Properties

Label 967.8
Modulus $967$
Conductor $967$
Order $161$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(967, base_ring=CyclotomicField(322))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([52]))
 
pari: [g,chi] = znchar(Mod(8,967))
 

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()
 
pari: order = charorder(g,chi)
 
pari: [ 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

\(5\) → \(e\left(\frac{26}{161}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(1\)\(1\)\(e\left(\frac{64}{161}\right)\)\(e\left(\frac{44}{161}\right)\)\(e\left(\frac{128}{161}\right)\)\(e\left(\frac{26}{161}\right)\)\(e\left(\frac{108}{161}\right)\)\(e\left(\frac{132}{161}\right)\)\(e\left(\frac{31}{161}\right)\)\(e\left(\frac{88}{161}\right)\)\(e\left(\frac{90}{161}\right)\)\(e\left(\frac{65}{161}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 967 }(8,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 967 }(8,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 967 }(8,·),\chi_{ 967 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 967 }(8,·)) \;\) at \(\; a,b = \) e.g. 1,2