Properties

Label 4560.1291
Modulus $4560$
Conductor $304$
Order $4$
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(4))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([2,1,0,0,2]))
 
pari: [g,chi] = znchar(Mod(1291,4560))
 

Basic properties

Modulus: \(4560\)
Conductor: \(304\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(4\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{304}(75,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4560.dc

\(\chi_{4560}(1291,\cdot)\) \(\chi_{4560}(3571,\cdot)\)

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: \(\Q(\sqrt{-1}) \)
Fixed field: 4.4.739328.1

Values on generators

\((1711,1141,3041,2737,1921)\) → \((-1,i,1,1,-1)\)

Values

\(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\(1\)\(1\)\(1\)\(-i\)\(i\)\(1\)\(1\)\(i\)\(1\)\(-i\)\(1\)\(-i\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4560 }(1291,a) \;\) at \(\;a = \) e.g. 2