Properties

Label 967.90
Modulus $967$
Conductor $967$
Order $322$
Real no
Primitive yes
Minimal yes
Parity odd

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([283]))
 
pari: [g,chi] = znchar(Mod(90,967))
 

Basic properties

Modulus: \(967\)
Conductor: \(967\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(322\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 967.n

\(\chi_{967}(3,\cdot)\) \(\chi_{967}(10,\cdot)\) \(\chi_{967}(23,\cdot)\) \(\chi_{967}(24,\cdot)\) \(\chi_{967}(26,\cdot)\) \(\chi_{967}(27,\cdot)\) \(\chi_{967}(29,\cdot)\) \(\chi_{967}(33,\cdot)\) \(\chi_{967}(67,\cdot)\) \(\chi_{967}(76,\cdot)\) \(\chi_{967}(80,\cdot)\) \(\chi_{967}(90,\cdot)\) \(\chi_{967}(109,\cdot)\) \(\chi_{967}(110,\cdot)\) \(\chi_{967}(112,\cdot)\) \(\chi_{967}(125,\cdot)\) \(\chi_{967}(126,\cdot)\) \(\chi_{967}(154,\cdot)\) \(\chi_{967}(158,\cdot)\) \(\chi_{967}(167,\cdot)\) \(\chi_{967}(170,\cdot)\) \(\chi_{967}(178,\cdot)\) \(\chi_{967}(184,\cdot)\) \(\chi_{967}(186,\cdot)\) \(\chi_{967}(191,\cdot)\) \(\chi_{967}(192,\cdot)\) \(\chi_{967}(207,\cdot)\) \(\chi_{967}(208,\cdot)\) \(\chi_{967}(213,\cdot)\) \(\chi_{967}(214,\cdot)\) ...

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 322 polynomial (not computed)

Values on generators

\(5\) → \(e\left(\frac{283}{322}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(-1\)\(1\)\(e\left(\frac{113}{161}\right)\)\(e\left(\frac{95}{322}\right)\)\(e\left(\frac{65}{161}\right)\)\(e\left(\frac{283}{322}\right)\)\(e\left(\frac{321}{322}\right)\)\(e\left(\frac{285}{322}\right)\)\(e\left(\frac{17}{161}\right)\)\(e\left(\frac{95}{161}\right)\)\(e\left(\frac{187}{322}\right)\)\(e\left(\frac{72}{161}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 967 }(90,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 967 }(90,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 967 }(90,·),\chi_{ 967 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 967 }(90,·)) \;\) at \(\; a,b = \) e.g. 1,2