Properties

Label 517.494
Modulus $517$
Conductor $517$
Order $46$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(517, base_ring=CyclotomicField(46))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([23,28]))
 
pari: [g,chi] = znchar(Mod(494,517))
 

Basic properties

Modulus: \(517\)
Conductor: \(517\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(46\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 517.k

\(\chi_{517}(21,\cdot)\) \(\chi_{517}(32,\cdot)\) \(\chi_{517}(54,\cdot)\) \(\chi_{517}(65,\cdot)\) \(\chi_{517}(98,\cdot)\) \(\chi_{517}(131,\cdot)\) \(\chi_{517}(153,\cdot)\) \(\chi_{517}(175,\cdot)\) \(\chi_{517}(197,\cdot)\) \(\chi_{517}(230,\cdot)\) \(\chi_{517}(241,\cdot)\) \(\chi_{517}(252,\cdot)\) \(\chi_{517}(263,\cdot)\) \(\chi_{517}(285,\cdot)\) \(\chi_{517}(296,\cdot)\) \(\chi_{517}(307,\cdot)\) \(\chi_{517}(318,\cdot)\) \(\chi_{517}(384,\cdot)\) \(\chi_{517}(439,\cdot)\) \(\chi_{517}(450,\cdot)\) \(\chi_{517}(472,\cdot)\) \(\chi_{517}(494,\cdot)\)

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\zeta_{23})\)
Fixed field: 46.0.33445485745541531749546267414718227385437022890768869389066429180265423777079649185104842349209411.1

Values on generators

\((189,287)\) → \((-1,e\left(\frac{14}{23}\right))\)

Values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 517 }(494, a) \) \(-1\)\(1\)\(e\left(\frac{21}{46}\right)\)\(e\left(\frac{4}{23}\right)\)\(e\left(\frac{21}{23}\right)\)\(e\left(\frac{14}{23}\right)\)\(e\left(\frac{29}{46}\right)\)\(e\left(\frac{45}{46}\right)\)\(e\left(\frac{17}{46}\right)\)\(e\left(\frac{8}{23}\right)\)\(e\left(\frac{3}{46}\right)\)\(e\left(\frac{2}{23}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 517 }(494,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 517 }(494,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 517 }(494,·),\chi_{ 517 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 517 }(494,·)) \;\) at \(\; a,b = \) e.g. 1,2