Properties

Label 4021.31
Modulus $4021$
Conductor $4021$
Order $4020$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4021, base_ring=CyclotomicField(4020))
 
M = H._module
 
chi = DirichletCharacter(H, M([2353]))
 
pari: [g,chi] = znchar(Mod(31,4021))
 

Basic properties

Modulus: \(4021\)
Conductor: \(4021\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(4020\)
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 4021.x

\(\chi_{4021}(2,\cdot)\) \(\chi_{4021}(11,\cdot)\) \(\chi_{4021}(18,\cdot)\) \(\chi_{4021}(19,\cdot)\) \(\chi_{4021}(21,\cdot)\) \(\chi_{4021}(23,\cdot)\) \(\chi_{4021}(24,\cdot)\) \(\chi_{4021}(26,\cdot)\) \(\chi_{4021}(30,\cdot)\) \(\chi_{4021}(31,\cdot)\) \(\chi_{4021}(34,\cdot)\) \(\chi_{4021}(35,\cdot)\) \(\chi_{4021}(40,\cdot)\) \(\chi_{4021}(41,\cdot)\) \(\chi_{4021}(43,\cdot)\) \(\chi_{4021}(50,\cdot)\) \(\chi_{4021}(54,\cdot)\) \(\chi_{4021}(63,\cdot)\) \(\chi_{4021}(72,\cdot)\) \(\chi_{4021}(76,\cdot)\) \(\chi_{4021}(79,\cdot)\) \(\chi_{4021}(84,\cdot)\) \(\chi_{4021}(90,\cdot)\) \(\chi_{4021}(96,\cdot)\) \(\chi_{4021}(98,\cdot)\) \(\chi_{4021}(105,\cdot)\) \(\chi_{4021}(112,\cdot)\) \(\chi_{4021}(120,\cdot)\) \(\chi_{4021}(122,\cdot)\) \(\chi_{4021}(128,\cdot)\) ...

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: $\Q(\zeta_{4020})$
Fixed field: Number field defined by a degree 4020 polynomial (not computed)

Values on generators

\(2\) → \(e\left(\frac{2353}{4020}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4021 }(31, a) \) \(-1\)\(1\)\(e\left(\frac{2353}{4020}\right)\)\(e\left(\frac{608}{1005}\right)\)\(e\left(\frac{343}{2010}\right)\)\(e\left(\frac{713}{1005}\right)\)\(e\left(\frac{51}{268}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{1013}{1340}\right)\)\(e\left(\frac{211}{1005}\right)\)\(e\left(\frac{79}{268}\right)\)\(e\left(\frac{2701}{4020}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4021 }(31,a) \;\) at \(\;a = \) e.g. 2