Properties

Label 539.20
Modulus $539$
Conductor $539$
Order $70$
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(70))
 
M = H._module
 
chi = DirichletCharacter(H, M([65,42]))
 
pari: [g,chi] = znchar(Mod(20,539))
 

Basic properties

Modulus: \(539\)
Conductor: \(539\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(70\)
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.bb

\(\chi_{539}(20,\cdot)\) \(\chi_{539}(27,\cdot)\) \(\chi_{539}(69,\cdot)\) \(\chi_{539}(104,\cdot)\) \(\chi_{539}(125,\cdot)\) \(\chi_{539}(174,\cdot)\) \(\chi_{539}(181,\cdot)\) \(\chi_{539}(202,\cdot)\) \(\chi_{539}(223,\cdot)\) \(\chi_{539}(251,\cdot)\) \(\chi_{539}(258,\cdot)\) \(\chi_{539}(279,\cdot)\) \(\chi_{539}(300,\cdot)\) \(\chi_{539}(328,\cdot)\) \(\chi_{539}(335,\cdot)\) \(\chi_{539}(356,\cdot)\) \(\chi_{539}(377,\cdot)\) \(\chi_{539}(405,\cdot)\) \(\chi_{539}(412,\cdot)\) \(\chi_{539}(433,\cdot)\) \(\chi_{539}(454,\cdot)\) \(\chi_{539}(482,\cdot)\) \(\chi_{539}(510,\cdot)\) \(\chi_{539}(531,\cdot)\)

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: $\Q(\zeta_{35})$
Fixed field: Number field defined by a degree 70 polynomial

Values on generators

\((199,442)\) → \((e\left(\frac{13}{14}\right),e\left(\frac{3}{5}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(12\)\(13\)
\( \chi_{ 539 }(20, a) \) \(-1\)\(1\)\(e\left(\frac{26}{35}\right)\)\(e\left(\frac{51}{70}\right)\)\(e\left(\frac{17}{35}\right)\)\(e\left(\frac{23}{70}\right)\)\(e\left(\frac{33}{70}\right)\)\(e\left(\frac{8}{35}\right)\)\(e\left(\frac{16}{35}\right)\)\(e\left(\frac{1}{14}\right)\)\(e\left(\frac{3}{14}\right)\)\(e\left(\frac{17}{70}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 539 }(20,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 539 }(20,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 539 }(20,·),\chi_{ 539 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 539 }(20,·)) \;\) at \(\; a,b = \) e.g. 1,2