Properties

Label 269.25
Modulus $269$
Conductor $269$
Order $67$
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([74]))
 
pari: [g,chi] = znchar(Mod(25,269))
 

Basic properties

Modulus: \(269\)
Conductor: \(269\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(67\)
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.d

\(\chi_{269}(5,\cdot)\) \(\chi_{269}(14,\cdot)\) \(\chi_{269}(16,\cdot)\) \(\chi_{269}(21,\cdot)\) \(\chi_{269}(23,\cdot)\) \(\chi_{269}(24,\cdot)\) \(\chi_{269}(25,\cdot)\) \(\chi_{269}(36,\cdot)\) \(\chi_{269}(37,\cdot)\) \(\chi_{269}(38,\cdot)\) \(\chi_{269}(41,\cdot)\) \(\chi_{269}(44,\cdot)\) \(\chi_{269}(47,\cdot)\) \(\chi_{269}(52,\cdot)\) \(\chi_{269}(53,\cdot)\) \(\chi_{269}(54,\cdot)\) \(\chi_{269}(57,\cdot)\) \(\chi_{269}(58,\cdot)\) \(\chi_{269}(61,\cdot)\) \(\chi_{269}(62,\cdot)\) \(\chi_{269}(66,\cdot)\) \(\chi_{269}(67,\cdot)\) \(\chi_{269}(70,\cdot)\) \(\chi_{269}(78,\cdot)\) \(\chi_{269}(80,\cdot)\) \(\chi_{269}(81,\cdot)\) \(\chi_{269}(87,\cdot)\) \(\chi_{269}(93,\cdot)\) \(\chi_{269}(99,\cdot)\) \(\chi_{269}(105,\cdot)\) ...

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 67 polynomial

Values on generators

\(2\) → \(e\left(\frac{37}{67}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 269 }(25, a) \) \(1\)\(1\)\(e\left(\frac{37}{67}\right)\)\(e\left(\frac{13}{67}\right)\)\(e\left(\frac{7}{67}\right)\)\(e\left(\frac{58}{67}\right)\)\(e\left(\frac{50}{67}\right)\)\(e\left(\frac{33}{67}\right)\)\(e\left(\frac{44}{67}\right)\)\(e\left(\frac{26}{67}\right)\)\(e\left(\frac{28}{67}\right)\)\(e\left(\frac{1}{67}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 269 }(25,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 269 }(25,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 269 }(25,·),\chi_{ 269 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 269 }(25,·)) \;\) at \(\; a,b = \) e.g. 1,2