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Properties

Label 65.8
Modulus $65$
Conductor $65$
Order $4$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(65, base_ring=CyclotomicField(4)) M = H._module chi = DirichletCharacter(H, M([3,1]))
 
Copy content pari:[g,chi] = znchar(Mod(8,65))
 

Basic properties

Modulus: \(65\)
Conductor: \(65\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(4\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: yes
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 65.k

\(\chi_{65}(8,\cdot)\) \(\chi_{65}(57,\cdot)\)

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

Related number fields

Field of values: \(\mathbb{Q}(i)\)
Fixed field: 4.4.274625.1

Values on generators

\((27,41)\) → \((-i,i)\)

First values

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

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 65 }(8,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 65 }(8,·),\chi_{ 65 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 65 }(8,·)) \;\) at \(\; a,b = \) e.g. 1,2