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Properties

Label 792.361
Modulus $792$
Conductor $11$
Order $5$
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(792, base_ring=CyclotomicField(10)) M = H._module chi = DirichletCharacter(H, M([0,0,0,6]))
 
Copy content pari:[g,chi] = znchar(Mod(361,792))
 

Basic properties

Modulus: \(792\)
Conductor: \(11\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(5\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{11}(9,\cdot)\)
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 792.r

\(\chi_{792}(289,\cdot)\) \(\chi_{792}(361,\cdot)\) \(\chi_{792}(433,\cdot)\) \(\chi_{792}(577,\cdot)\)

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: \(\Q(\zeta_{5})\)
Fixed field: \(\Q(\zeta_{11})^+\)

Values on generators

\((199,397,353,145)\) → \((1,1,1,e\left(\frac{3}{5}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 792 }(361, a) \) \(1\)\(1\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{4}{5}\right)\)\(1\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{3}{5}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 792 }(361,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 792 }(361,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 792 }(361,·),\chi_{ 792 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 792 }(361,·)) \;\) at \(\; a,b = \) e.g. 1,2