Properties

Label 4080.227
Modulus $4080$
Conductor $4080$
Order $16$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4080, base_ring=CyclotomicField(16))
 
M = H._module
 
chi = DirichletCharacter(H, M([8,12,8,4,15]))
 
pari: [g,chi] = znchar(Mod(227,4080))
 

Basic properties

Modulus: \(4080\)
Conductor: \(4080\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(16\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4080.jq

\(\chi_{4080}(227,\cdot)\) \(\chi_{4080}(683,\cdot)\) \(\chi_{4080}(707,\cdot)\) \(\chi_{4080}(923,\cdot)\) \(\chi_{4080}(2387,\cdot)\) \(\chi_{4080}(2867,\cdot)\) \(\chi_{4080}(3803,\cdot)\) \(\chi_{4080}(4043,\cdot)\)

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: \(\Q(\zeta_{16})\)
Fixed field: 16.16.80661021875089306767964564881408000000000000.2

Values on generators

\((511,3061,1361,817,241)\) → \((-1,-i,-1,i,e\left(\frac{15}{16}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 4080 }(227, a) \) \(1\)\(1\)\(e\left(\frac{9}{16}\right)\)\(e\left(\frac{5}{16}\right)\)\(-i\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{5}{16}\right)\)\(e\left(\frac{7}{16}\right)\)\(e\left(\frac{15}{16}\right)\)\(e\left(\frac{15}{16}\right)\)\(e\left(\frac{5}{16}\right)\)\(e\left(\frac{7}{8}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4080 }(227,a) \;\) at \(\;a = \) e.g. 2