Properties

Label 3120.677
Modulus $3120$
Conductor $240$
Order $4$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3120, base_ring=CyclotomicField(4))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,1,2,1,0]))
 
pari: [g,chi] = znchar(Mod(677,3120))
 

Basic properties

Modulus: \(3120\)
Conductor: \(240\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(4\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{240}(197,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3120.do

\(\chi_{3120}(677,\cdot)\) \(\chi_{3120}(1613,\cdot)\)

sage: chi.galois_orbit()
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\sqrt{-1}) \)
Fixed field: 4.4.2304000.1

Values on generators

\((1951,2341,2081,2497,2641)\) → \((1,i,-1,i,1)\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 3120 }(677, a) \) \(1\)\(1\)\(-i\)\(-i\)\(-i\)\(i\)\(-i\)\(-i\)\(1\)\(-1\)\(1\)\(1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3120 }(677,a) \;\) at \(\;a = \) e.g. 2