Properties

Label 8640.1081
Modulus $8640$
Conductor $32$
Order $8$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8640, base_ring=CyclotomicField(8))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,5,0,0]))
 
pari: [g,chi] = znchar(Mod(1081,8640))
 

Basic properties

Modulus: \(8640\)
Conductor: \(32\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(8\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{32}(21,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8640.cl

\(\chi_{8640}(1081,\cdot)\) \(\chi_{8640}(3241,\cdot)\) \(\chi_{8640}(5401,\cdot)\) \(\chi_{8640}(7561,\cdot)\)

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: \(\Q(\zeta_{8})\)
Fixed field: \(\Q(\zeta_{32})^+\)

Values on generators

\((2431,3781,6401,3457)\) → \((1,e\left(\frac{5}{8}\right),1,1)\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 8640 }(1081, a) \) \(1\)\(1\)\(i\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{3}{8}\right)\)\(-1\)\(e\left(\frac{3}{8}\right)\)\(-i\)\(e\left(\frac{7}{8}\right)\)\(1\)\(e\left(\frac{5}{8}\right)\)\(-i\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8640 }(1081,a) \;\) at \(\;a = \) e.g. 2