Properties

Label 1755.586
Modulus $1755$
Conductor $9$
Order $3$
Real no
Primitive no
Minimal no
Parity even

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Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1755, base_ring=CyclotomicField(6))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([4,0,0]))
 
pari: [g,chi] = znchar(Mod(586,1755))
 

Basic properties

Modulus: \(1755\)
Conductor: \(9\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(3\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{9}(7,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1755.i

\(\chi_{1755}(586,\cdot)\) \(\chi_{1755}(1171,\cdot)\)

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: \(\Q(\sqrt{-3}) \)
Fixed field: \(\Q(\zeta_{9})^+\)

Values on generators

\((326,352,1081)\) → \((e\left(\frac{2}{3}\right),1,1)\)

Values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(14\)\(16\)\(17\)\(19\)\(22\)
\( \chi_{ 1755 }(586, a) \) \(1\)\(1\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{2}{3}\right)\)\(1\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{2}{3}\right)\)\(1\)\(1\)\(e\left(\frac{1}{3}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1755 }(586,a) \;\) at \(\;a = \) e.g. 2