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Properties

Label 494.277
Modulus $494$
Conductor $247$
Order $6$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(494, base_ring=CyclotomicField(6)) M = H._module chi = DirichletCharacter(H, M([1,4]))
 
Copy content pari:[g,chi] = znchar(Mod(277,494))
 

Basic properties

Modulus: \(494\)
Conductor: \(247\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(6\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{247}(30,\cdot)\)
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 494.o

\(\chi_{494}(277,\cdot)\) \(\chi_{494}(387,\cdot)\)

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: \(\mathbb{Q}(\zeta_3)\)
Fixed field: 6.6.48387275053.2

Values on generators

\((457,287)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{2}{3}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(15\)\(17\)\(21\)\(23\)\(25\)
\( \chi_{ 494 }(277, a) \) \(1\)\(1\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{6}\right)\)\(-1\)\(1\)\(e\left(\frac{1}{6}\right)\)\(1\)\(e\left(\frac{1}{3}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 494 }(277,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 494 }(277,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 494 }(277,·),\chi_{ 494 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 494 }(277,·)) \;\) at \(\; a,b = \) e.g. 1,2