Properties

Label 1183.111
Modulus $1183$
Conductor $1183$
Order $156$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1183, base_ring=CyclotomicField(156))
 
M = H._module
 
chi = DirichletCharacter(H, M([78,119]))
 
pari: [g,chi] = znchar(Mod(111,1183))
 

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.cb

\(\chi_{1183}(6,\cdot)\) \(\chi_{1183}(20,\cdot)\) \(\chi_{1183}(41,\cdot)\) \(\chi_{1183}(76,\cdot)\) \(\chi_{1183}(97,\cdot)\) \(\chi_{1183}(111,\cdot)\) \(\chi_{1183}(132,\cdot)\) \(\chi_{1183}(167,\cdot)\) \(\chi_{1183}(202,\cdot)\) \(\chi_{1183}(223,\cdot)\) \(\chi_{1183}(279,\cdot)\) \(\chi_{1183}(293,\cdot)\) \(\chi_{1183}(314,\cdot)\) \(\chi_{1183}(349,\cdot)\) \(\chi_{1183}(370,\cdot)\) \(\chi_{1183}(384,\cdot)\) \(\chi_{1183}(405,\cdot)\) \(\chi_{1183}(440,\cdot)\) \(\chi_{1183}(461,\cdot)\) \(\chi_{1183}(475,\cdot)\) \(\chi_{1183}(496,\cdot)\) \(\chi_{1183}(531,\cdot)\) \(\chi_{1183}(552,\cdot)\) \(\chi_{1183}(566,\cdot)\) \(\chi_{1183}(622,\cdot)\) \(\chi_{1183}(643,\cdot)\) \(\chi_{1183}(678,\cdot)\) \(\chi_{1183}(713,\cdot)\) \(\chi_{1183}(734,\cdot)\) \(\chi_{1183}(748,\cdot)\) ...

sage: chi.galois_orbit()
 
order = charorder(g,chi)
 
[ 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

\((339,1016)\) → \((-1,e\left(\frac{119}{156}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 1183 }(111, a) \) \(1\)\(1\)\(e\left(\frac{119}{156}\right)\)\(e\left(\frac{7}{78}\right)\)\(e\left(\frac{41}{78}\right)\)\(e\left(\frac{19}{52}\right)\)\(e\left(\frac{133}{156}\right)\)\(e\left(\frac{15}{52}\right)\)\(e\left(\frac{7}{39}\right)\)\(e\left(\frac{5}{39}\right)\)\(e\left(\frac{89}{156}\right)\)\(e\left(\frac{8}{13}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1183 }(111,a) \;\) at \(\;a = \) e.g. 2