Properties

Label 539.219
Modulus $539$
Conductor $539$
Order $42$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(539, base_ring=CyclotomicField(42))
 
M = H._module
 
chi = DirichletCharacter(H, M([38,21]))
 
pari: [g,chi] = znchar(Mod(219,539))
 

Basic properties

Modulus: \(539\)
Conductor: \(539\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
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 539.x

\(\chi_{539}(32,\cdot)\) \(\chi_{539}(65,\cdot)\) \(\chi_{539}(109,\cdot)\) \(\chi_{539}(142,\cdot)\) \(\chi_{539}(186,\cdot)\) \(\chi_{539}(219,\cdot)\) \(\chi_{539}(296,\cdot)\) \(\chi_{539}(340,\cdot)\) \(\chi_{539}(417,\cdot)\) \(\chi_{539}(450,\cdot)\) \(\chi_{539}(494,\cdot)\) \(\chi_{539}(527,\cdot)\)

sage: chi.galois_orbit()
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\zeta_{21})\)
Fixed field: Number field defined by a degree 42 polynomial

Values on generators

\((199,442)\) → \((e\left(\frac{19}{21}\right),-1)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(12\)\(13\)
\( \chi_{ 539 }(219, a) \) \(-1\)\(1\)\(e\left(\frac{1}{42}\right)\)\(e\left(\frac{19}{21}\right)\)\(e\left(\frac{1}{21}\right)\)\(e\left(\frac{5}{21}\right)\)\(e\left(\frac{13}{14}\right)\)\(e\left(\frac{1}{14}\right)\)\(e\left(\frac{17}{21}\right)\)\(e\left(\frac{11}{42}\right)\)\(e\left(\frac{20}{21}\right)\)\(e\left(\frac{5}{14}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 539 }(219,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 539 }(219,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 539 }(219,·),\chi_{ 539 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 539 }(219,·)) \;\) at \(\; a,b = \) e.g. 1,2