Properties

Label 4560.863
Modulus $4560$
Conductor $1140$
Order $12$
Real no
Primitive no
Minimal yes
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(12))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([6,0,6,9,2]))
 
pari: [g,chi] = znchar(Mod(863,4560))
 

Basic properties

Modulus: \(4560\)
Conductor: \(1140\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(12\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1140}(863,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4560.ft

\(\chi_{4560}(863,\cdot)\) \(\chi_{4560}(1247,\cdot)\) \(\chi_{4560}(2687,\cdot)\) \(\chi_{4560}(3983,\cdot)\)

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: \(\Q(\zeta_{12})\)
Fixed field: 12.12.35756378415495432000000000.1

Values on generators

\((1711,1141,3041,2737,1921)\) → \((-1,1,-1,-i,e\left(\frac{1}{6}\right))\)

Values

\(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\(1\)\(1\)\(i\)\(1\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{5}{6}\right)\)\(1\)\(i\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{5}{12}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4560 }(863,a) \;\) at \(\;a = \) e.g. 2