Properties

Label 539.17
Modulus $539$
Conductor $539$
Order $210$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(539, base_ring=CyclotomicField(210))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([125,189]))
 
pari: [g,chi] = znchar(Mod(17,539))
 

Basic properties

Modulus: \(539\)
Conductor: \(539\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(210\)
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 539.bf

\(\chi_{539}(17,\cdot)\) \(\chi_{539}(24,\cdot)\) \(\chi_{539}(40,\cdot)\) \(\chi_{539}(52,\cdot)\) \(\chi_{539}(61,\cdot)\) \(\chi_{539}(73,\cdot)\) \(\chi_{539}(94,\cdot)\) \(\chi_{539}(96,\cdot)\) \(\chi_{539}(101,\cdot)\) \(\chi_{539}(138,\cdot)\) \(\chi_{539}(145,\cdot)\) \(\chi_{539}(150,\cdot)\) \(\chi_{539}(171,\cdot)\) \(\chi_{539}(173,\cdot)\) \(\chi_{539}(194,\cdot)\) \(\chi_{539}(206,\cdot)\) \(\chi_{539}(222,\cdot)\) \(\chi_{539}(248,\cdot)\) \(\chi_{539}(250,\cdot)\) \(\chi_{539}(255,\cdot)\) \(\chi_{539}(271,\cdot)\) \(\chi_{539}(283,\cdot)\) \(\chi_{539}(292,\cdot)\) \(\chi_{539}(299,\cdot)\) \(\chi_{539}(304,\cdot)\) \(\chi_{539}(327,\cdot)\) \(\chi_{539}(332,\cdot)\) \(\chi_{539}(348,\cdot)\) \(\chi_{539}(360,\cdot)\) \(\chi_{539}(369,\cdot)\) ...

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: $\Q(\zeta_{105})$
Fixed field: Number field defined by a degree 210 polynomial (not computed)

Values on generators

\((199,442)\) → \((e\left(\frac{25}{42}\right),e\left(\frac{9}{10}\right))\)

Values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(12\)\(13\)
\( \chi_{ 539 }(17, a) \) \(1\)\(1\)\(e\left(\frac{79}{210}\right)\)\(e\left(\frac{167}{210}\right)\)\(e\left(\frac{79}{105}\right)\)\(e\left(\frac{181}{210}\right)\)\(e\left(\frac{6}{35}\right)\)\(e\left(\frac{9}{70}\right)\)\(e\left(\frac{62}{105}\right)\)\(e\left(\frac{5}{21}\right)\)\(e\left(\frac{23}{42}\right)\)\(e\left(\frac{19}{35}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 539 }(17,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 539 }(17,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 539 }(17,·),\chi_{ 539 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 539 }(17,·)) \;\) at \(\; a,b = \) e.g. 1,2