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Properties

Label 4.3
Modulus $4$
Conductor $4$
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(4, base_ring=CyclotomicField(2)) M = H._module chi = DirichletCharacter(H, M([1]))
 
Copy content pari:[g,chi] = znchar(Mod(3,4))
 

Kronecker symbol representation

Copy content sage:kronecker_character(-4)
 
Copy content pari:znchartokronecker(g,chi)
 

\(\displaystyle\left(\frac{-4}{\bullet}\right)\)

Basic properties

Modulus: \(4\)
Conductor: \(4\)
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 4.b

\(\chi_{4}(3,\cdot)\)

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: \(\Q(\sqrt{-1}) \)

Values on generators

\(3\) → \(-1\)

Values

\(a\) \(-1\)\(1\)
\( \chi_{ 4 }(3, a) \) \(-1\)\(1\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 4 }(3,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 4 }(3,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 4 }(3,·),\chi_{ 4 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 4 }(3,·)) \;\) at \(\; a,b = \) e.g. 1,2

Additional information

This is the first Dirichlet character with composite conductor.