Properties

Label 500.7
Modulus $500$
Conductor $100$
Order $20$
Real no
Primitive no
Minimal no
Parity even

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Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(500, base_ring=CyclotomicField(20))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([10,17]))
 
pari: [g,chi] = znchar(Mod(7,500))
 

Basic properties

Modulus: \(500\)
Conductor: \(100\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(20\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{100}(47,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 500.l

\(\chi_{500}(7,\cdot)\) \(\chi_{500}(43,\cdot)\) \(\chi_{500}(107,\cdot)\) \(\chi_{500}(143,\cdot)\) \(\chi_{500}(207,\cdot)\) \(\chi_{500}(243,\cdot)\) \(\chi_{500}(343,\cdot)\) \(\chi_{500}(407,\cdot)\)

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: \(\Q(\zeta_{20})\)
Fixed field: \(\Q(\zeta_{100})^+\)

Values on generators

\((251,377)\) → \((-1,e\left(\frac{17}{20}\right))\)

Values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 500 }(7, a) \) \(1\)\(1\)\(e\left(\frac{9}{20}\right)\)\(-i\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{7}{20}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 500 }(7,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 500 }(7,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 500 }(7,·),\chi_{ 500 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 500 }(7,·)) \;\) at \(\; a,b = \) e.g. 1,2