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Properties

Label 736.191
Modulus $736$
Conductor $92$
Order $22$
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(736, base_ring=CyclotomicField(22)) M = H._module chi = DirichletCharacter(H, M([11,0,19]))
 
Copy content pari:[g,chi] = znchar(Mod(191,736))
 

Basic properties

Modulus: \(736\)
Conductor: \(92\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(22\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{92}(7,\cdot)\)
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 736.w

\(\chi_{736}(63,\cdot)\) \(\chi_{736}(159,\cdot)\) \(\chi_{736}(191,\cdot)\) \(\chi_{736}(287,\cdot)\) \(\chi_{736}(319,\cdot)\) \(\chi_{736}(383,\cdot)\) \(\chi_{736}(447,\cdot)\) \(\chi_{736}(479,\cdot)\) \(\chi_{736}(511,\cdot)\) \(\chi_{736}(543,\cdot)\)

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: \(\Q(\zeta_{11})\)
Fixed field: \(\Q(\zeta_{92})^+\)

Values on generators

\((415,645,97)\) → \((-1,1,e\left(\frac{19}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 736 }(191, a) \) \(1\)\(1\)\(e\left(\frac{7}{22}\right)\)\(e\left(\frac{19}{22}\right)\)\(e\left(\frac{10}{11}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{1}{11}\right)\)\(e\left(\frac{2}{11}\right)\)\(e\left(\frac{1}{22}\right)\)\(e\left(\frac{5}{11}\right)\)\(e\left(\frac{5}{22}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 736 }(191,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 736 }(191,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 736 }(191,·),\chi_{ 736 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 736 }(191,·)) \;\) at \(\; a,b = \) e.g. 1,2