Properties

Label 265.8
Modulus $265$
Conductor $265$
Order $52$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(265, base_ring=CyclotomicField(52))
 
M = H._module
 
chi = DirichletCharacter(H, M([39,3]))
 
pari: [g,chi] = znchar(Mod(8,265))
 

Basic properties

Modulus: \(265\)
Conductor: \(265\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(52\)
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 265.t

\(\chi_{265}(2,\cdot)\) \(\chi_{265}(8,\cdot)\) \(\chi_{265}(18,\cdot)\) \(\chi_{265}(32,\cdot)\) \(\chi_{265}(33,\cdot)\) \(\chi_{265}(58,\cdot)\) \(\chi_{265}(72,\cdot)\) \(\chi_{265}(92,\cdot)\) \(\chi_{265}(103,\cdot)\) \(\chi_{265}(118,\cdot)\) \(\chi_{265}(128,\cdot)\) \(\chi_{265}(132,\cdot)\) \(\chi_{265}(133,\cdot)\) \(\chi_{265}(137,\cdot)\) \(\chi_{265}(147,\cdot)\) \(\chi_{265}(162,\cdot)\) \(\chi_{265}(173,\cdot)\) \(\chi_{265}(193,\cdot)\) \(\chi_{265}(207,\cdot)\) \(\chi_{265}(232,\cdot)\) \(\chi_{265}(233,\cdot)\) \(\chi_{265}(247,\cdot)\) \(\chi_{265}(257,\cdot)\) \(\chi_{265}(263,\cdot)\)

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: $\Q(\zeta_{52})$
Fixed field: Number field defined by a degree 52 polynomial

Values on generators

\((107,161)\) → \((-i,e\left(\frac{3}{52}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 265 }(8, a) \) \(1\)\(1\)\(e\left(\frac{21}{26}\right)\)\(e\left(\frac{3}{13}\right)\)\(e\left(\frac{8}{13}\right)\)\(e\left(\frac{1}{26}\right)\)\(e\left(\frac{29}{52}\right)\)\(e\left(\frac{11}{26}\right)\)\(e\left(\frac{6}{13}\right)\)\(e\left(\frac{9}{26}\right)\)\(e\left(\frac{11}{13}\right)\)\(e\left(\frac{33}{52}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 265 }(8,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 265 }(8,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 265 }(8,·),\chi_{ 265 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 265 }(8,·)) \;\) at \(\; a,b = \) e.g. 1,2