Properties

Conductor 27
Order 18
Real No
Primitive No
Parity Odd
Orbit Label 54.f

Related objects

Learn more about

Show commands for: SageMath / Pari/GP
sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
sage: H = DirichletGroup_conrey(54)
sage: chi = H[47]
pari: [g,chi] = znchar(Mod(47,54))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 27
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 18
Real = No
sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
Primitive = No
sage: chi.is_odd()
pari: zncharisodd(g,chi)
Parity = Odd
Orbit label = 54.f
Orbit index = 6

Galois orbit

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

\(\chi_{54}(5,\cdot)\) \(\chi_{54}(11,\cdot)\) \(\chi_{54}(23,\cdot)\) \(\chi_{54}(29,\cdot)\) \(\chi_{54}(41,\cdot)\) \(\chi_{54}(47,\cdot)\)

Inducing primitive character

\(\chi_{27}(20,\cdot)\)

Values on generators

\(29\) → \(e\left(\frac{7}{18}\right)\)

Values

-11571113171923252931
\(-1\)\(1\)\(e\left(\frac{17}{18}\right)\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{5}{18}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{7}{18}\right)\)\(e\left(\frac{7}{9}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{9})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 54 }(47,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{54}(47,\cdot)) = \sum_{r\in \Z/54\Z} \chi_{54}(47,r) e\left(\frac{r}{27}\right) = -3.7795443099+3.5658161492i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 54 }(47,·),\chi_{ 54 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{54}(47,\cdot),\chi_{54}(1,\cdot)) = \sum_{r\in \Z/54\Z} \chi_{54}(47,r) \chi_{54}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 54 }(47,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{54}(47,·)) = \sum_{r \in \Z/54\Z} \chi_{54}(47,r) e\left(\frac{1 r + 2 r^{-1}}{54}\right) = -8.4572335871+3.0781812899i \)