Properties

Label 269.13
Modulus $269$
Conductor $269$
Order $134$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(269, base_ring=CyclotomicField(134))
 
M = H._module
 
chi = DirichletCharacter(H, M([71]))
 
pari: [g,chi] = znchar(Mod(13,269))
 

Basic properties

Modulus: \(269\)
Conductor: \(269\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(134\)
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 269.e

\(\chi_{269}(4,\cdot)\) \(\chi_{269}(6,\cdot)\) \(\chi_{269}(9,\cdot)\) \(\chi_{269}(11,\cdot)\) \(\chi_{269}(13,\cdot)\) \(\chi_{269}(20,\cdot)\) \(\chi_{269}(30,\cdot)\) \(\chi_{269}(34,\cdot)\) \(\chi_{269}(43,\cdot)\) \(\chi_{269}(45,\cdot)\) \(\chi_{269}(49,\cdot)\) \(\chi_{269}(51,\cdot)\) \(\chi_{269}(55,\cdot)\) \(\chi_{269}(56,\cdot)\) \(\chi_{269}(64,\cdot)\) \(\chi_{269}(65,\cdot)\) \(\chi_{269}(73,\cdot)\) \(\chi_{269}(79,\cdot)\) \(\chi_{269}(84,\cdot)\) \(\chi_{269}(89,\cdot)\) \(\chi_{269}(92,\cdot)\) \(\chi_{269}(96,\cdot)\) \(\chi_{269}(97,\cdot)\) \(\chi_{269}(100,\cdot)\) \(\chi_{269}(103,\cdot)\) \(\chi_{269}(126,\cdot)\) \(\chi_{269}(127,\cdot)\) \(\chi_{269}(133,\cdot)\) \(\chi_{269}(138,\cdot)\) \(\chi_{269}(144,\cdot)\) ...

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: $\Q(\zeta_{67})$
Fixed field: Number field defined by a degree 134 polynomial (not computed)

Values on generators

\(2\) → \(e\left(\frac{71}{134}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 269 }(13, a) \) \(1\)\(1\)\(e\left(\frac{71}{134}\right)\)\(e\left(\frac{101}{134}\right)\)\(e\left(\frac{4}{67}\right)\)\(e\left(\frac{14}{67}\right)\)\(e\left(\frac{19}{67}\right)\)\(e\left(\frac{9}{134}\right)\)\(e\left(\frac{79}{134}\right)\)\(e\left(\frac{34}{67}\right)\)\(e\left(\frac{99}{134}\right)\)\(e\left(\frac{58}{67}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 269 }(13,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 269 }(13,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 269 }(13,·),\chi_{ 269 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 269 }(13,·)) \;\) at \(\; a,b = \) e.g. 1,2