Properties

Label 4080.3523
Modulus $4080$
Conductor $1360$
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(4080, base_ring=CyclotomicField(4))
 
M = H._module
 
chi = DirichletCharacter(H, M([2,3,0,3,3]))
 
pari: [g,chi] = znchar(Mod(3523,4080))
 

Basic properties

Modulus: \(4080\)
Conductor: \(1360\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(4\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1360}(803,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4080.eq

\(\chi_{4080}(3523,\cdot)\) \(\chi_{4080}(3787,\cdot)\)

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: 4.4.1257728000.3

Values on generators

\((511,3061,1361,817,241)\) → \((-1,-i,1,-i,-i)\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 4080 }(3523, a) \) \(1\)\(1\)\(1\)\(-1\)\(-1\)\(-i\)\(-1\)\(-1\)\(i\)\(i\)\(-i\)\(1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4080 }(3523,a) \;\) at \(\;a = \) e.g. 2