Properties

Label 833.61
Modulus $833$
Conductor $833$
Order $336$
Real no
Primitive yes
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(833, base_ring=CyclotomicField(336)) M = H._module chi = DirichletCharacter(H, M([88,63]))
 
Copy content pari:[g,chi] = znchar(Mod(61,833))
 

Basic properties

Modulus: \(833\)
Conductor: \(833\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(336\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: yes
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 833.bn

\(\chi_{833}(3,\cdot)\) \(\chi_{833}(5,\cdot)\) \(\chi_{833}(10,\cdot)\) \(\chi_{833}(12,\cdot)\) \(\chi_{833}(24,\cdot)\) \(\chi_{833}(40,\cdot)\) \(\chi_{833}(45,\cdot)\) \(\chi_{833}(54,\cdot)\) \(\chi_{833}(61,\cdot)\) \(\chi_{833}(73,\cdot)\) \(\chi_{833}(75,\cdot)\) \(\chi_{833}(82,\cdot)\) \(\chi_{833}(96,\cdot)\) \(\chi_{833}(108,\cdot)\) \(\chi_{833}(122,\cdot)\) \(\chi_{833}(124,\cdot)\) \(\chi_{833}(131,\cdot)\) \(\chi_{833}(143,\cdot)\) \(\chi_{833}(150,\cdot)\) \(\chi_{833}(159,\cdot)\) \(\chi_{833}(164,\cdot)\) \(\chi_{833}(173,\cdot)\) \(\chi_{833}(180,\cdot)\) \(\chi_{833}(192,\cdot)\) \(\chi_{833}(194,\cdot)\) \(\chi_{833}(199,\cdot)\) \(\chi_{833}(201,\cdot)\) \(\chi_{833}(241,\cdot)\) \(\chi_{833}(243,\cdot)\) \(\chi_{833}(248,\cdot)\) ...

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: $\Q(\zeta_{336})$
Fixed field: Number field defined by a degree 336 polynomial (not computed)

Values on generators

\((52,785)\) → \((e\left(\frac{11}{42}\right),e\left(\frac{3}{16}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 833 }(61, a) \) \(1\)\(1\)\(e\left(\frac{73}{168}\right)\)\(e\left(\frac{151}{336}\right)\)\(e\left(\frac{73}{84}\right)\)\(e\left(\frac{179}{336}\right)\)\(e\left(\frac{99}{112}\right)\)\(e\left(\frac{17}{56}\right)\)\(e\left(\frac{151}{168}\right)\)\(e\left(\frac{325}{336}\right)\)\(e\left(\frac{265}{336}\right)\)\(e\left(\frac{107}{336}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 833 }(61,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 833 }(61,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 833 }(61,·),\chi_{ 833 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 833 }(61,·)) \;\) at \(\; a,b = \) e.g. 1,2