Properties

Conductor 19
Order 9
Real No
Primitive Yes
Parity Even
Orbit Label 19.e

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(19)
sage: chi = H[16]
pari: [g,chi] = znchar(Mod(16,19))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 19
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 9
Real = No
sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
Primitive = Yes
sage: chi.is_odd()
pari: zncharisodd(g,chi)
Parity = Even
Orbit label = 19.e
Orbit index = 5

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 ]

\(\chi_{19}(4,\cdot)\) \(\chi_{19}(5,\cdot)\) \(\chi_{19}(6,\cdot)\) \(\chi_{19}(9,\cdot)\) \(\chi_{19}(16,\cdot)\) \(\chi_{19}(17,\cdot)\)

Values on generators

\(2\) → \(e\left(\frac{2}{9}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{5}{9}\right)\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{7}{9}\right)\)\(e\left(\frac{7}{9}\right)\)\(e\left(\frac{2}{3}\right)\)
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)
\( \tau_{ a }( \chi_{ 19 }(16,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{19}(16,\cdot)) = \sum_{r\in \Z/19\Z} \chi_{19}(16,r) e\left(\frac{2r}{19}\right) = 3.3811815023+-2.7509292336i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 19 }(16,·),\chi_{ 19 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{19}(16,\cdot),\chi_{19}(1,\cdot)) = \sum_{r\in \Z/19\Z} \chi_{19}(16,r) \chi_{19}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 19 }(16,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{19}(16,·)) = \sum_{r \in \Z/19\Z} \chi_{19}(16,r) e\left(\frac{1 r + 2 r^{-1}}{19}\right) = 3.5742611918+2.9991612478i \)