Properties

Label 7488.647
Modulus $7488$
Conductor $1248$
Order $24$
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(7488, base_ring=CyclotomicField(24))
 
M = H._module
 
chi = DirichletCharacter(H, M([12,15,12,20]))
 
pari: [g,chi] = znchar(Mod(647,7488))
 

Basic properties

Modulus: \(7488\)
Conductor: \(1248\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(24\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1248}(491,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 7488.kd

\(\chi_{7488}(647,\cdot)\) \(\chi_{7488}(1655,\cdot)\) \(\chi_{7488}(2519,\cdot)\) \(\chi_{7488}(3527,\cdot)\) \(\chi_{7488}(4391,\cdot)\) \(\chi_{7488}(5399,\cdot)\) \(\chi_{7488}(6263,\cdot)\) \(\chi_{7488}(7271,\cdot)\)

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: \(\Q(\zeta_{24})\)
Fixed field: 24.24.100025723073455953280051851768449931504820595788844367872.1

Values on generators

\((703,6085,5825,5761)\) → \((-1,e\left(\frac{5}{8}\right),-1,e\left(\frac{5}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 7488 }(647, a) \) \(1\)\(1\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{23}{24}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{24}\right)\)\(e\left(\frac{1}{12}\right)\)\(i\)\(e\left(\frac{17}{24}\right)\)\(1\)\(e\left(\frac{13}{24}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 7488 }(647,a) \;\) at \(\;a = \) e.g. 2