Properties

Label 281.280
Modulus $281$
Conductor $281$
Order $2$
Real yes
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(281, base_ring=CyclotomicField(2))
 
M = H._module
 
chi = DirichletCharacter(H, M([1]))
 
pari: [g,chi] = znchar(Mod(280,281))
 

Kronecker symbol representation

sage: kronecker_character(281)
 
pari: znchartokronecker(g,chi)
 

\(\displaystyle\left(\frac{281}{\bullet}\right)\)

Basic properties

Modulus: \(281\)
Conductor: \(281\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(2\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: yes
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 281.b

\(\chi_{281}(280,\cdot)\)

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: \(\Q\)
Fixed field: \(\Q(\sqrt{281}) \)

Values on generators

\(3\) → \(-1\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 281 }(280, a) \) \(1\)\(1\)\(1\)\(-1\)\(1\)\(1\)\(-1\)\(1\)\(1\)\(1\)\(1\)\(-1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 281 }(280,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 281 }(280,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 281 }(280,·),\chi_{ 281 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 281 }(280,·)) \;\) at \(\; a,b = \) e.g. 1,2