Properties

Label 643.21
Modulus $643$
Conductor $643$
Order $642$
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(642))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([197]))
 
pari: [g,chi] = znchar(Mod(21,643))
 

Basic properties

Modulus: \(643\)
Conductor: \(643\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(642\)
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.h

\(\chi_{643}(11,\cdot)\) \(\chi_{643}(13,\cdot)\) \(\chi_{643}(14,\cdot)\) \(\chi_{643}(17,\cdot)\) \(\chi_{643}(19,\cdot)\) \(\chi_{643}(21,\cdot)\) \(\chi_{643}(35,\cdot)\) \(\chi_{643}(37,\cdot)\) \(\chi_{643}(41,\cdot)\) \(\chi_{643}(44,\cdot)\) \(\chi_{643}(46,\cdot)\) \(\chi_{643}(47,\cdot)\) \(\chi_{643}(52,\cdot)\) \(\chi_{643}(56,\cdot)\) \(\chi_{643}(58,\cdot)\) \(\chi_{643}(59,\cdot)\) \(\chi_{643}(61,\cdot)\) \(\chi_{643}(62,\cdot)\) \(\chi_{643}(66,\cdot)\) \(\chi_{643}(68,\cdot)\) \(\chi_{643}(69,\cdot)\) \(\chi_{643}(73,\cdot)\) \(\chi_{643}(76,\cdot)\) \(\chi_{643}(78,\cdot)\) \(\chi_{643}(79,\cdot)\) \(\chi_{643}(84,\cdot)\) \(\chi_{643}(87,\cdot)\) \(\chi_{643}(91,\cdot)\) \(\chi_{643}(93,\cdot)\) \(\chi_{643}(98,\cdot)\) ...

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 642 polynomial (not computed)

Values on generators

\(11\) → \(e\left(\frac{197}{642}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(-1\)\(1\)\(e\left(\frac{1}{214}\right)\)\(e\left(\frac{175}{214}\right)\)\(e\left(\frac{1}{107}\right)\)\(e\left(\frac{61}{214}\right)\)\(e\left(\frac{88}{107}\right)\)\(e\left(\frac{203}{321}\right)\)\(e\left(\frac{3}{214}\right)\)\(e\left(\frac{68}{107}\right)\)\(e\left(\frac{31}{107}\right)\)\(e\left(\frac{197}{642}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 643 }(21,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 643 }(21,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 643 }(21,·),\chi_{ 643 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 643 }(21,·)) \;\) at \(\; a,b = \) e.g. 1,2