Properties

Label 1216.303
Modulus $1216$
Conductor $304$
Order $4$
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(1216, base_ring=CyclotomicField(4))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([2,3,2]))
 
pari: [g,chi] = znchar(Mod(303,1216))
 

Basic properties

Modulus: \(1216\)
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 \(\chi_{304}(227,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1216.m

\(\chi_{1216}(303,\cdot)\) \(\chi_{1216}(911,\cdot)\)

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

\((191,837,705)\) → \((-1,-i,-1)\)

Values

\(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(21\)\(23\)
\(1\)\(1\)\(i\)\(-i\)\(1\)\(-1\)\(i\)\(-i\)\(1\)\(1\)\(i\)\(1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1216 }(303,a) \;\) at \(\;a = \) e.g. 2