Properties

Label 1596.1595
Modulus $1596$
Conductor $1596$
Order $2$
Real yes
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1596, base_ring=CyclotomicField(2))
 
M = H._module
 
chi = DirichletCharacter(H, M([1,1,1,1]))
 
pari: [g,chi] = znchar(Mod(1595,1596))
 

Kronecker symbol representation

sage: kronecker_character(1596)
 
pari: znchartokronecker(g,chi)
 

\(\displaystyle\left(\frac{1596}{\bullet}\right)\)

Basic properties

Modulus: \(1596\)
Conductor: \(1596\)
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 1596.p

\(\chi_{1596}(1595,\cdot)\)

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: \(\Q(\sqrt{399}) \)

Values on generators

\((799,533,913,1009)\) → \((-1,-1,-1,-1)\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(23\)\(25\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 1596 }(1595, a) \) \(1\)\(1\)\(1\)\(1\)\(1\)\(1\)\(1\)\(1\)\(1\)\(-1\)\(-1\)\(-1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1596 }(1595,a) \;\) at \(\;a = \) e.g. 2