Properties

Label 3332.587
Modulus $3332$
Conductor $476$
Order $8$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3332, base_ring=CyclotomicField(8))
 
M = H._module
 
chi = DirichletCharacter(H, M([4,4,1]))
 
pari: [g,chi] = znchar(Mod(587,3332))
 

Basic properties

Modulus: \(3332\)
Conductor: \(476\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(8\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{476}(111,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3332.x

\(\chi_{3332}(195,\cdot)\) \(\chi_{3332}(587,\cdot)\) \(\chi_{3332}(1175,\cdot)\) \(\chi_{3332}(2939,\cdot)\)

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: \(\Q(\zeta_{8})\)
Fixed field: 8.8.252217127391488.1

Values on generators

\((1667,885,785)\) → \((-1,-1,e\left(\frac{1}{8}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(19\)\(23\)\(25\)\(27\)
\( \chi_{ 3332 }(587, a) \) \(1\)\(1\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{1}{8}\right)\)\(i\)\(e\left(\frac{3}{8}\right)\)\(1\)\(i\)\(-i\)\(e\left(\frac{3}{8}\right)\)\(i\)\(e\left(\frac{3}{8}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3332 }(587,a) \;\) at \(\;a = \) e.g. 2