Properties

Label 265.3
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([39,17]))
 
pari: [g,chi] = znchar(Mod(3,265))
 

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.o

\(\chi_{265}(3,\cdot)\) \(\chi_{265}(12,\cdot)\) \(\chi_{265}(22,\cdot)\) \(\chi_{265}(27,\cdot)\) \(\chi_{265}(48,\cdot)\) \(\chi_{265}(67,\cdot)\) \(\chi_{265}(73,\cdot)\) \(\chi_{265}(87,\cdot)\) \(\chi_{265}(88,\cdot)\) \(\chi_{265}(98,\cdot)\) \(\chi_{265}(108,\cdot)\) \(\chi_{265}(127,\cdot)\) \(\chi_{265}(138,\cdot)\) \(\chi_{265}(157,\cdot)\) \(\chi_{265}(167,\cdot)\) \(\chi_{265}(177,\cdot)\) \(\chi_{265}(178,\cdot)\) \(\chi_{265}(192,\cdot)\) \(\chi_{265}(198,\cdot)\) \(\chi_{265}(217,\cdot)\) \(\chi_{265}(238,\cdot)\) \(\chi_{265}(243,\cdot)\) \(\chi_{265}(253,\cdot)\) \(\chi_{265}(262,\cdot)\)

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

\((107,161)\) → \((-i,e\left(\frac{17}{52}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 265 }(3, a) \) \(1\)\(1\)\(e\left(\frac{1}{13}\right)\)\(e\left(\frac{21}{26}\right)\)\(e\left(\frac{2}{13}\right)\)\(e\left(\frac{23}{26}\right)\)\(e\left(\frac{17}{52}\right)\)\(e\left(\frac{3}{13}\right)\)\(e\left(\frac{8}{13}\right)\)\(e\left(\frac{25}{26}\right)\)\(e\left(\frac{25}{26}\right)\)\(e\left(\frac{5}{52}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 265 }(3,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 265 }(3,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 265 }(3,·),\chi_{ 265 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 265 }(3,·)) \;\) at \(\; a,b = \) e.g. 1,2