Properties

Label 3.2
Modulus $3$
Conductor $3$
Order $2$
Real yes
Primitive yes
Minimal yes
Parity odd

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3, base_ring=CyclotomicField(2))
 
M = H._module
 
chi = DirichletCharacter(H, M([1]))
 
pari: [g,chi] = znchar(Mod(2,3))
 

Kronecker symbol representation

sage: kronecker_character(-3)
 
pari: znchartokronecker(g,chi)
 

\(\displaystyle\left(\frac{-3}{\bullet}\right)\)

Basic properties

Modulus: \(3\)
Conductor: \(3\)
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: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3.b

\(\chi_{3}(2,\cdot)\)

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: \(\Q(\sqrt{-3}) \)

Values on generators

\(2\) → \(-1\)

Values

\(a\) \(-1\)\(1\)
\( \chi_{ 3 }(2, a) \) \(-1\)\(1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3 }(2,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 3 }(2,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 3 }(2,·),\chi_{ 3 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 3 }(2,·)) \;\) at \(\; a,b = \) e.g. 1,2

Additional information

This is the first Dirichlet character taking a value other than $1$. It is also the first Dirichlet character with odd parity.