Properties

Label 23.11
Modulus $23$
Conductor $23$
Order $22$
Real no
Primitive yes
Minimal yes
Parity odd

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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([9]))
 
pari: [g,chi] = znchar(Mod(11,23))
 

Basic properties

Modulus: \(23\)
Conductor: \(23\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(22\)
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 23.d

\(\chi_{23}(5,\cdot)\) \(\chi_{23}(7,\cdot)\) \(\chi_{23}(10,\cdot)\) \(\chi_{23}(11,\cdot)\) \(\chi_{23}(14,\cdot)\) \(\chi_{23}(15,\cdot)\) \(\chi_{23}(17,\cdot)\) \(\chi_{23}(19,\cdot)\) \(\chi_{23}(20,\cdot)\) \(\chi_{23}(21,\cdot)\)

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: Number field defined by a degree 22 polynomial

Values on generators

\(5\) → \(e\left(\frac{9}{22}\right)\)

Values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 23 }(11, a) \) \(-1\)\(1\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{9}{22}\right)\)\(e\left(\frac{4}{11}\right)\)\(e\left(\frac{17}{22}\right)\)\(e\left(\frac{5}{11}\right)\)\(e\left(\frac{1}{11}\right)\)\(e\left(\frac{5}{22}\right)\)\(e\left(\frac{15}{22}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 23 }(11,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 23 }(11,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 23 }(11,·),\chi_{ 23 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 23 }(11,·)) \;\) at \(\; a,b = \) e.g. 1,2