Properties

Label 4757.4756
Modulus $4757$
Conductor $4757$
Order $2$
Real yes
Primitive yes
Minimal yes
Parity even

Related objects

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4757, base_ring=CyclotomicField(2))
 
M = H._module
 
chi = DirichletCharacter(H, M([1,1]))
 
pari: [g,chi] = znchar(Mod(4756,4757))
 

Kronecker symbol representation

sage: kronecker_character(4757)
 
pari: znchartokronecker(g,chi)
 

\(\displaystyle\left(\frac{4757}{\bullet}\right)\)

Basic properties

Modulus: \(4757\)
Conductor: \(4757\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(2\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: yes
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4757.c

\(\chi_{4757}(4756,\cdot)\)

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: \(\Q(\sqrt{4757}) \)

Values on generators

\((1208,2279)\) → \((-1,-1)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4757 }(4756, a) \) \(1\)\(1\)\(-1\)\(-1\)\(1\)\(-1\)\(1\)\(1\)\(-1\)\(1\)\(1\)\(1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4757 }(4756,a) \;\) at \(\;a = \) e.g. 2