Properties

Label 500.47
Modulus $500$
Conductor $500$
Order $100$
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(500, base_ring=CyclotomicField(100))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([50,97]))
 
pari: [g,chi] = znchar(Mod(47,500))
 

Basic properties

Modulus: \(500\)
Conductor: \(500\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(100\)
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 500.r

\(\chi_{500}(3,\cdot)\) \(\chi_{500}(23,\cdot)\) \(\chi_{500}(27,\cdot)\) \(\chi_{500}(47,\cdot)\) \(\chi_{500}(63,\cdot)\) \(\chi_{500}(67,\cdot)\) \(\chi_{500}(83,\cdot)\) \(\chi_{500}(87,\cdot)\) \(\chi_{500}(103,\cdot)\) \(\chi_{500}(123,\cdot)\) \(\chi_{500}(127,\cdot)\) \(\chi_{500}(147,\cdot)\) \(\chi_{500}(163,\cdot)\) \(\chi_{500}(167,\cdot)\) \(\chi_{500}(183,\cdot)\) \(\chi_{500}(187,\cdot)\) \(\chi_{500}(203,\cdot)\) \(\chi_{500}(223,\cdot)\) \(\chi_{500}(227,\cdot)\) \(\chi_{500}(247,\cdot)\) \(\chi_{500}(263,\cdot)\) \(\chi_{500}(267,\cdot)\) \(\chi_{500}(283,\cdot)\) \(\chi_{500}(287,\cdot)\) \(\chi_{500}(303,\cdot)\) \(\chi_{500}(323,\cdot)\) \(\chi_{500}(327,\cdot)\) \(\chi_{500}(347,\cdot)\) \(\chi_{500}(363,\cdot)\) \(\chi_{500}(367,\cdot)\) ...

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: $\Q(\zeta_{100})$
Fixed field: Number field defined by a degree 100 polynomial

Values on generators

\((251,377)\) → \((-1,e\left(\frac{97}{100}\right))\)

Values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 500 }(47, a) \) \(1\)\(1\)\(e\left(\frac{29}{100}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{29}{50}\right)\)\(e\left(\frac{11}{50}\right)\)\(e\left(\frac{83}{100}\right)\)\(e\left(\frac{81}{100}\right)\)\(e\left(\frac{24}{25}\right)\)\(e\left(\frac{6}{25}\right)\)\(e\left(\frac{57}{100}\right)\)\(e\left(\frac{87}{100}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 500 }(47,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 500 }(47,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 500 }(47,·),\chi_{ 500 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 500 }(47,·)) \;\) at \(\; a,b = \) e.g. 1,2