Properties

Label 516.239
Modulus $516$
Conductor $516$
Order $42$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(516, base_ring=CyclotomicField(42))
 
M = H._module
 
chi = DirichletCharacter(H, M([21,21,40]))
 
pari: [g,chi] = znchar(Mod(239,516))
 

Basic properties

Modulus: \(516\)
Conductor: \(516\)
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: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 516.bb

\(\chi_{516}(23,\cdot)\) \(\chi_{516}(83,\cdot)\) \(\chi_{516}(95,\cdot)\) \(\chi_{516}(143,\cdot)\) \(\chi_{516}(167,\cdot)\) \(\chi_{516}(203,\cdot)\) \(\chi_{516}(239,\cdot)\) \(\chi_{516}(275,\cdot)\) \(\chi_{516}(311,\cdot)\) \(\chi_{516}(359,\cdot)\) \(\chi_{516}(443,\cdot)\) \(\chi_{516}(455,\cdot)\)

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

\((259,173,433)\) → \((-1,-1,e\left(\frac{20}{21}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 516 }(239, a) \) \(1\)\(1\)\(e\left(\frac{13}{42}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{10}{21}\right)\)\(e\left(\frac{29}{42}\right)\)\(e\left(\frac{25}{42}\right)\)\(e\left(\frac{5}{21}\right)\)\(e\left(\frac{13}{21}\right)\)\(e\left(\frac{23}{42}\right)\)\(e\left(\frac{37}{42}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 516 }(239,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 516 }(239,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 516 }(239,·),\chi_{ 516 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 516 }(239,·)) \;\) at \(\; a,b = \) e.g. 1,2