Properties

Label 2016.83
Modulus $2016$
Conductor $2016$
Order $24$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2016, base_ring=CyclotomicField(24))
 
M = H._module
 
chi = DirichletCharacter(H, M([12,21,4,12]))
 
pari: [g,chi] = znchar(Mod(83,2016))
 

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: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2016.ey

\(\chi_{2016}(83,\cdot)\) \(\chi_{2016}(419,\cdot)\) \(\chi_{2016}(587,\cdot)\) \(\chi_{2016}(923,\cdot)\) \(\chi_{2016}(1091,\cdot)\) \(\chi_{2016}(1427,\cdot)\) \(\chi_{2016}(1595,\cdot)\) \(\chi_{2016}(1931,\cdot)\)

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: 24.0.20574592712627357116606755731219706996406776278896607232.1

Values on generators

\((127,1765,1793,577)\) → \((-1,e\left(\frac{7}{8}\right),e\left(\frac{1}{6}\right),-1)\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\( \chi_{ 2016 }(83, a) \) \(-1\)\(1\)\(e\left(\frac{5}{24}\right)\)\(e\left(\frac{1}{24}\right)\)\(e\left(\frac{23}{24}\right)\)\(-1\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{19}{24}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{7}{8}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2016 }(83,a) \;\) at \(\;a = \) e.g. 2