Properties

Label 4928.419
Modulus $4928$
Conductor $448$
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(4928, base_ring=CyclotomicField(16))
 
M = H._module
 
chi = DirichletCharacter(H, M([8,11,8,0]))
 
pari: [g,chi] = znchar(Mod(419,4928))
 

Basic properties

Modulus: \(4928\)
Conductor: \(448\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(16\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{448}(419,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4928.cy

\(\chi_{4928}(419,\cdot)\) \(\chi_{4928}(1035,\cdot)\) \(\chi_{4928}(1651,\cdot)\) \(\chi_{4928}(2267,\cdot)\) \(\chi_{4928}(2883,\cdot)\) \(\chi_{4928}(3499,\cdot)\) \(\chi_{4928}(4115,\cdot)\) \(\chi_{4928}(4731,\cdot)\)

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: 16.16.3484608386920116940487669055488.4

Values on generators

\((4159,1541,2817,3137)\) → \((-1,e\left(\frac{11}{16}\right),-1,1)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)\(27\)
\( \chi_{ 4928 }(419, a) \) \(1\)\(1\)\(e\left(\frac{1}{16}\right)\)\(e\left(\frac{3}{16}\right)\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{13}{16}\right)\)\(i\)\(-i\)\(e\left(\frac{13}{16}\right)\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{3}{16}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4928 }(419,a) \;\) at \(\;a = \) e.g. 2