Properties

Label 6840.709
Modulus $6840$
Conductor $6840$
Order $18$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6840, base_ring=CyclotomicField(18))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,9,12,9,14]))
 
pari: [g,chi] = znchar(Mod(709,6840))
 

Basic properties

Modulus: \(6840\)
Conductor: \(6840\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(18\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6840.lj

\(\chi_{6840}(709,\cdot)\) \(\chi_{6840}(1309,\cdot)\) \(\chi_{6840}(1429,\cdot)\) \(\chi_{6840}(1669,\cdot)\) \(\chi_{6840}(5629,\cdot)\) \(\chi_{6840}(6469,\cdot)\)

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: \(\Q(\zeta_{9})\)
Fixed field: Number field defined by a degree 18 polynomial

Values on generators

\((1711,3421,5321,2737,6481)\) → \((1,-1,e\left(\frac{2}{3}\right),-1,e\left(\frac{7}{9}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 6840 }(709, a) \) \(1\)\(1\)\(e\left(\frac{5}{6}\right)\)\(-1\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{5}{18}\right)\)\(e\left(\frac{7}{18}\right)\)\(e\left(\frac{7}{18}\right)\)\(1\)\(1\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{1}{9}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6840 }(709,a) \;\) at \(\;a = \) e.g. 2