Properties

Label 1183.872
Modulus $1183$
Conductor $1183$
Order $39$
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([52,30]))
 
pari: [g,chi] = znchar(Mod(872,1183))
 

Basic properties

Modulus: \(1183\)
Conductor: \(1183\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(39\)
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.bl

\(\chi_{1183}(53,\cdot)\) \(\chi_{1183}(79,\cdot)\) \(\chi_{1183}(144,\cdot)\) \(\chi_{1183}(235,\cdot)\) \(\chi_{1183}(261,\cdot)\) \(\chi_{1183}(326,\cdot)\) \(\chi_{1183}(352,\cdot)\) \(\chi_{1183}(417,\cdot)\) \(\chi_{1183}(443,\cdot)\) \(\chi_{1183}(534,\cdot)\) \(\chi_{1183}(599,\cdot)\) \(\chi_{1183}(625,\cdot)\) \(\chi_{1183}(690,\cdot)\) \(\chi_{1183}(716,\cdot)\) \(\chi_{1183}(781,\cdot)\) \(\chi_{1183}(807,\cdot)\) \(\chi_{1183}(872,\cdot)\) \(\chi_{1183}(898,\cdot)\) \(\chi_{1183}(963,\cdot)\) \(\chi_{1183}(989,\cdot)\) \(\chi_{1183}(1054,\cdot)\) \(\chi_{1183}(1080,\cdot)\) \(\chi_{1183}(1145,\cdot)\) \(\chi_{1183}(1171,\cdot)\)

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: 39.39.1501310100540182816122902385277086845494955654696971364374430555896401217548869308777432830007194746769.1

Values on generators

\((339,1016)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{5}{13}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\(1\)\(1\)\(e\left(\frac{28}{39}\right)\)\(e\left(\frac{14}{39}\right)\)\(e\left(\frac{17}{39}\right)\)\(e\left(\frac{31}{39}\right)\)\(e\left(\frac{1}{13}\right)\)\(e\left(\frac{2}{13}\right)\)\(e\left(\frac{28}{39}\right)\)\(e\left(\frac{20}{39}\right)\)\(e\left(\frac{11}{39}\right)\)\(e\left(\frac{31}{39}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1183 }(872,a) \;\) at \(\;a = \) e.g. 2