Properties

Label 643.119
Modulus $643$
Conductor $643$
Order $214$
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(643, base_ring=CyclotomicField(214))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([163]))
 
pari: [g,chi] = znchar(Mod(119,643))
 

Basic properties

Modulus: \(643\)
Conductor: \(643\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(214\)
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 643.f

\(\chi_{643}(2,\cdot)\) \(\chi_{643}(3,\cdot)\) \(\chi_{643}(5,\cdot)\) \(\chi_{643}(8,\cdot)\) \(\chi_{643}(12,\cdot)\) \(\chi_{643}(18,\cdot)\) \(\chi_{643}(20,\cdot)\) \(\chi_{643}(27,\cdot)\) \(\chi_{643}(30,\cdot)\) \(\chi_{643}(32,\cdot)\) \(\chi_{643}(43,\cdot)\) \(\chi_{643}(45,\cdot)\) \(\chi_{643}(48,\cdot)\) \(\chi_{643}(50,\cdot)\) \(\chi_{643}(67,\cdot)\) \(\chi_{643}(71,\cdot)\) \(\chi_{643}(72,\cdot)\) \(\chi_{643}(75,\cdot)\) \(\chi_{643}(77,\cdot)\) \(\chi_{643}(80,\cdot)\) \(\chi_{643}(103,\cdot)\) \(\chi_{643}(107,\cdot)\) \(\chi_{643}(108,\cdot)\) \(\chi_{643}(119,\cdot)\) \(\chi_{643}(120,\cdot)\) \(\chi_{643}(125,\cdot)\) \(\chi_{643}(127,\cdot)\) \(\chi_{643}(128,\cdot)\) \(\chi_{643}(157,\cdot)\) \(\chi_{643}(162,\cdot)\) ...

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: $\Q(\zeta_{107})$
Fixed field: Number field defined by a degree 214 polynomial (not computed)

Values on generators

\(11\) → \(e\left(\frac{163}{214}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(-1\)\(1\)\(e\left(\frac{9}{214}\right)\)\(e\left(\frac{77}{214}\right)\)\(e\left(\frac{9}{107}\right)\)\(e\left(\frac{121}{214}\right)\)\(e\left(\frac{43}{107}\right)\)\(e\left(\frac{74}{107}\right)\)\(e\left(\frac{27}{214}\right)\)\(e\left(\frac{77}{107}\right)\)\(e\left(\frac{65}{107}\right)\)\(e\left(\frac{163}{214}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 643 }(119,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 643 }(119,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 643 }(119,·),\chi_{ 643 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 643 }(119,·)) \;\) at \(\; a,b = \) e.g. 1,2