Properties

Label 643.22
Modulus $643$
Conductor $643$
Order $321$
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(642))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([454]))
 
pari: [g,chi] = znchar(Mod(22,643))
 

Basic properties

Modulus: \(643\)
Conductor: \(643\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(321\)
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.g

\(\chi_{643}(7,\cdot)\) \(\chi_{643}(22,\cdot)\) \(\chi_{643}(23,\cdot)\) \(\chi_{643}(26,\cdot)\) \(\chi_{643}(28,\cdot)\) \(\chi_{643}(29,\cdot)\) \(\chi_{643}(31,\cdot)\) \(\chi_{643}(33,\cdot)\) \(\chi_{643}(34,\cdot)\) \(\chi_{643}(38,\cdot)\) \(\chi_{643}(39,\cdot)\) \(\chi_{643}(42,\cdot)\) \(\chi_{643}(49,\cdot)\) \(\chi_{643}(51,\cdot)\) \(\chi_{643}(53,\cdot)\) \(\chi_{643}(55,\cdot)\) \(\chi_{643}(57,\cdot)\) \(\chi_{643}(63,\cdot)\) \(\chi_{643}(65,\cdot)\) \(\chi_{643}(70,\cdot)\) \(\chi_{643}(74,\cdot)\) \(\chi_{643}(82,\cdot)\) \(\chi_{643}(83,\cdot)\) \(\chi_{643}(85,\cdot)\) \(\chi_{643}(88,\cdot)\) \(\chi_{643}(89,\cdot)\) \(\chi_{643}(92,\cdot)\) \(\chi_{643}(94,\cdot)\) \(\chi_{643}(95,\cdot)\) \(\chi_{643}(97,\cdot)\) ...

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: $\Q(\zeta_{321})$
Fixed field: Number field defined by a degree 321 polynomial (not computed)

Values on generators

\(11\) → \(e\left(\frac{227}{321}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(1\)\(1\)\(e\left(\frac{37}{107}\right)\)\(e\left(\frac{55}{107}\right)\)\(e\left(\frac{74}{107}\right)\)\(e\left(\frac{10}{107}\right)\)\(e\left(\frac{92}{107}\right)\)\(e\left(\frac{256}{321}\right)\)\(e\left(\frac{4}{107}\right)\)\(e\left(\frac{3}{107}\right)\)\(e\left(\frac{47}{107}\right)\)\(e\left(\frac{227}{321}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 643 }(22,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 643 }(22,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 643 }(22,·),\chi_{ 643 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 643 }(22,·)) \;\) at \(\; a,b = \) e.g. 1,2