Properties

Label 6023.1586
Modulus $6023$
Conductor $19$
Order $9$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6023, base_ring=CyclotomicField(18))
 
M = H._module
 
chi = DirichletCharacter(H, M([8,0]))
 
pari: [g,chi] = znchar(Mod(1586,6023))
 

Basic properties

Modulus: \(6023\)
Conductor: \(19\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(9\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{19}(9,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6023.k

\(\chi_{6023}(1586,\cdot)\) \(\chi_{6023}(2220,\cdot)\) \(\chi_{6023}(2854,\cdot)\) \(\chi_{6023}(3171,\cdot)\) \(\chi_{6023}(3805,\cdot)\) \(\chi_{6023}(4756,\cdot)\)

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: \(\Q(\zeta_{19})^+\)

Values on generators

\((952,5074)\) → \((e\left(\frac{4}{9}\right),1)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 6023 }(1586, a) \) \(1\)\(1\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{7}{9}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{5}{9}\right)\)\(e\left(\frac{5}{9}\right)\)\(e\left(\frac{1}{3}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6023 }(1586,a) \;\) at \(\;a = \) e.g. 2