Properties

Label 4760.2041
Modulus $4760$
Conductor $7$
Order $3$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4760, base_ring=CyclotomicField(6))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,0,0,4,0]))
 
pari: [g,chi] = znchar(Mod(2041,4760))
 

Basic properties

Modulus: \(4760\)
Conductor: \(7\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(3\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{7}(4,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4760.bg

\(\chi_{4760}(681,\cdot)\) \(\chi_{4760}(2041,\cdot)\)

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: \(\mathbb{Q}(\zeta_3)\)
Fixed field: \(\Q(\zeta_{7})^+\)

Values on generators

\((1191,2381,2857,1361,3641)\) → \((1,1,1,e\left(\frac{2}{3}\right),1)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(9\)\(11\)\(13\)\(19\)\(23\)\(27\)\(29\)\(31\)\(33\)
\( \chi_{ 4760 }(2041, a) \) \(1\)\(1\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{2}{3}\right)\)\(1\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(1\)\(1\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{3}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4760 }(2041,a) \;\) at \(\;a = \) e.g. 2