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Properties

Label 82.57
Modulus $82$
Conductor $41$
Order $5$
Real no
Primitive no
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(82, base_ring=CyclotomicField(10)) M = H._module chi = DirichletCharacter(H, M([6]))
 
Copy content pari:[g,chi] = znchar(Mod(57,82))
 

Basic properties

Modulus: \(82\)
Conductor: \(41\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(5\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{41}(16,\cdot)\)
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 82.d

\(\chi_{82}(37,\cdot)\) \(\chi_{82}(51,\cdot)\) \(\chi_{82}(57,\cdot)\) \(\chi_{82}(59,\cdot)\)

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: \(\Q(\zeta_{5})\)
Fixed field: 5.5.2825761.1

Values on generators

\(47\) → \(e\left(\frac{3}{5}\right)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 82 }(57, a) \) \(1\)\(1\)\(1\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{2}{5}\right)\)\(1\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{2}{5}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 82 }(57,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 82 }(57,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 82 }(57,·),\chi_{ 82 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 82 }(57,·)) \;\) at \(\; a,b = \) e.g. 1,2