Properties

Label 65.64
Modulus $65$
Conductor $65$
Order $2$
Real yes
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(65, base_ring=CyclotomicField(2))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([1,1]))
 
pari: [g,chi] = znchar(Mod(64,65))
 

Kronecker symbol representation

sage: kronecker_character(65)
 
pari: znchartokronecker(g,chi)
 

\(\displaystyle\left(\frac{65}{\bullet}\right)\)

Basic properties

Modulus: \(65\)
Conductor: \(65\)
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 65.d

\(\chi_{65}(64,\cdot)\)

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: \(\Q(\sqrt{65}) \)

Values on generators

\((27,41)\) → \((-1,-1)\)

Values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(14\)
\( \chi_{ 65 }(64, a) \) \(1\)\(1\)\(1\)\(-1\)\(1\)\(-1\)\(1\)\(1\)\(1\)\(-1\)\(-1\)\(1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 65 }(64,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 65 }(64,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 65 }(64,·),\chi_{ 65 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 65 }(64,·)) \;\) at \(\; a,b = \) e.g. 1,2