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Properties

Label 693.353
Modulus $693$
Conductor $63$
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(693, base_ring=CyclotomicField(6)) M = H._module chi = DirichletCharacter(H, M([1,1,0]))
 
Copy content pari:[g,chi] = znchar(Mod(353,693))
 

Basic properties

Modulus: \(693\)
Conductor: \(63\)
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 \(\chi_{63}(38,\cdot)\)
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 693.n

\(\chi_{693}(320,\cdot)\) \(\chi_{693}(353,\cdot)\)

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: 6.6.330812181.2

Values on generators

\((155,199,442)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{1}{6}\right),1)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(13\)\(16\)\(17\)\(19\)\(20\)
\( \chi_{ 693 }(353, a) \) \(1\)\(1\)\(-1\)\(1\)\(e\left(\frac{2}{3}\right)\)\(-1\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{5}{6}\right)\)\(1\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{2}{3}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 693 }(353,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 693 }(353,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 693 }(353,·),\chi_{ 693 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 693 }(353,·)) \;\) at \(\; a,b = \) e.g. 1,2