Properties

Label 9360.2989
Modulus $9360$
Conductor $1040$
Order $4$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9360, base_ring=CyclotomicField(4))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,3,0,2,2]))
 
pari: [g,chi] = znchar(Mod(2989,9360))
 

Basic properties

Modulus: \(9360\)
Conductor: \(1040\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(4\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1040}(909,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 9360.bq

\(\chi_{9360}(2989,\cdot)\) \(\chi_{9360}(7669,\cdot)\)

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: 4.4.8652800.1

Values on generators

\((8191,2341,2081,5617,5761)\) → \((1,-i,1,-1,-1)\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 9360 }(2989, a) \) \(1\)\(1\)\(-1\)\(i\)\(-1\)\(-i\)\(1\)\(i\)\(-1\)\(-i\)\(1\)\(i\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 9360 }(2989,a) \;\) at \(\;a = \) e.g. 2