Properties

Label 3520.439
Modulus $3520$
Conductor $1760$
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(3520, base_ring=CyclotomicField(8))
 
M = H._module
 
chi = DirichletCharacter(H, M([4,3,4,4]))
 
pari: [g,chi] = znchar(Mod(439,3520))
 

Basic properties

Modulus: \(3520\)
Conductor: \(1760\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(8\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1760}(1539,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3520.bt

\(\chi_{3520}(439,\cdot)\) \(\chi_{3520}(1319,\cdot)\) \(\chi_{3520}(2199,\cdot)\) \(\chi_{3520}(3079,\cdot)\)

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: 8.8.19650817556480000.1

Values on generators

\((2751,1541,2817,321)\) → \((-1,e\left(\frac{3}{8}\right),-1,-1)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)\(29\)
\( \chi_{ 3520 }(439, a) \) \(1\)\(1\)\(e\left(\frac{1}{8}\right)\)\(i\)\(i\)\(e\left(\frac{5}{8}\right)\)\(-1\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{3}{8}\right)\)\(i\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{5}{8}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3520 }(439,a) \;\) at \(\;a = \) e.g. 2