Properties

Label 4021.53
Modulus $4021$
Conductor $4021$
Order $402$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4021, base_ring=CyclotomicField(402))
 
M = H._module
 
chi = DirichletCharacter(H, M([35]))
 
pari: [g,chi] = znchar(Mod(53,4021))
 

Basic properties

Modulus: \(4021\)
Conductor: \(4021\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(402\)
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.r

\(\chi_{4021}(53,\cdot)\) \(\chi_{4021}(147,\cdot)\) \(\chi_{4021}(153,\cdot)\) \(\chi_{4021}(168,\cdot)\) \(\chi_{4021}(174,\cdot)\) \(\chi_{4021}(192,\cdot)\) \(\chi_{4021}(245,\cdot)\) \(\chi_{4021}(249,\cdot)\) \(\chi_{4021}(251,\cdot)\) \(\chi_{4021}(255,\cdot)\) \(\chi_{4021}(280,\cdot)\) \(\chi_{4021}(287,\cdot)\) \(\chi_{4021}(290,\cdot)\) \(\chi_{4021}(298,\cdot)\) \(\chi_{4021}(320,\cdot)\) \(\chi_{4021}(328,\cdot)\) \(\chi_{4021}(331,\cdot)\) \(\chi_{4021}(341,\cdot)\) \(\chi_{4021}(381,\cdot)\) \(\chi_{4021}(415,\cdot)\) \(\chi_{4021}(425,\cdot)\) \(\chi_{4021}(439,\cdot)\) \(\chi_{4021}(543,\cdot)\) \(\chi_{4021}(549,\cdot)\) \(\chi_{4021}(557,\cdot)\) \(\chi_{4021}(594,\cdot)\) \(\chi_{4021}(599,\cdot)\) \(\chi_{4021}(635,\cdot)\) \(\chi_{4021}(678,\cdot)\) \(\chi_{4021}(686,\cdot)\) ...

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: $\Q(\zeta_{201})$
Fixed field: Number field defined by a degree 402 polynomial (not computed)

Values on generators

\(2\) → \(e\left(\frac{35}{402}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4021 }(53, a) \) \(1\)\(1\)\(e\left(\frac{35}{402}\right)\)\(e\left(\frac{161}{201}\right)\)\(e\left(\frac{35}{201}\right)\)\(e\left(\frac{125}{201}\right)\)\(e\left(\frac{119}{134}\right)\)\(-1\)\(e\left(\frac{35}{134}\right)\)\(e\left(\frac{121}{201}\right)\)\(e\left(\frac{95}{134}\right)\)\(e\left(\frac{251}{402}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4021 }(53,a) \;\) at \(\;a = \) e.g. 2