Properties

Label 12.11
Modulus $12$
Conductor $12$
Order $2$
Real yes
Primitive yes
Minimal yes
Parity even

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(12)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([1,1]))
 
pari: [g,chi] = znchar(Mod(11,12))
 

Kronecker symbol representation

sage: kronecker_character(12)
 
pari: znchartokronecker(g,chi)
 

\(\displaystyle\left(\frac{12}{\bullet}\right)\)

Basic properties

Modulus: \(12\)
Conductor: \(12\)
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 12.b

\(\chi_{12}(11,\cdot)\)

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

\((7,5)\) → \((-1,-1)\)

Values

\(-1\)\(1\)\(5\)\(7\)
\(1\)\(1\)\(-1\)\(-1\)
value at e.g. 2

Related number fields

Field of values: \(\Q\)
Fixed field: \(\Q(\sqrt{3}) \)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 12 }(11,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{12}(11,\cdot)) = \sum_{r\in \Z/12\Z} \chi_{12}(11,r) e\left(\frac{r}{6}\right) = 0.0 \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 12 }(11,·),\chi_{ 12 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{12}(11,\cdot),\chi_{12}(1,\cdot)) = \sum_{r\in \Z/12\Z} \chi_{12}(11,r) \chi_{12}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 12 }(11,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{12}(11,·)) = \sum_{r \in \Z/12\Z} \chi_{12}(11,r) e\left(\frac{1 r + 2 r^{-1}}{12}\right) = 0.0 \)