Properties

Label 967.59
Modulus $967$
Conductor $967$
Order $483$
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(966))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([724]))
 
pari: [g,chi] = znchar(Mod(59,967))
 

Basic properties

Modulus: \(967\)
Conductor: \(967\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(483\)
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.o

\(\chi_{967}(2,\cdot)\) \(\chi_{967}(4,\cdot)\) \(\chi_{967}(16,\cdot)\) \(\chi_{967}(18,\cdot)\) \(\chi_{967}(21,\cdot)\) \(\chi_{967}(22,\cdot)\) \(\chi_{967}(25,\cdot)\) \(\chi_{967}(31,\cdot)\) \(\chi_{967}(32,\cdot)\) \(\chi_{967}(34,\cdot)\) \(\chi_{967}(35,\cdot)\) \(\chi_{967}(36,\cdot)\) \(\chi_{967}(44,\cdot)\) \(\chi_{967}(49,\cdot)\) \(\chi_{967}(50,\cdot)\) \(\chi_{967}(57,\cdot)\) \(\chi_{967}(59,\cdot)\) \(\chi_{967}(60,\cdot)\) \(\chi_{967}(65,\cdot)\) \(\chi_{967}(70,\cdot)\) \(\chi_{967}(83,\cdot)\) \(\chi_{967}(84,\cdot)\) \(\chi_{967}(91,\cdot)\) \(\chi_{967}(98,\cdot)\) \(\chi_{967}(101,\cdot)\) \(\chi_{967}(103,\cdot)\) \(\chi_{967}(106,\cdot)\) \(\chi_{967}(111,\cdot)\) \(\chi_{967}(114,\cdot)\) \(\chi_{967}(115,\cdot)\) ...

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 483 polynomial (not computed)

Values on generators

\(5\) → \(e\left(\frac{362}{483}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(1\)\(1\)\(e\left(\frac{148}{483}\right)\)\(e\left(\frac{101}{161}\right)\)\(e\left(\frac{296}{483}\right)\)\(e\left(\frac{362}{483}\right)\)\(e\left(\frac{451}{483}\right)\)\(e\left(\frac{104}{483}\right)\)\(e\left(\frac{148}{161}\right)\)\(e\left(\frac{41}{161}\right)\)\(e\left(\frac{9}{161}\right)\)\(e\left(\frac{87}{161}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 967 }(59,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 967 }(59,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 967 }(59,·),\chi_{ 967 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 967 }(59,·)) \;\) at \(\; a,b = \) e.g. 1,2