Properties

Label 2016.197
Modulus $2016$
Conductor $96$
Order $8$
Real no
Primitive no
Minimal no
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2016, base_ring=CyclotomicField(8))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,1,4,0]))
 
pari: [g,chi] = znchar(Mod(197,2016))
 

Basic properties

Modulus: \(2016\)
Conductor: \(96\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(8\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{96}(5,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2016.dr

\(\chi_{2016}(197,\cdot)\) \(\chi_{2016}(701,\cdot)\) \(\chi_{2016}(1205,\cdot)\) \(\chi_{2016}(1709,\cdot)\)

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: 8.0.173946175488.1

Values on generators

\((127,1765,1793,577)\) → \((1,e\left(\frac{1}{8}\right),-1,1)\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\( \chi_{ 2016 }(197, a) \) \(-1\)\(1\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{7}{8}\right)\)\(1\)\(e\left(\frac{7}{8}\right)\)\(i\)\(i\)\(e\left(\frac{7}{8}\right)\)\(1\)\(e\left(\frac{1}{8}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2016 }(197,a) \;\) at \(\;a = \) e.g. 2