Properties

Label 1021.194
Modulus $1021$
Conductor $1021$
Order $510$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1021, base_ring=CyclotomicField(510))
 
M = H._module
 
chi = DirichletCharacter(H, M([509]))
 
pari: [g,chi] = znchar(Mod(194,1021))
 

Basic properties

Modulus: \(1021\)
Conductor: \(1021\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(510\)
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 1021.w

\(\chi_{1021}(15,\cdot)\) \(\chi_{1021}(17,\cdot)\) \(\chi_{1021}(20,\cdot)\) \(\chi_{1021}(29,\cdot)\) \(\chi_{1021}(33,\cdot)\) \(\chi_{1021}(38,\cdot)\) \(\chi_{1021}(44,\cdot)\) \(\chi_{1021}(61,\cdot)\) \(\chi_{1021}(69,\cdot)\) \(\chi_{1021}(74,\cdot)\) \(\chi_{1021}(75,\cdot)\) \(\chi_{1021}(92,\cdot)\) \(\chi_{1021}(100,\cdot)\) \(\chi_{1021}(106,\cdot)\) \(\chi_{1021}(130,\cdot)\) \(\chi_{1021}(135,\cdot)\) \(\chi_{1021}(141,\cdot)\) \(\chi_{1021}(149,\cdot)\) \(\chi_{1021}(153,\cdot)\) \(\chi_{1021}(163,\cdot)\) \(\chi_{1021}(165,\cdot)\) \(\chi_{1021}(179,\cdot)\) \(\chi_{1021}(180,\cdot)\) \(\chi_{1021}(188,\cdot)\) \(\chi_{1021}(190,\cdot)\) \(\chi_{1021}(194,\cdot)\) \(\chi_{1021}(201,\cdot)\) \(\chi_{1021}(204,\cdot)\) \(\chi_{1021}(205,\cdot)\) \(\chi_{1021}(210,\cdot)\) ...

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: $\Q(\zeta_{255})$
Fixed field: Number field defined by a degree 510 polynomial (not computed)

Values on generators

\(10\) → \(e\left(\frac{509}{510}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 1021 }(194, a) \) \(1\)\(1\)\(e\left(\frac{59}{170}\right)\)\(e\left(\frac{8}{17}\right)\)\(e\left(\frac{59}{85}\right)\)\(e\left(\frac{166}{255}\right)\)\(e\left(\frac{139}{170}\right)\)\(e\left(\frac{89}{170}\right)\)\(e\left(\frac{7}{170}\right)\)\(e\left(\frac{16}{17}\right)\)\(e\left(\frac{509}{510}\right)\)\(e\left(\frac{247}{255}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1021 }(194,a) \;\) at \(\;a = \) e.g. 2