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Properties

Label 513.413
Modulus $513$
Conductor $171$
Order $18$
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(513, base_ring=CyclotomicField(18)) M = H._module chi = DirichletCharacter(H, M([3,7]))
 
Copy content pari:[g,chi] = znchar(Mod(413,513))
 

Basic properties

Modulus: \(513\)
Conductor: \(171\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(18\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{171}(128,\cdot)\)
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 513.cd

\(\chi_{513}(116,\cdot)\) \(\chi_{513}(260,\cdot)\) \(\chi_{513}(314,\cdot)\) \(\chi_{513}(395,\cdot)\) \(\chi_{513}(413,\cdot)\) \(\chi_{513}(440,\cdot)\)

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: \(\Q(\zeta_{9})\)
Fixed field: Number field defined by a degree 18 polynomial

Values on generators

\((191,325)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{7}{18}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 513 }(413, a) \) \(1\)\(1\)\(e\left(\frac{5}{9}\right)\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{1}{18}\right)\)\(1\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{11}{18}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{5}{18}\right)\)\(e\left(\frac{5}{9}\right)\)\(e\left(\frac{2}{9}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 513 }(413,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 513 }(413,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 513 }(413,·),\chi_{ 513 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 513 }(413,·)) \;\) at \(\; a,b = \) e.g. 1,2