Properties

Label 23.8
Modulus $23$
Conductor $23$
Order $11$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(23, base_ring=CyclotomicField(22))
 
M = H._module
 
chi = DirichletCharacter(H, M([6]))
 
pari: [g,chi] = znchar(Mod(8,23))
 

Basic properties

Modulus: \(23\)
Conductor: \(23\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(11\)
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 23.c

\(\chi_{23}(2,\cdot)\) \(\chi_{23}(3,\cdot)\) \(\chi_{23}(4,\cdot)\) \(\chi_{23}(6,\cdot)\) \(\chi_{23}(8,\cdot)\) \(\chi_{23}(9,\cdot)\) \(\chi_{23}(12,\cdot)\) \(\chi_{23}(13,\cdot)\) \(\chi_{23}(16,\cdot)\) \(\chi_{23}(18,\cdot)\)

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

Related number fields

Field of values: \(\Q(\zeta_{11})\)
Fixed field: \(\Q(\zeta_{23})^+\)

Values on generators

\(5\) → \(e\left(\frac{3}{11}\right)\)

Values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 23 }(8, a) \) \(1\)\(1\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{4}{11}\right)\)\(e\left(\frac{1}{11}\right)\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{10}{11}\right)\)\(e\left(\frac{2}{11}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{8}{11}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{5}{11}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 23 }(8,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 23 }(8,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 23 }(8,·),\chi_{ 23 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 23 }(8,·)) \;\) at \(\; a,b = \) e.g. 1,2