Properties

Label 500.319
Modulus $500$
Conductor $500$
Order $50$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Downloads

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(500, base_ring=CyclotomicField(50))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([25,19]))
 
pari: [g,chi] = znchar(Mod(319,500))
 

Basic properties

Modulus: \(500\)
Conductor: \(500\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(50\)
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 500.n

\(\chi_{500}(19,\cdot)\) \(\chi_{500}(39,\cdot)\) \(\chi_{500}(59,\cdot)\) \(\chi_{500}(79,\cdot)\) \(\chi_{500}(119,\cdot)\) \(\chi_{500}(139,\cdot)\) \(\chi_{500}(159,\cdot)\) \(\chi_{500}(179,\cdot)\) \(\chi_{500}(219,\cdot)\) \(\chi_{500}(239,\cdot)\) \(\chi_{500}(259,\cdot)\) \(\chi_{500}(279,\cdot)\) \(\chi_{500}(319,\cdot)\) \(\chi_{500}(339,\cdot)\) \(\chi_{500}(359,\cdot)\) \(\chi_{500}(379,\cdot)\) \(\chi_{500}(419,\cdot)\) \(\chi_{500}(439,\cdot)\) \(\chi_{500}(459,\cdot)\) \(\chi_{500}(479,\cdot)\)

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: \(\Q(\zeta_{25})\)
Fixed field: Number field defined by a degree 50 polynomial

Values on generators

\((251,377)\) → \((-1,e\left(\frac{19}{50}\right))\)

Values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 500 }(319, a) \) \(-1\)\(1\)\(e\left(\frac{4}{25}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{8}{25}\right)\)\(e\left(\frac{19}{50}\right)\)\(e\left(\frac{41}{50}\right)\)\(e\left(\frac{37}{50}\right)\)\(e\left(\frac{17}{50}\right)\)\(e\left(\frac{24}{25}\right)\)\(e\left(\frac{7}{25}\right)\)\(e\left(\frac{12}{25}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 500 }(319,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 500 }(319,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 500 }(319,·),\chi_{ 500 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 500 }(319,·)) \;\) at \(\; a,b = \) e.g. 1,2