Properties

Label 643.6
Modulus $643$
Conductor $643$
Order $107$
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(643, base_ring=CyclotomicField(214))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([40]))
 
pari: [g,chi] = znchar(Mod(6,643))
 

Basic properties

Modulus: \(643\)
Conductor: \(643\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(107\)
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 643.e

\(\chi_{643}(4,\cdot)\) \(\chi_{643}(6,\cdot)\) \(\chi_{643}(9,\cdot)\) \(\chi_{643}(10,\cdot)\) \(\chi_{643}(15,\cdot)\) \(\chi_{643}(16,\cdot)\) \(\chi_{643}(24,\cdot)\) \(\chi_{643}(25,\cdot)\) \(\chi_{643}(36,\cdot)\) \(\chi_{643}(40,\cdot)\) \(\chi_{643}(54,\cdot)\) \(\chi_{643}(60,\cdot)\) \(\chi_{643}(64,\cdot)\) \(\chi_{643}(81,\cdot)\) \(\chi_{643}(86,\cdot)\) \(\chi_{643}(90,\cdot)\) \(\chi_{643}(96,\cdot)\) \(\chi_{643}(100,\cdot)\) \(\chi_{643}(129,\cdot)\) \(\chi_{643}(131,\cdot)\) \(\chi_{643}(134,\cdot)\) \(\chi_{643}(135,\cdot)\) \(\chi_{643}(142,\cdot)\) \(\chi_{643}(143,\cdot)\) \(\chi_{643}(144,\cdot)\) \(\chi_{643}(150,\cdot)\) \(\chi_{643}(154,\cdot)\) \(\chi_{643}(160,\cdot)\) \(\chi_{643}(161,\cdot)\) \(\chi_{643}(163,\cdot)\) ...

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 107 polynomial (not computed)

Values on generators

\(11\) → \(e\left(\frac{20}{107}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(1\)\(1\)\(e\left(\frac{72}{107}\right)\)\(e\left(\frac{81}{107}\right)\)\(e\left(\frac{37}{107}\right)\)\(e\left(\frac{5}{107}\right)\)\(e\left(\frac{46}{107}\right)\)\(e\left(\frac{7}{107}\right)\)\(e\left(\frac{2}{107}\right)\)\(e\left(\frac{55}{107}\right)\)\(e\left(\frac{77}{107}\right)\)\(e\left(\frac{20}{107}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 643 }(6,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 643 }(6,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 643 }(6,·),\chi_{ 643 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 643 }(6,·)) \;\) at \(\; a,b = \) e.g. 1,2