Properties

Label 1183.3
Modulus $1183$
Conductor $1183$
Order $78$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1183, base_ring=CyclotomicField(78))
 
M = H._module
 
chi = DirichletCharacter(H, M([13,62]))
 
pari: [g,chi] = znchar(Mod(3,1183))
 

Basic properties

Modulus: \(1183\)
Conductor: \(1183\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(78\)
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 1183.bx

\(\chi_{1183}(3,\cdot)\) \(\chi_{1183}(61,\cdot)\) \(\chi_{1183}(94,\cdot)\) \(\chi_{1183}(152,\cdot)\) \(\chi_{1183}(185,\cdot)\) \(\chi_{1183}(243,\cdot)\) \(\chi_{1183}(276,\cdot)\) \(\chi_{1183}(334,\cdot)\) \(\chi_{1183}(367,\cdot)\) \(\chi_{1183}(425,\cdot)\) \(\chi_{1183}(458,\cdot)\) \(\chi_{1183}(516,\cdot)\) \(\chi_{1183}(549,\cdot)\) \(\chi_{1183}(607,\cdot)\) \(\chi_{1183}(640,\cdot)\) \(\chi_{1183}(731,\cdot)\) \(\chi_{1183}(789,\cdot)\) \(\chi_{1183}(880,\cdot)\) \(\chi_{1183}(913,\cdot)\) \(\chi_{1183}(971,\cdot)\) \(\chi_{1183}(1004,\cdot)\) \(\chi_{1183}(1062,\cdot)\) \(\chi_{1183}(1095,\cdot)\) \(\chi_{1183}(1153,\cdot)\)

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: $\Q(\zeta_{39})$
Fixed field: Number field defined by a degree 78 polynomial

Values on generators

\((339,1016)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{31}{39}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 1183 }(3, a) \) \(-1\)\(1\)\(e\left(\frac{5}{39}\right)\)\(e\left(\frac{19}{26}\right)\)\(e\left(\frac{10}{39}\right)\)\(e\left(\frac{77}{78}\right)\)\(e\left(\frac{67}{78}\right)\)\(e\left(\frac{5}{13}\right)\)\(e\left(\frac{6}{13}\right)\)\(e\left(\frac{3}{26}\right)\)\(e\left(\frac{7}{13}\right)\)\(e\left(\frac{77}{78}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1183 }(3,a) \;\) at \(\;a = \) e.g. 2