Properties

Label 500.69
Modulus $500$
Conductor $125$
Order $50$
Real no
Primitive no
Minimal yes
Parity even

Related objects

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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([0,19]))
 
pari: [g,chi] = znchar(Mod(69,500))
 

Basic properties

Modulus: \(500\)
Conductor: \(125\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(50\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{125}(69,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 500.o

\(\chi_{500}(9,\cdot)\) \(\chi_{500}(29,\cdot)\) \(\chi_{500}(69,\cdot)\) \(\chi_{500}(89,\cdot)\) \(\chi_{500}(109,\cdot)\) \(\chi_{500}(129,\cdot)\) \(\chi_{500}(169,\cdot)\) \(\chi_{500}(189,\cdot)\) \(\chi_{500}(209,\cdot)\) \(\chi_{500}(229,\cdot)\) \(\chi_{500}(269,\cdot)\) \(\chi_{500}(289,\cdot)\) \(\chi_{500}(309,\cdot)\) \(\chi_{500}(329,\cdot)\) \(\chi_{500}(369,\cdot)\) \(\chi_{500}(389,\cdot)\) \(\chi_{500}(409,\cdot)\) \(\chi_{500}(429,\cdot)\) \(\chi_{500}(469,\cdot)\) \(\chi_{500}(489,\cdot)\)

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

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 500 }(69, a) \) \(1\)\(1\)\(e\left(\frac{33}{50}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{8}{25}\right)\)\(e\left(\frac{22}{25}\right)\)\(e\left(\frac{41}{50}\right)\)\(e\left(\frac{37}{50}\right)\)\(e\left(\frac{21}{25}\right)\)\(e\left(\frac{24}{25}\right)\)\(e\left(\frac{39}{50}\right)\)\(e\left(\frac{49}{50}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 500 }(69,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 500 }(69,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 500 }(69,·),\chi_{ 500 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 500 }(69,·)) \;\) at \(\; a,b = \) e.g. 1,2