Properties

Label 500.11
Modulus $500$
Conductor $500$
Order $50$
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(500, base_ring=CyclotomicField(50))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([25,38]))
 
pari: [g,chi] = znchar(Mod(11,500))
 

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.p

\(\chi_{500}(11,\cdot)\) \(\chi_{500}(31,\cdot)\) \(\chi_{500}(71,\cdot)\) \(\chi_{500}(91,\cdot)\) \(\chi_{500}(111,\cdot)\) \(\chi_{500}(131,\cdot)\) \(\chi_{500}(171,\cdot)\) \(\chi_{500}(191,\cdot)\) \(\chi_{500}(211,\cdot)\) \(\chi_{500}(231,\cdot)\) \(\chi_{500}(271,\cdot)\) \(\chi_{500}(291,\cdot)\) \(\chi_{500}(311,\cdot)\) \(\chi_{500}(331,\cdot)\) \(\chi_{500}(371,\cdot)\) \(\chi_{500}(391,\cdot)\) \(\chi_{500}(411,\cdot)\) \(\chi_{500}(431,\cdot)\) \(\chi_{500}(471,\cdot)\) \(\chi_{500}(491,\cdot)\)

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

\((251,377)\) → \((-1,e\left(\frac{19}{25}\right))\)

Values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 500 }(11, a) \) \(-1\)\(1\)\(e\left(\frac{41}{50}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{16}{25}\right)\)\(e\left(\frac{13}{50}\right)\)\(e\left(\frac{16}{25}\right)\)\(e\left(\frac{12}{25}\right)\)\(e\left(\frac{9}{50}\right)\)\(e\left(\frac{23}{25}\right)\)\(e\left(\frac{3}{50}\right)\)\(e\left(\frac{23}{50}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 500 }(11,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 500 }(11,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 500 }(11,·),\chi_{ 500 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 500 }(11,·)) \;\) at \(\; a,b = \) e.g. 1,2