Properties

Label 539.102
Modulus $539$
Conductor $539$
Order $105$
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(539, base_ring=CyclotomicField(210))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([50,168]))
 
pari: [g,chi] = znchar(Mod(102,539))
 

Basic properties

Modulus: \(539\)
Conductor: \(539\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(105\)
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.bc

\(\chi_{539}(4,\cdot)\) \(\chi_{539}(9,\cdot)\) \(\chi_{539}(16,\cdot)\) \(\chi_{539}(25,\cdot)\) \(\chi_{539}(37,\cdot)\) \(\chi_{539}(53,\cdot)\) \(\chi_{539}(58,\cdot)\) \(\chi_{539}(60,\cdot)\) \(\chi_{539}(81,\cdot)\) \(\chi_{539}(86,\cdot)\) \(\chi_{539}(93,\cdot)\) \(\chi_{539}(102,\cdot)\) \(\chi_{539}(114,\cdot)\) \(\chi_{539}(130,\cdot)\) \(\chi_{539}(135,\cdot)\) \(\chi_{539}(137,\cdot)\) \(\chi_{539}(158,\cdot)\) \(\chi_{539}(163,\cdot)\) \(\chi_{539}(170,\cdot)\) \(\chi_{539}(179,\cdot)\) \(\chi_{539}(191,\cdot)\) \(\chi_{539}(207,\cdot)\) \(\chi_{539}(212,\cdot)\) \(\chi_{539}(235,\cdot)\) \(\chi_{539}(240,\cdot)\) \(\chi_{539}(247,\cdot)\) \(\chi_{539}(256,\cdot)\) \(\chi_{539}(268,\cdot)\) \(\chi_{539}(284,\cdot)\) \(\chi_{539}(289,\cdot)\) ...

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 105 polynomial (not computed)

Values on generators

\((199,442)\) → \((e\left(\frac{5}{21}\right),e\left(\frac{4}{5}\right))\)

Values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(12\)\(13\)
\( \chi_{ 539 }(102, a) \) \(1\)\(1\)\(e\left(\frac{104}{105}\right)\)\(e\left(\frac{67}{105}\right)\)\(e\left(\frac{103}{105}\right)\)\(e\left(\frac{11}{105}\right)\)\(e\left(\frac{22}{35}\right)\)\(e\left(\frac{34}{35}\right)\)\(e\left(\frac{29}{105}\right)\)\(e\left(\frac{2}{21}\right)\)\(e\left(\frac{13}{21}\right)\)\(e\left(\frac{23}{35}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 539 }(102,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 539 }(102,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 539 }(102,·),\chi_{ 539 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 539 }(102,·)) \;\) at \(\; a,b = \) e.g. 1,2