Properties

Label 111.47
Modulus $111$
Conductor $111$
Order $6$
Real no
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(111)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([3,4]))
 
pari: [g,chi] = znchar(Mod(47,111))
 

Basic properties

Modulus: \(111\)
Conductor: \(111\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(6\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
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 111.i

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

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

\((38,76)\) → \((-1,e\left(\frac{2}{3}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\(-1\)\(1\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{1}{3}\right)\)\(-1\)\(1\)\(-1\)\(e\left(\frac{1}{3}\right)\)\(-1\)\(e\left(\frac{2}{3}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\sqrt{-3}) \)
Fixed field: 6.0.50602347.1

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 111 }(47,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{111}(47,\cdot)) = \sum_{r\in \Z/111\Z} \chi_{111}(47,r) e\left(\frac{2r}{111}\right) = -8.2534858849+-6.5482799839i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 111 }(47,·),\chi_{ 111 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{111}(47,\cdot),\chi_{111}(1,\cdot)) = \sum_{r\in \Z/111\Z} \chi_{111}(47,r) \chi_{111}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 111 }(47,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{111}(47,·)) = \sum_{r \in \Z/111\Z} \chi_{111}(47,r) e\left(\frac{1 r + 2 r^{-1}}{111}\right) = -0.0 \)