Properties

Label 4176.755
Modulus $4176$
Conductor $48$
Order $4$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4176, base_ring=CyclotomicField(4))
 
M = H._module
 
chi = DirichletCharacter(H, M([2,3,2,0]))
 
pari: [g,chi] = znchar(Mod(755,4176))
 

Basic properties

Modulus: \(4176\)
Conductor: \(48\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(4\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{48}(35,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4176.bf

\(\chi_{4176}(755,\cdot)\) \(\chi_{4176}(2843,\cdot)\)

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: \(\Q(\sqrt{-1}) \)
Fixed field: 4.4.18432.1

Values on generators

\((1567,1045,929,4033)\) → \((-1,-i,-1,1)\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(31\)\(35\)
\( \chi_{ 4176 }(755, a) \) \(1\)\(1\)\(i\)\(1\)\(-i\)\(i\)\(-1\)\(-i\)\(-1\)\(-1\)\(-1\)\(i\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4176 }(755,a) \;\) at \(\;a = \) e.g. 2