Properties

Modulus 111
Conductor 111
Order 6
Real no
Primitive yes
Minimal yes
Parity odd
Orbit label 111.i

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(111)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([3,2]))
 
pari: [g,chi] = znchar(Mod(26,111))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Modulus = 111
Conductor = 111
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 6
Real = no
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
 
Primitive = yes
Minimal = yes
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = odd
Orbit label = 111.i
Orbit index = 9

Galois orbit

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

\(\chi_{111}(26,\cdot)\) \(\chi_{111}(47,\cdot)\)

Values on generators

\((38,76)\) → \((-1,e\left(\frac{1}{3}\right))\)

Values

-11245781011131416
\(-1\)\(1\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{2}{3}\right)\)\(-1\)\(1\)\(-1\)\(e\left(\frac{2}{3}\right)\)\(-1\)\(e\left(\frac{1}{3}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{3})\)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 111 }(26,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{111}(26,\cdot)) = \sum_{r\in \Z/111\Z} \chi_{111}(26,r) e\left(\frac{2r}{111}\right) = 8.2534858849+-6.5482799839i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 111 }(26,·),\chi_{ 111 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{111}(26,\cdot),\chi_{111}(1,\cdot)) = \sum_{r\in \Z/111\Z} \chi_{111}(26,r) \chi_{111}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 111 }(26,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{111}(26,·)) = \sum_{r \in \Z/111\Z} \chi_{111}(26,r) e\left(\frac{1 r + 2 r^{-1}}{111}\right) = 0.0 \)