Properties

Label 4022.3
Modulus $4022$
Conductor $2011$
Order $2010$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4022, base_ring=CyclotomicField(2010))
 
M = H._module
 
chi = DirichletCharacter(H, M([1]))
 
pari: [g,chi] = znchar(Mod(3,4022))
 

Basic properties

Modulus: \(4022\)
Conductor: \(2011\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(2010\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2011}(3,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4022.p

\(\chi_{4022}(3,\cdot)\) \(\chi_{4022}(7,\cdot)\) \(\chi_{4022}(11,\cdot)\) \(\chi_{4022}(17,\cdot)\) \(\chi_{4022}(19,\cdot)\) \(\chi_{4022}(29,\cdot)\) \(\chi_{4022}(35,\cdot)\) \(\chi_{4022}(39,\cdot)\) \(\chi_{4022}(61,\cdot)\) \(\chi_{4022}(69,\cdot)\) \(\chi_{4022}(73,\cdot)\) \(\chi_{4022}(79,\cdot)\) \(\chi_{4022}(93,\cdot)\) \(\chi_{4022}(99,\cdot)\) \(\chi_{4022}(107,\cdot)\) \(\chi_{4022}(123,\cdot)\) \(\chi_{4022}(129,\cdot)\) \(\chi_{4022}(131,\cdot)\) \(\chi_{4022}(135,\cdot)\) \(\chi_{4022}(145,\cdot)\) \(\chi_{4022}(153,\cdot)\) \(\chi_{4022}(161,\cdot)\) \(\chi_{4022}(167,\cdot)\) \(\chi_{4022}(173,\cdot)\) \(\chi_{4022}(179,\cdot)\) \(\chi_{4022}(185,\cdot)\) \(\chi_{4022}(195,\cdot)\) \(\chi_{4022}(223,\cdot)\) \(\chi_{4022}(227,\cdot)\) \(\chi_{4022}(229,\cdot)\) ...

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: $\Q(\zeta_{1005})$
Fixed field: Number field defined by a degree 2010 polynomial (not computed)

Values on generators

\(3\) → \(e\left(\frac{1}{2010}\right)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 4022 }(3, a) \) \(-1\)\(1\)\(e\left(\frac{1}{2010}\right)\)\(e\left(\frac{512}{1005}\right)\)\(e\left(\frac{1807}{2010}\right)\)\(e\left(\frac{1}{1005}\right)\)\(e\left(\frac{1037}{2010}\right)\)\(e\left(\frac{323}{335}\right)\)\(e\left(\frac{205}{402}\right)\)\(e\left(\frac{347}{2010}\right)\)\(e\left(\frac{1163}{2010}\right)\)\(e\left(\frac{904}{1005}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4022 }(3,a) \;\) at \(\;a = \) e.g. 2