Properties

Label 9576.1427
Modulus $9576$
Conductor $9576$
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(9576, base_ring=CyclotomicField(18))
 
M = H._module
 
chi = DirichletCharacter(H, M([9,9,15,9,1]))
 
pari: [g,chi] = znchar(Mod(1427,9576))
 

Basic properties

Modulus: \(9576\)
Conductor: \(9576\)
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 9576.rv

\(\chi_{9576}(1427,\cdot)\) \(\chi_{9576}(2435,\cdot)\) \(\chi_{9576}(3107,\cdot)\) \(\chi_{9576}(3947,\cdot)\) \(\chi_{9576}(6131,\cdot)\) \(\chi_{9576}(8147,\cdot)\)

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

\((7183,4789,5321,4105,1009)\) → \((-1,-1,e\left(\frac{5}{6}\right),-1,e\left(\frac{1}{18}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(23\)\(25\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 9576 }(1427, a) \) \(1\)\(1\)\(e\left(\frac{1}{18}\right)\)\(-1\)\(e\left(\frac{17}{18}\right)\)\(e\left(\frac{5}{9}\right)\)\(e\left(\frac{7}{9}\right)\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{5}{18}\right)\)\(-1\)\(1\)\(e\left(\frac{7}{18}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 9576 }(1427,a) \;\) at \(\;a = \) e.g. 2