Properties

Label 4176.331
Modulus $4176$
Conductor $4176$
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(4176, base_ring=CyclotomicField(12))
 
M = H._module
 
chi = DirichletCharacter(H, M([6,3,8,3]))
 
pari: [g,chi] = znchar(Mod(331,4176))
 

Basic properties

Modulus: \(4176\)
Conductor: \(4176\)
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 4176.cj

\(\chi_{4176}(331,\cdot)\) \(\chi_{4176}(1699,\cdot)\) \(\chi_{4176}(1723,\cdot)\) \(\chi_{4176}(3091,\cdot)\)

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: 12.12.5364285864861299821665224491008.1

Values on generators

\((1567,1045,929,4033)\) → \((-1,i,e\left(\frac{2}{3}\right),i)\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(31\)\(35\)
\( \chi_{ 4176 }(331, a) \) \(1\)\(1\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{7}{12}\right)\)\(i\)\(-1\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{1}{12}\right)\)\(-i\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4176 }(331,a) \;\) at \(\;a = \) e.g. 2