Properties

Label 11.4
Modulus $11$
Conductor $11$
Order $5$
Real no
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(11, base_ring=CyclotomicField(10))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([2]))
 
pari: [g,chi] = znchar(Mod(4,11))
 

Basic properties

Modulus: \(11\)
Conductor: \(11\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(5\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
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 11.c

\(\chi_{11}(3,\cdot)\) \(\chi_{11}(4,\cdot)\) \(\chi_{11}(5,\cdot)\) \(\chi_{11}(9,\cdot)\)

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

\(2\) → \(e\left(\frac{1}{5}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)
\(1\)\(1\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{1}{5}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{5})\)
Fixed field: \(\Q(\zeta_{11})^+\)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 11 }(4,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{11}(4,\cdot)) = \sum_{r\in \Z/11\Z} \chi_{11}(4,r) e\left(\frac{2r}{11}\right) = 2.7287895997+-1.8851279322i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 11 }(4,·),\chi_{ 11 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{11}(4,\cdot),\chi_{11}(1,\cdot)) = \sum_{r\in \Z/11\Z} \chi_{11}(4,r) \chi_{11}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 11 }(4,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{11}(4,·)) = \sum_{r \in \Z/11\Z} \chi_{11}(4,r) e\left(\frac{1 r + 2 r^{-1}}{11}\right) = -2.8318726053+-2.0574758816i \)