Properties

Label 967.118
Modulus $967$
Conductor $967$
Order $161$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(967, base_ring=CyclotomicField(322))
 
M = H._module
 
chi = DirichletCharacter(H, M([44]))
 
pari: [g,chi] = znchar(Mod(118,967))
 

Basic properties

Modulus: \(967\)
Conductor: \(967\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(161\)
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 967.m

\(\chi_{967}(8,\cdot)\) \(\chi_{967}(9,\cdot)\) \(\chi_{967}(11,\cdot)\) \(\chi_{967}(17,\cdot)\) \(\chi_{967}(30,\cdot)\) \(\chi_{967}(42,\cdot)\) \(\chi_{967}(62,\cdot)\) \(\chi_{967}(64,\cdot)\) \(\chi_{967}(71,\cdot)\) \(\chi_{967}(78,\cdot)\) \(\chi_{967}(81,\cdot)\) \(\chi_{967}(87,\cdot)\) \(\chi_{967}(88,\cdot)\) \(\chi_{967}(95,\cdot)\) \(\chi_{967}(99,\cdot)\) \(\chi_{967}(100,\cdot)\) \(\chi_{967}(118,\cdot)\) \(\chi_{967}(121,\cdot)\) \(\chi_{967}(122,\cdot)\) \(\chi_{967}(123,\cdot)\) \(\chi_{967}(131,\cdot)\) \(\chi_{967}(136,\cdot)\) \(\chi_{967}(140,\cdot)\) \(\chi_{967}(146,\cdot)\) \(\chi_{967}(153,\cdot)\) \(\chi_{967}(201,\cdot)\) \(\chi_{967}(206,\cdot)\) \(\chi_{967}(212,\cdot)\) \(\chi_{967}(215,\cdot)\) \(\chi_{967}(222,\cdot)\) ...

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: $\Q(\zeta_{161})$
Fixed field: Number field defined by a degree 161 polynomial (not computed)

Values on generators

\(5\) → \(e\left(\frac{22}{161}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 967 }(118, a) \) \(1\)\(1\)\(e\left(\frac{17}{161}\right)\)\(e\left(\frac{62}{161}\right)\)\(e\left(\frac{34}{161}\right)\)\(e\left(\frac{22}{161}\right)\)\(e\left(\frac{79}{161}\right)\)\(e\left(\frac{25}{161}\right)\)\(e\left(\frac{51}{161}\right)\)\(e\left(\frac{124}{161}\right)\)\(e\left(\frac{39}{161}\right)\)\(e\left(\frac{55}{161}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 967 }(118,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 967 }(118,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 967 }(118,·),\chi_{ 967 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 967 }(118,·)) \;\) at \(\; a,b = \) e.g. 1,2