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Properties

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

Related objects

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

Kronecker symbol representation

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

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

Basic properties

Modulus: \(3\)
Conductor: \(3\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(2\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: yes
Primitive: yes
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 3.b

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

Copy content sage:chi.galois_orbit()
 
Copy content pari: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

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

Gauss sum

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

Jacobi sum

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

Kloosterman sum

Copy content 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.