Properties

Label 8640.2753
Modulus $8640$
Conductor $15$
Order $4$
Real no
Primitive no
Minimal no
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8640, base_ring=CyclotomicField(4))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,0,2,3]))
 
pari: [g,chi] = znchar(Mod(2753,8640))
 

Basic properties

Modulus: \(8640\)
Conductor: \(15\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(4\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{15}(8,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8640.w

\(\chi_{8640}(2753,\cdot)\) \(\chi_{8640}(7937,\cdot)\)

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: \(\Q(\zeta_{15})^+\)

Values on generators

\((2431,3781,6401,3457)\) → \((1,1,-1,-i)\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 8640 }(2753, a) \) \(1\)\(1\)\(-i\)\(-1\)\(i\)\(i\)\(-1\)\(-i\)\(1\)\(1\)\(-i\)\(-1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8640 }(2753,a) \;\) at \(\;a = \) e.g. 2