Properties

Label 1183.570
Modulus $1183$
Conductor $1183$
Order $156$
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(1183, base_ring=CyclotomicField(156))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([26,43]))
 
pari: [g,chi] = znchar(Mod(570,1183))
 

Basic properties

Modulus: \(1183\)
Conductor: \(1183\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(156\)
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 1183.ch

\(\chi_{1183}(24,\cdot)\) \(\chi_{1183}(33,\cdot)\) \(\chi_{1183}(110,\cdot)\) \(\chi_{1183}(115,\cdot)\) \(\chi_{1183}(124,\cdot)\) \(\chi_{1183}(171,\cdot)\) \(\chi_{1183}(201,\cdot)\) \(\chi_{1183}(206,\cdot)\) \(\chi_{1183}(215,\cdot)\) \(\chi_{1183}(262,\cdot)\) \(\chi_{1183}(292,\cdot)\) \(\chi_{1183}(297,\cdot)\) \(\chi_{1183}(306,\cdot)\) \(\chi_{1183}(353,\cdot)\) \(\chi_{1183}(383,\cdot)\) \(\chi_{1183}(388,\cdot)\) \(\chi_{1183}(397,\cdot)\) \(\chi_{1183}(444,\cdot)\) \(\chi_{1183}(474,\cdot)\) \(\chi_{1183}(479,\cdot)\) \(\chi_{1183}(535,\cdot)\) \(\chi_{1183}(565,\cdot)\) \(\chi_{1183}(570,\cdot)\) \(\chi_{1183}(579,\cdot)\) \(\chi_{1183}(626,\cdot)\) \(\chi_{1183}(656,\cdot)\) \(\chi_{1183}(661,\cdot)\) \(\chi_{1183}(670,\cdot)\) \(\chi_{1183}(717,\cdot)\) \(\chi_{1183}(747,\cdot)\) ...

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: $\Q(\zeta_{156})$
Fixed field: Number field defined by a degree 156 polynomial (not computed)

Values on generators

\((339,1016)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{43}{156}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\(1\)\(1\)\(e\left(\frac{95}{156}\right)\)\(e\left(\frac{9}{26}\right)\)\(e\left(\frac{17}{78}\right)\)\(e\left(\frac{49}{156}\right)\)\(e\left(\frac{149}{156}\right)\)\(e\left(\frac{43}{52}\right)\)\(e\left(\frac{9}{13}\right)\)\(e\left(\frac{12}{13}\right)\)\(e\left(\frac{3}{52}\right)\)\(e\left(\frac{22}{39}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1183 }(570,a) \;\) at \(\;a = \) e.g. 2