Properties

Label 517.3
Modulus $517$
Conductor $517$
Order $115$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(517, base_ring=CyclotomicField(230))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([184,100]))
 
pari: [g,chi] = znchar(Mod(3,517))
 

Basic properties

Modulus: \(517\)
Conductor: \(517\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(115\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 517.m

\(\chi_{517}(3,\cdot)\) \(\chi_{517}(4,\cdot)\) \(\chi_{517}(9,\cdot)\) \(\chi_{517}(14,\cdot)\) \(\chi_{517}(16,\cdot)\) \(\chi_{517}(25,\cdot)\) \(\chi_{517}(27,\cdot)\) \(\chi_{517}(36,\cdot)\) \(\chi_{517}(37,\cdot)\) \(\chi_{517}(42,\cdot)\) \(\chi_{517}(49,\cdot)\) \(\chi_{517}(53,\cdot)\) \(\chi_{517}(59,\cdot)\) \(\chi_{517}(64,\cdot)\) \(\chi_{517}(71,\cdot)\) \(\chi_{517}(75,\cdot)\) \(\chi_{517}(81,\cdot)\) \(\chi_{517}(97,\cdot)\) \(\chi_{517}(102,\cdot)\) \(\chi_{517}(103,\cdot)\) \(\chi_{517}(108,\cdot)\) \(\chi_{517}(115,\cdot)\) \(\chi_{517}(119,\cdot)\) \(\chi_{517}(126,\cdot)\) \(\chi_{517}(130,\cdot)\) \(\chi_{517}(136,\cdot)\) \(\chi_{517}(147,\cdot)\) \(\chi_{517}(148,\cdot)\) \(\chi_{517}(157,\cdot)\) \(\chi_{517}(158,\cdot)\) ...

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: $\Q(\zeta_{115})$
Fixed field: Number field defined by a degree 115 polynomial (not computed)

Values on generators

\((189,287)\) → \((e\left(\frac{4}{5}\right),e\left(\frac{10}{23}\right))\)

Values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 517 }(3, a) \) \(1\)\(1\)\(e\left(\frac{72}{115}\right)\)\(e\left(\frac{11}{115}\right)\)\(e\left(\frac{29}{115}\right)\)\(e\left(\frac{73}{115}\right)\)\(e\left(\frac{83}{115}\right)\)\(e\left(\frac{59}{115}\right)\)\(e\left(\frac{101}{115}\right)\)\(e\left(\frac{22}{115}\right)\)\(e\left(\frac{6}{23}\right)\)\(e\left(\frac{8}{23}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 517 }(3,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 517 }(3,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 517 }(3,·),\chi_{ 517 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 517 }(3,·)) \;\) at \(\; a,b = \) e.g. 1,2