Properties

Label 265.26
Modulus $265$
Conductor $53$
Order $52$
Real no
Primitive no
Minimal yes
Parity odd

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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([0,25]))
 
pari: [g,chi] = znchar(Mod(26,265))
 

Basic properties

Modulus: \(265\)
Conductor: \(53\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(52\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{53}(26,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 265.s

\(\chi_{265}(21,\cdot)\) \(\chi_{265}(26,\cdot)\) \(\chi_{265}(31,\cdot)\) \(\chi_{265}(41,\cdot)\) \(\chi_{265}(51,\cdot)\) \(\chi_{265}(56,\cdot)\) \(\chi_{265}(61,\cdot)\) \(\chi_{265}(71,\cdot)\) \(\chi_{265}(86,\cdot)\) \(\chi_{265}(101,\cdot)\) \(\chi_{265}(111,\cdot)\) \(\chi_{265}(126,\cdot)\) \(\chi_{265}(141,\cdot)\) \(\chi_{265}(151,\cdot)\) \(\chi_{265}(156,\cdot)\) \(\chi_{265}(161,\cdot)\) \(\chi_{265}(171,\cdot)\) \(\chi_{265}(181,\cdot)\) \(\chi_{265}(186,\cdot)\) \(\chi_{265}(191,\cdot)\) \(\chi_{265}(226,\cdot)\) \(\chi_{265}(231,\cdot)\) \(\chi_{265}(246,\cdot)\) \(\chi_{265}(251,\cdot)\)

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

\((107,161)\) → \((1,e\left(\frac{25}{52}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 265 }(26, a) \) \(-1\)\(1\)\(e\left(\frac{25}{52}\right)\)\(e\left(\frac{9}{52}\right)\)\(e\left(\frac{25}{26}\right)\)\(e\left(\frac{17}{26}\right)\)\(e\left(\frac{19}{26}\right)\)\(e\left(\frac{23}{52}\right)\)\(e\left(\frac{9}{26}\right)\)\(e\left(\frac{23}{26}\right)\)\(e\left(\frac{7}{52}\right)\)\(e\left(\frac{7}{13}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 265 }(26,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 265 }(26,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 265 }(26,·),\chi_{ 265 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 265 }(26,·)) \;\) at \(\; a,b = \) e.g. 1,2