Properties

Label 1021.8
Modulus $1021$
Conductor $1021$
Order $340$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1021, base_ring=CyclotomicField(340))
 
M = H._module
 
chi = DirichletCharacter(H, M([333]))
 
pari: [g,chi] = znchar(Mod(8,1021))
 

Basic properties

Modulus: \(1021\)
Conductor: \(1021\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(340\)
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 1021.v

\(\chi_{1021}(2,\cdot)\) \(\chi_{1021}(6,\cdot)\) \(\chi_{1021}(7,\cdot)\) \(\chi_{1021}(8,\cdot)\) \(\chi_{1021}(18,\cdot)\) \(\chi_{1021}(21,\cdot)\) \(\chi_{1021}(24,\cdot)\) \(\chi_{1021}(28,\cdot)\) \(\chi_{1021}(52,\cdot)\) \(\chi_{1021}(54,\cdot)\) \(\chi_{1021}(63,\cdot)\) \(\chi_{1021}(72,\cdot)\) \(\chi_{1021}(84,\cdot)\) \(\chi_{1021}(98,\cdot)\) \(\chi_{1021}(128,\cdot)\) \(\chi_{1021}(131,\cdot)\) \(\chi_{1021}(139,\cdot)\) \(\chi_{1021}(142,\cdot)\) \(\chi_{1021}(146,\cdot)\) \(\chi_{1021}(151,\cdot)\) \(\chi_{1021}(156,\cdot)\) \(\chi_{1021}(158,\cdot)\) \(\chi_{1021}(162,\cdot)\) \(\chi_{1021}(170,\cdot)\) \(\chi_{1021}(182,\cdot)\) \(\chi_{1021}(189,\cdot)\) \(\chi_{1021}(191,\cdot)\) \(\chi_{1021}(208,\cdot)\) \(\chi_{1021}(216,\cdot)\) \(\chi_{1021}(230,\cdot)\) ...

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: $\Q(\zeta_{340})$
Fixed field: Number field defined by a degree 340 polynomial (not computed)

Values on generators

\(10\) → \(e\left(\frac{333}{340}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 1021 }(8, a) \) \(-1\)\(1\)\(e\left(\frac{49}{340}\right)\)\(e\left(\frac{15}{34}\right)\)\(e\left(\frac{49}{170}\right)\)\(e\left(\frac{71}{85}\right)\)\(e\left(\frac{199}{340}\right)\)\(e\left(\frac{339}{340}\right)\)\(e\left(\frac{147}{340}\right)\)\(e\left(\frac{15}{17}\right)\)\(e\left(\frac{333}{340}\right)\)\(e\left(\frac{57}{85}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1021 }(8,a) \;\) at \(\;a = \) e.g. 2