Properties

Label 4560.2681
Modulus $4560$
Conductor $456$
Order $18$
Real no
Primitive no
Minimal no
Parity even

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(4560, base_ring=CyclotomicField(18))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,9,9,0,1]))
 
pari: [g,chi] = znchar(Mod(2681,4560))
 

Basic properties

Modulus: \(4560\)
Conductor: \(456\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(18\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{456}(173,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4560.gk

\(\chi_{4560}(41,\cdot)\) \(\chi_{4560}(281,\cdot)\) \(\chi_{4560}(1001,\cdot)\) \(\chi_{4560}(2681,\cdot)\) \(\chi_{4560}(2921,\cdot)\) \(\chi_{4560}(4361,\cdot)\)

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: \(\Q(\zeta_{9})\)
Fixed field: 18.18.14478127324240404768365927869710336.1

Values on generators

\((1711,1141,3041,2737,1921)\) → \((1,-1,-1,1,e\left(\frac{1}{18}\right))\)

Values

\(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\(1\)\(1\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{7}{9}\right)\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{11}{18}\right)\)\(e\left(\frac{17}{18}\right)\)\(e\left(\frac{5}{6}\right)\)\(1\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{7}{18}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4560 }(2681,a) \;\) at \(\;a = \) e.g. 2