Properties

Label 2016.139
Modulus $2016$
Conductor $2016$
Order $24$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2016, base_ring=CyclotomicField(24))
 
M = H._module
 
chi = DirichletCharacter(H, M([12,15,8,12]))
 
pari: [g,chi] = znchar(Mod(139,2016))
 

Basic properties

Modulus: \(2016\)
Conductor: \(2016\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(24\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2016.gb

\(\chi_{2016}(139,\cdot)\) \(\chi_{2016}(475,\cdot)\) \(\chi_{2016}(643,\cdot)\) \(\chi_{2016}(979,\cdot)\) \(\chi_{2016}(1147,\cdot)\) \(\chi_{2016}(1483,\cdot)\) \(\chi_{2016}(1651,\cdot)\) \(\chi_{2016}(1987,\cdot)\)

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: \(\Q(\zeta_{24})\)
Fixed field: Number field defined by a degree 24 polynomial

Values on generators

\((127,1765,1793,577)\) → \((-1,e\left(\frac{5}{8}\right),e\left(\frac{1}{3}\right),-1)\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\( \chi_{ 2016 }(139, a) \) \(1\)\(1\)\(e\left(\frac{19}{24}\right)\)\(e\left(\frac{23}{24}\right)\)\(e\left(\frac{13}{24}\right)\)\(1\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{5}{24}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{5}{8}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2016 }(139,a) \;\) at \(\;a = \) e.g. 2