Properties

Label 4021.3
Modulus $4021$
Conductor $4021$
Order $1005$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4021, base_ring=CyclotomicField(2010))
 
M = H._module
 
chi = DirichletCharacter(H, M([1492]))
 
pari: [g,chi] = znchar(Mod(3,4021))
 

Basic properties

Modulus: \(4021\)
Conductor: \(4021\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(1005\)
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 4021.u

\(\chi_{4021}(3,\cdot)\) \(\chi_{4021}(5,\cdot)\) \(\chi_{4021}(9,\cdot)\) \(\chi_{4021}(14,\cdot)\) \(\chi_{4021}(15,\cdot)\) \(\chi_{4021}(16,\cdot)\) \(\chi_{4021}(25,\cdot)\) \(\chi_{4021}(46,\cdot)\) \(\chi_{4021}(52,\cdot)\) \(\chi_{4021}(68,\cdot)\) \(\chi_{4021}(81,\cdot)\) \(\chi_{4021}(82,\cdot)\) \(\chi_{4021}(111,\cdot)\) \(\chi_{4021}(114,\cdot)\) \(\chi_{4021}(126,\cdot)\) \(\chi_{4021}(135,\cdot)\) \(\chi_{4021}(138,\cdot)\) \(\chi_{4021}(144,\cdot)\) \(\chi_{4021}(156,\cdot)\) \(\chi_{4021}(185,\cdot)\) \(\chi_{4021}(190,\cdot)\) \(\chi_{4021}(191,\cdot)\) \(\chi_{4021}(196,\cdot)\) \(\chi_{4021}(201,\cdot)\) \(\chi_{4021}(203,\cdot)\) \(\chi_{4021}(204,\cdot)\) \(\chi_{4021}(210,\cdot)\) \(\chi_{4021}(212,\cdot)\) \(\chi_{4021}(217,\cdot)\) \(\chi_{4021}(223,\cdot)\) ...

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 1005 polynomial (not computed)

Values on generators

\(2\) → \(e\left(\frac{746}{1005}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4021 }(3, a) \) \(1\)\(1\)\(e\left(\frac{746}{1005}\right)\)\(e\left(\frac{994}{1005}\right)\)\(e\left(\frac{487}{1005}\right)\)\(e\left(\frac{964}{1005}\right)\)\(e\left(\frac{49}{67}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{76}{335}\right)\)\(e\left(\frac{983}{1005}\right)\)\(e\left(\frac{47}{67}\right)\)\(e\left(\frac{32}{1005}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4021 }(3,a) \;\) at \(\;a = \) e.g. 2