Properties

Label 4508.183
Modulus $4508$
Conductor $4508$
Order $14$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4508, base_ring=CyclotomicField(14))
 
M = H._module
 
chi = DirichletCharacter(H, M([7,4,7]))
 
pari: [g,chi] = znchar(Mod(183,4508))
 

Basic properties

Modulus: \(4508\)
Conductor: \(4508\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(14\)
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 4508.x

\(\chi_{4508}(183,\cdot)\) \(\chi_{4508}(827,\cdot)\) \(\chi_{4508}(2115,\cdot)\) \(\chi_{4508}(2759,\cdot)\) \(\chi_{4508}(3403,\cdot)\) \(\chi_{4508}(4047,\cdot)\)

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: \(\Q(\zeta_{7})\)
Fixed field: Number field defined by a degree 14 polynomial

Values on generators

\((2255,1473,1569)\) → \((-1,e\left(\frac{2}{7}\right),-1)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(25\)\(27\)
\( \chi_{ 4508 }(183, a) \) \(1\)\(1\)\(e\left(\frac{11}{14}\right)\)\(e\left(\frac{11}{14}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{9}{14}\right)\)\(1\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{5}{14}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4508 }(183,a) \;\) at \(\;a = \) e.g. 2