Properties

Label 6009.35
Modulus $6009$
Conductor $6009$
Order $2002$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6009, base_ring=CyclotomicField(2002))
 
M = H._module
 
chi = DirichletCharacter(H, M([1001,124]))
 
pari: [g,chi] = znchar(Mod(35,6009))
 

Basic properties

Modulus: \(6009\)
Conductor: \(6009\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(2002\)
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 6009.bd

\(\chi_{6009}(14,\cdot)\) \(\chi_{6009}(35,\cdot)\) \(\chi_{6009}(47,\cdot)\) \(\chi_{6009}(59,\cdot)\) \(\chi_{6009}(62,\cdot)\) \(\chi_{6009}(74,\cdot)\) \(\chi_{6009}(77,\cdot)\) \(\chi_{6009}(86,\cdot)\) \(\chi_{6009}(101,\cdot)\) \(\chi_{6009}(107,\cdot)\) \(\chi_{6009}(119,\cdot)\) \(\chi_{6009}(122,\cdot)\) \(\chi_{6009}(140,\cdot)\) \(\chi_{6009}(161,\cdot)\) \(\chi_{6009}(167,\cdot)\) \(\chi_{6009}(173,\cdot)\) \(\chi_{6009}(179,\cdot)\) \(\chi_{6009}(188,\cdot)\) \(\chi_{6009}(194,\cdot)\) \(\chi_{6009}(197,\cdot)\) \(\chi_{6009}(203,\cdot)\) \(\chi_{6009}(206,\cdot)\) \(\chi_{6009}(212,\cdot)\) \(\chi_{6009}(215,\cdot)\) \(\chi_{6009}(218,\cdot)\) \(\chi_{6009}(224,\cdot)\) \(\chi_{6009}(227,\cdot)\) \(\chi_{6009}(248,\cdot)\) \(\chi_{6009}(251,\cdot)\) \(\chi_{6009}(254,\cdot)\) ...

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: $\Q(\zeta_{1001})$
Fixed field: Number field defined by a degree 2002 polynomial (not computed)

Values on generators

\((4007,2008)\) → \((-1,e\left(\frac{62}{1001}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 6009 }(35, a) \) \(-1\)\(1\)\(e\left(\frac{89}{286}\right)\)\(e\left(\frac{89}{143}\right)\)\(e\left(\frac{1125}{2002}\right)\)\(e\left(\frac{619}{1001}\right)\)\(e\left(\frac{267}{286}\right)\)\(e\left(\frac{874}{1001}\right)\)\(e\left(\frac{153}{286}\right)\)\(e\left(\frac{536}{1001}\right)\)\(e\left(\frac{1861}{2002}\right)\)\(e\left(\frac{35}{143}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6009 }(35,a) \;\) at \(\;a = \) e.g. 2