Properties

Label 4.3
Modulus $4$
Conductor $4$
Order $2$
Real yes
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(4)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([1]))
 
pari: [g,chi] = znchar(Mod(3,4))
 

Kronecker symbol representation

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

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

Basic properties

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

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

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Values on generators

\(3\) → \(-1\)

Values

\(-1\)\(1\)
\(-1\)\(1\)
value at e.g. 2

Related number fields

Field of values: \(\Q\)
Fixed field: \(\Q(\sqrt{-1}) \)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 4 }(3,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{4}(3,\cdot)) = \sum_{r\in \Z/4\Z} \chi_{4}(3,r) e\left(\frac{r}{2}\right) = 0.0 \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 4 }(3,·),\chi_{ 4 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{4}(3,\cdot),\chi_{4}(1,\cdot)) = \sum_{r\in \Z/4\Z} \chi_{4}(3,r) \chi_{4}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 4 }(3,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{4}(3,·)) = \sum_{r \in \Z/4\Z} \chi_{4}(3,r) e\left(\frac{1 r + 2 r^{-1}}{4}\right) = -2.0i \)

Additional information

This is the first Dirichlet character with composite conductor.