Properties

Label 4560.401
Modulus $4560$
Conductor $57$
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([0,0,9,0,1]))
 
pari: [g,chi] = znchar(Mod(401,4560))
 

Basic properties

Modulus: \(4560\)
Conductor: \(57\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(18\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{57}(2,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4560.gr

\(\chi_{4560}(401,\cdot)\) \(\chi_{4560}(641,\cdot)\) \(\chi_{4560}(2081,\cdot)\) \(\chi_{4560}(2321,\cdot)\) \(\chi_{4560}(2561,\cdot)\) \(\chi_{4560}(3281,\cdot)\)

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: \(\Q(\zeta_{57})^+\)

Values on generators

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

Values

\(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\(1\)\(1\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{5}{18}\right)\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{11}{18}\right)\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{5}{6}\right)\)\(-1\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{8}{9}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4560 }(401,a) \;\) at \(\;a = \) e.g. 2