Properties

Label 2016.403
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([12,21,16,16]))
 
pari: [g,chi] = znchar(Mod(403,2016))
 

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.fp

\(\chi_{2016}(403,\cdot)\) \(\chi_{2016}(499,\cdot)\) \(\chi_{2016}(907,\cdot)\) \(\chi_{2016}(1003,\cdot)\) \(\chi_{2016}(1411,\cdot)\) \(\chi_{2016}(1507,\cdot)\) \(\chi_{2016}(1915,\cdot)\) \(\chi_{2016}(2011,\cdot)\)

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

\((127,1765,1793,577)\) → \((-1,e\left(\frac{7}{8}\right),e\left(\frac{2}{3}\right),e\left(\frac{2}{3}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\( \chi_{ 2016 }(403, a) \) \(-1\)\(1\)\(e\left(\frac{13}{24}\right)\)\(e\left(\frac{5}{24}\right)\)\(e\left(\frac{11}{24}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{23}{24}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{7}{24}\right)\)\(-1\)\(e\left(\frac{5}{24}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2016 }(403,a) \;\) at \(\;a = \) e.g. 2