Properties

Label 7488.3457
Modulus $7488$
Conductor $13$
Order $2$
Real yes
Primitive no
Minimal no
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7488, base_ring=CyclotomicField(2))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,0,0,1]))
 
pari: [g,chi] = znchar(Mod(3457,7488))
 

Basic properties

Modulus: \(7488\)
Conductor: \(13\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(2\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: yes
Primitive: no, induced from \(\chi_{13}(12,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 7488.c

\(\chi_{7488}(3457,\cdot)\)

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: \(\Q\)
Fixed field: \(\Q(\sqrt{13}) \)

Values on generators

\((703,6085,5825,5761)\) → \((1,1,1,-1)\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 7488 }(3457, a) \) \(1\)\(1\)\(-1\)\(-1\)\(-1\)\(1\)\(-1\)\(1\)\(1\)\(1\)\(-1\)\(1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 7488 }(3457,a) \;\) at \(\;a = \) e.g. 2