Properties

Label 164.163
Modulus $164$
Conductor $164$
Order $2$
Real yes
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(164, base_ring=CyclotomicField(2))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([1,1]))
 
pari: [g,chi] = znchar(Mod(163,164))
 

Kronecker symbol representation

sage: kronecker_character(-164)
 
pari: znchartokronecker(g,chi)
 

\(\displaystyle\left(\frac{-164}{\bullet}\right)\)

Basic properties

Modulus: \(164\)
Conductor: \(164\)
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: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 164.d

\(\chi_{164}(163,\cdot)\)

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q\)
Fixed field: \(\Q(\sqrt{-41}) \)

Values on generators

\((83,129)\) → \((-1,-1)\)

Values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 164 }(163, a) \) \(-1\)\(1\)\(1\)\(1\)\(1\)\(1\)\(1\)\(-1\)\(1\)\(-1\)\(1\)\(1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 164 }(163,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 164 }(163,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 164 }(163,·),\chi_{ 164 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 164 }(163,·)) \;\) at \(\; a,b = \) e.g. 1,2