Properties

Label 9576.649
Modulus $9576$
Conductor $133$
Order $18$
Real no
Primitive no
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([0,0,0,15,13]))
 
pari: [g,chi] = znchar(Mod(649,9576))
 

Basic properties

Modulus: \(9576\)
Conductor: \(133\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(18\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{133}(117,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 9576.bbj

\(\chi_{9576}(649,\cdot)\) \(\chi_{9576}(2161,\cdot)\) \(\chi_{9576}(3601,\cdot)\) \(\chi_{9576}(6625,\cdot)\) \(\chi_{9576}(8641,\cdot)\) \(\chi_{9576}(9217,\cdot)\)

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: 18.18.26018473705616588670034125883291477.1

Values on generators

\((7183,4789,5321,4105,1009)\) → \((1,1,1,e\left(\frac{5}{6}\right),e\left(\frac{13}{18}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(23\)\(25\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 9576 }(649, a) \) \(1\)\(1\)\(e\left(\frac{13}{18}\right)\)\(1\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{5}{18}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{8}{9}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 9576 }(649,a) \;\) at \(\;a = \) e.g. 2