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Properties

Label 522.343
Modulus $522$
Conductor $29$
Order $7$
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(522, base_ring=CyclotomicField(14)) M = H._module chi = DirichletCharacter(H, M([0,4]))
 
Copy content pari:[g,chi] = znchar(Mod(343,522))
 

Basic properties

Modulus: \(522\)
Conductor: \(29\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(7\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{29}(24,\cdot)\)
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 522.k

\(\chi_{522}(181,\cdot)\) \(\chi_{522}(199,\cdot)\) \(\chi_{522}(343,\cdot)\) \(\chi_{522}(397,\cdot)\) \(\chi_{522}(451,\cdot)\) \(\chi_{522}(487,\cdot)\)

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: \(\Q(\zeta_{7})\)
Fixed field: 7.7.594823321.1

Values on generators

\((407,379)\) → \((1,e\left(\frac{2}{7}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(31\)\(35\)
\( \chi_{ 522 }(343, a) \) \(1\)\(1\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{1}{7}\right)\)\(1\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{5}{7}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 522 }(343,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 522 }(343,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 522 }(343,·),\chi_{ 522 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 522 }(343,·)) \;\) at \(\; a,b = \) e.g. 1,2