Properties

Label 4368.59
Modulus $4368$
Conductor $4368$
Order $12$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4368, base_ring=CyclotomicField(12))
 
M = H._module
 
chi = DirichletCharacter(H, M([6,3,6,2,11]))
 
pari: [g,chi] = znchar(Mod(59,4368))
 

Basic properties

Modulus: \(4368\)
Conductor: \(4368\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(12\)
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 4368.oi

\(\chi_{4368}(59,\cdot)\) \(\chi_{4368}(227,\cdot)\) \(\chi_{4368}(635,\cdot)\) \(\chi_{4368}(2147,\cdot)\)

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: \(\Q(\zeta_{12})\)
Fixed field: 12.12.3170112398857312997665091547561984.2

Values on generators

\((3823,1093,1457,1249,2017)\) → \((-1,i,-1,e\left(\frac{1}{6}\right),e\left(\frac{11}{12}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 4368 }(59, a) \) \(1\)\(1\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{1}{3}\right)\)\(-1\)\(e\left(\frac{2}{3}\right)\)\(1\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{11}{12}\right)\)\(1\)\(e\left(\frac{5}{12}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4368 }(59,a) \;\) at \(\;a = \) e.g. 2