Properties

Label 1183.1006
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([130,81]))
 
pari: [g,chi] = znchar(Mod(1006,1183))
 

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.cc

\(\chi_{1183}(5,\cdot)\) \(\chi_{1183}(31,\cdot)\) \(\chi_{1183}(47,\cdot)\) \(\chi_{1183}(73,\cdot)\) \(\chi_{1183}(96,\cdot)\) \(\chi_{1183}(122,\cdot)\) \(\chi_{1183}(138,\cdot)\) \(\chi_{1183}(164,\cdot)\) \(\chi_{1183}(187,\cdot)\) \(\chi_{1183}(213,\cdot)\) \(\chi_{1183}(229,\cdot)\) \(\chi_{1183}(255,\cdot)\) \(\chi_{1183}(278,\cdot)\) \(\chi_{1183}(304,\cdot)\) \(\chi_{1183}(320,\cdot)\) \(\chi_{1183}(346,\cdot)\) \(\chi_{1183}(369,\cdot)\) \(\chi_{1183}(395,\cdot)\) \(\chi_{1183}(411,\cdot)\) \(\chi_{1183}(460,\cdot)\) \(\chi_{1183}(486,\cdot)\) \(\chi_{1183}(502,\cdot)\) \(\chi_{1183}(528,\cdot)\) \(\chi_{1183}(551,\cdot)\) \(\chi_{1183}(593,\cdot)\) \(\chi_{1183}(619,\cdot)\) \(\chi_{1183}(642,\cdot)\) \(\chi_{1183}(668,\cdot)\) \(\chi_{1183}(684,\cdot)\) \(\chi_{1183}(710,\cdot)\) ...

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

\((339,1016)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{27}{52}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 1183 }(1006, a) \) \(1\)\(1\)\(e\left(\frac{29}{156}\right)\)\(e\left(\frac{17}{78}\right)\)\(e\left(\frac{29}{78}\right)\)\(e\left(\frac{131}{156}\right)\)\(e\left(\frac{21}{52}\right)\)\(e\left(\frac{29}{52}\right)\)\(e\left(\frac{17}{39}\right)\)\(e\left(\frac{1}{39}\right)\)\(e\left(\frac{127}{156}\right)\)\(e\left(\frac{23}{39}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1183 }(1006,a) \;\) at \(\;a = \) e.g. 2