Properties

Label 253.166
Modulus $253$
Conductor $23$
Order $22$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(253, base_ring=CyclotomicField(22))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,1]))
 
pari: [g,chi] = znchar(Mod(166,253))
 

Basic properties

Modulus: \(253\)
Conductor: \(23\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(22\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{23}(5,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 253.j

\(\chi_{253}(34,\cdot)\) \(\chi_{253}(56,\cdot)\) \(\chi_{253}(67,\cdot)\) \(\chi_{253}(89,\cdot)\) \(\chi_{253}(111,\cdot)\) \(\chi_{253}(122,\cdot)\) \(\chi_{253}(155,\cdot)\) \(\chi_{253}(166,\cdot)\) \(\chi_{253}(199,\cdot)\) \(\chi_{253}(221,\cdot)\)

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

\((24,166)\) → \((1,e\left(\frac{1}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 253 }(166, a) \) \(-1\)\(1\)\(e\left(\frac{1}{11}\right)\)\(e\left(\frac{8}{11}\right)\)\(e\left(\frac{2}{11}\right)\)\(e\left(\frac{1}{22}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{19}{22}\right)\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{5}{11}\right)\)\(e\left(\frac{3}{22}\right)\)\(e\left(\frac{10}{11}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 253 }(166,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 253 }(166,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 253 }(166,·),\chi_{ 253 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 253 }(166,·)) \;\) at \(\; a,b = \) e.g. 1,2