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Properties

Label 343.116
Modulus $343$
Conductor $49$
Order $21$
Real no
Primitive no
Minimal no
Parity even

Related objects

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(343, base_ring=CyclotomicField(42)) M = H._module chi = DirichletCharacter(H, M([4]))
 
Copy content pari:[g,chi] = znchar(Mod(116,343))
 

Basic properties

Modulus: \(343\)
Conductor: \(49\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(21\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{49}(32,\cdot)\)
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 343.g

\(\chi_{343}(30,\cdot)\) \(\chi_{343}(67,\cdot)\) \(\chi_{343}(79,\cdot)\) \(\chi_{343}(116,\cdot)\) \(\chi_{343}(128,\cdot)\) \(\chi_{343}(165,\cdot)\) \(\chi_{343}(177,\cdot)\) \(\chi_{343}(214,\cdot)\) \(\chi_{343}(226,\cdot)\) \(\chi_{343}(263,\cdot)\) \(\chi_{343}(275,\cdot)\) \(\chi_{343}(312,\cdot)\)

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: \(\Q(\zeta_{21})\)
Fixed field: Number field defined by a degree 21 polynomial

Values on generators

\(3\) → \(e\left(\frac{2}{21}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 343 }(116, a) \) \(1\)\(1\)\(e\left(\frac{10}{21}\right)\)\(e\left(\frac{2}{21}\right)\)\(e\left(\frac{20}{21}\right)\)\(e\left(\frac{16}{21}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{4}{21}\right)\)\(e\left(\frac{5}{21}\right)\)\(e\left(\frac{17}{21}\right)\)\(e\left(\frac{1}{21}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 343 }(116,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 343 }(116,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 343 }(116,·),\chi_{ 343 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 343 }(116,·)) \;\) at \(\; a,b = \) e.g. 1,2