Properties

Label 1183.88
Modulus $1183$
Conductor $1183$
Order $78$
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(78))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([52,53]))
 
pari: [g,chi] = znchar(Mod(88,1183))
 

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: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1183.bp

\(\chi_{1183}(30,\cdot)\) \(\chi_{1183}(88,\cdot)\) \(\chi_{1183}(121,\cdot)\) \(\chi_{1183}(179,\cdot)\) \(\chi_{1183}(212,\cdot)\) \(\chi_{1183}(270,\cdot)\) \(\chi_{1183}(303,\cdot)\) \(\chi_{1183}(394,\cdot)\) \(\chi_{1183}(452,\cdot)\) \(\chi_{1183}(543,\cdot)\) \(\chi_{1183}(576,\cdot)\) \(\chi_{1183}(634,\cdot)\) \(\chi_{1183}(667,\cdot)\) \(\chi_{1183}(725,\cdot)\) \(\chi_{1183}(758,\cdot)\) \(\chi_{1183}(816,\cdot)\) \(\chi_{1183}(849,\cdot)\) \(\chi_{1183}(907,\cdot)\) \(\chi_{1183}(940,\cdot)\) \(\chi_{1183}(998,\cdot)\) \(\chi_{1183}(1031,\cdot)\) \(\chi_{1183}(1089,\cdot)\) \(\chi_{1183}(1122,\cdot)\) \(\chi_{1183}(1180,\cdot)\)

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

Values on generators

\((339,1016)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{53}{78}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\(1\)\(1\)\(e\left(\frac{1}{78}\right)\)\(e\left(\frac{12}{13}\right)\)\(e\left(\frac{1}{39}\right)\)\(e\left(\frac{35}{78}\right)\)\(e\left(\frac{73}{78}\right)\)\(e\left(\frac{1}{26}\right)\)\(e\left(\frac{11}{13}\right)\)\(e\left(\frac{6}{13}\right)\)\(e\left(\frac{17}{26}\right)\)\(e\left(\frac{37}{39}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1183 }(88,a) \;\) at \(\;a = \) e.g. 2