Properties

Label 3584.1665
Modulus $3584$
Conductor $112$
Order $4$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3584, base_ring=CyclotomicField(4))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,3,2]))
 
pari: [g,chi] = znchar(Mod(1665,3584))
 

Basic properties

Modulus: \(3584\)
Conductor: \(112\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(4\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{112}(13,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3584.l

\(\chi_{3584}(1665,\cdot)\) \(\chi_{3584}(3457,\cdot)\)

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: 4.0.100352.5

Values on generators

\((1023,1541,1025)\) → \((1,-i,-1)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\( \chi_{ 3584 }(1665, a) \) \(-1\)\(1\)\(-i\)\(i\)\(-1\)\(-i\)\(-i\)\(1\)\(-1\)\(-i\)\(-1\)\(-1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3584 }(1665,a) \;\) at \(\;a = \) e.g. 2