Properties

Label 1183.870
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([26,9]))
 
pari: [g,chi] = znchar(Mod(870,1183))
 

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.bs

\(\chi_{1183}(25,\cdot)\) \(\chi_{1183}(51,\cdot)\) \(\chi_{1183}(116,\cdot)\) \(\chi_{1183}(142,\cdot)\) \(\chi_{1183}(207,\cdot)\) \(\chi_{1183}(233,\cdot)\) \(\chi_{1183}(298,\cdot)\) \(\chi_{1183}(324,\cdot)\) \(\chi_{1183}(389,\cdot)\) \(\chi_{1183}(415,\cdot)\) \(\chi_{1183}(480,\cdot)\) \(\chi_{1183}(571,\cdot)\) \(\chi_{1183}(597,\cdot)\) \(\chi_{1183}(662,\cdot)\) \(\chi_{1183}(688,\cdot)\) \(\chi_{1183}(753,\cdot)\) \(\chi_{1183}(779,\cdot)\) \(\chi_{1183}(870,\cdot)\) \(\chi_{1183}(935,\cdot)\) \(\chi_{1183}(961,\cdot)\) \(\chi_{1183}(1026,\cdot)\) \(\chi_{1183}(1052,\cdot)\) \(\chi_{1183}(1117,\cdot)\) \(\chi_{1183}(1143,\cdot)\)

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

\((339,1016)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{3}{26}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\(1\)\(1\)\(e\left(\frac{61}{78}\right)\)\(e\left(\frac{25}{39}\right)\)\(e\left(\frac{22}{39}\right)\)\(e\left(\frac{55}{78}\right)\)\(e\left(\frac{11}{26}\right)\)\(e\left(\frac{9}{26}\right)\)\(e\left(\frac{11}{39}\right)\)\(e\left(\frac{19}{39}\right)\)\(e\left(\frac{17}{78}\right)\)\(e\left(\frac{8}{39}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1183 }(870,a) \;\) at \(\;a = \) e.g. 2