Properties

Label 9792.263
Modulus $9792$
Conductor $4896$
Order $24$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9792, base_ring=CyclotomicField(24))
 
M = H._module
 
chi = DirichletCharacter(H, M([12,15,4,15]))
 
pari: [g,chi] = znchar(Mod(263,9792))
 

Basic properties

Modulus: \(9792\)
Conductor: \(4896\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(24\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{4896}(875,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 9792.kc

\(\chi_{9792}(263,\cdot)\) \(\chi_{9792}(695,\cdot)\) \(\chi_{9792}(1175,\cdot)\) \(\chi_{9792}(4439,\cdot)\) \(\chi_{9792}(6503,\cdot)\) \(\chi_{9792}(6791,\cdot)\) \(\chi_{9792}(7223,\cdot)\) \(\chi_{9792}(9767,\cdot)\)

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: \(\Q(\zeta_{24})\)
Fixed field: 24.24.102702760180080482238862265758511365777986854716768014313536599353196544.2

Values on generators

\((7039,5509,8705,9217)\) → \((-1,e\left(\frac{5}{8}\right),e\left(\frac{1}{6}\right),e\left(\frac{5}{8}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 9792 }(263, a) \) \(1\)\(1\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{7}{24}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{5}{24}\right)\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{11}{24}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{11}{24}\right)\)\(e\left(\frac{7}{8}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 9792 }(263,a) \;\) at \(\;a = \) e.g. 2