) \)
Fixed field: 6.6.574998829461.4

Values on generators

\((1289,829,1891)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{2}{3}\right),-1)\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(25\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 2898 }(1103, a) \) \(1\)\(1\)\(1\)\(1\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{5}{6}\right)\)\(1\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{1}{6}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2898 }(1103,a) \;\) at \(\;a = \) e.g. 2