Properties

Label 967.75
Modulus $967$
Conductor $967$
Order $966$
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(966))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([239]))
 
pari: [g,chi] = znchar(Mod(75,967))
 

Basic properties

Modulus: \(967\)
Conductor: \(967\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(966\)
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.p

\(\chi_{967}(5,\cdot)\) \(\chi_{967}(6,\cdot)\) \(\chi_{967}(7,\cdot)\) \(\chi_{967}(12,\cdot)\) \(\chi_{967}(13,\cdot)\) \(\chi_{967}(19,\cdot)\) \(\chi_{967}(28,\cdot)\) \(\chi_{967}(37,\cdot)\) \(\chi_{967}(38,\cdot)\) \(\chi_{967}(40,\cdot)\) \(\chi_{967}(43,\cdot)\) \(\chi_{967}(45,\cdot)\) \(\chi_{967}(46,\cdot)\) \(\chi_{967}(47,\cdot)\) \(\chi_{967}(48,\cdot)\) \(\chi_{967}(56,\cdot)\) \(\chi_{967}(58,\cdot)\) \(\chi_{967}(63,\cdot)\) \(\chi_{967}(66,\cdot)\) \(\chi_{967}(75,\cdot)\) \(\chi_{967}(77,\cdot)\) \(\chi_{967}(79,\cdot)\) \(\chi_{967}(82,\cdot)\) \(\chi_{967}(85,\cdot)\) \(\chi_{967}(86,\cdot)\) \(\chi_{967}(89,\cdot)\) \(\chi_{967}(102,\cdot)\) \(\chi_{967}(104,\cdot)\) \(\chi_{967}(105,\cdot)\) \(\chi_{967}(107,\cdot)\) ...

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: $\Q(\zeta_{483})$
Fixed field: Number field defined by a degree 966 polynomial (not computed)

Values on generators

\(5\) → \(e\left(\frac{239}{966}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(-1\)\(1\)\(e\left(\frac{257}{483}\right)\)\(e\left(\frac{205}{322}\right)\)\(e\left(\frac{31}{483}\right)\)\(e\left(\frac{239}{966}\right)\)\(e\left(\frac{163}{966}\right)\)\(e\left(\frac{557}{966}\right)\)\(e\left(\frac{96}{161}\right)\)\(e\left(\frac{44}{161}\right)\)\(e\left(\frac{251}{322}\right)\)\(e\left(\frac{113}{161}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 967 }(75,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 967 }(75,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 967 }(75,·),\chi_{ 967 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 967 }(75,·)) \;\) at \(\; a,b = \) e.g. 1,2