Properties

Label 5610.5477
Modulus $5610$
Conductor $2805$
Order $16$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5610, base_ring=CyclotomicField(16))
 
M = H._module
 
chi = DirichletCharacter(H, M([8,4,8,1]))
 
pari: [g,chi] = znchar(Mod(5477,5610))
 

Basic properties

Modulus: \(5610\)
Conductor: \(2805\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(16\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2805}(2672,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 5610.cx

\(\chi_{5610}(1187,\cdot)\) \(\chi_{5610}(2573,\cdot)\) \(\chi_{5610}(3167,\cdot)\) \(\chi_{5610}(3497,\cdot)\) \(\chi_{5610}(3563,\cdot)\) \(\chi_{5610}(4223,\cdot)\) \(\chi_{5610}(5213,\cdot)\) \(\chi_{5610}(5477,\cdot)\)

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: Number field defined by a degree 16 polynomial

Values on generators

\((1871,3367,1531,3301)\) → \((-1,i,-1,e\left(\frac{1}{16}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(13\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)\(47\)
\( \chi_{ 5610 }(5477, a) \) \(1\)\(1\)\(e\left(\frac{7}{16}\right)\)\(-1\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{3}{16}\right)\)\(e\left(\frac{5}{16}\right)\)\(e\left(\frac{9}{16}\right)\)\(e\left(\frac{5}{16}\right)\)\(e\left(\frac{11}{16}\right)\)\(e\left(\frac{3}{8}\right)\)\(1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 5610 }(5477,a) \;\) at \(\;a = \) e.g. 2