Properties

Label 6039.4223
Modulus $6039$
Conductor $6039$
Order $6$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6039, base_ring=CyclotomicField(6))
 
M = H._module
 
chi = DirichletCharacter(H, M([1,3,5]))
 
pari: [g,chi] = znchar(Mod(4223,6039))
 

Basic properties

Modulus: \(6039\)
Conductor: \(6039\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(6\)
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 6039.z

\(\chi_{6039}(4223,\cdot)\) \(\chi_{6039}(5477,\cdot)\)

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: Number field defined by a degree 6 polynomial

Values on generators

\((1343,5491,5248)\) → \((e\left(\frac{1}{6}\right),-1,e\left(\frac{5}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(13\)\(14\)\(16\)\(17\)
\( \chi_{ 6039 }(4223, a) \) \(1\)\(1\)\(-1\)\(1\)\(e\left(\frac{1}{6}\right)\)\(1\)\(-1\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{6}\right)\)\(-1\)\(1\)\(e\left(\frac{1}{6}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6039 }(4223,a) \;\) at \(\;a = \) e.g. 2