Properties

Conductor 27
Order 18
Real No
Primitive Yes
Parity Odd
Orbit Label 27.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(27)
 
sage: chi = H[2]
 
pari: [g,chi] = znchar(Mod(2,27))
 

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 = Yes
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = Odd
Orbit label = 27.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 ]
 

\(\chi_{27}(2,\cdot)\) \(\chi_{27}(5,\cdot)\) \(\chi_{27}(11,\cdot)\) \(\chi_{27}(14,\cdot)\) \(\chi_{27}(20,\cdot)\) \(\chi_{27}(23,\cdot)\)

Values on generators

\(2\) → \(e\left(\frac{1}{18}\right)\)

Values

-11245781011131416
\(-1\)\(1\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{5}{18}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{13}{18}\right)\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{17}{18}\right)\)\(e\left(\frac{2}{9}\right)\)
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)
 
\( \tau_{ a }( \chi_{ 27 }(2,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{27}(2,\cdot)) = \sum_{r\in \Z/27\Z} \chi_{27}(2,r) e\left(\frac{2r}{27}\right) = -2.3320289475+4.643451409i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 27 }(2,·),\chi_{ 27 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{27}(2,\cdot),\chi_{27}(1,\cdot)) = \sum_{r\in \Z/27\Z} \chi_{27}(2,r) \chi_{27}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 27 }(2,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{27}(2,·)) = \sum_{r \in \Z/27\Z} \chi_{27}(2,r) e\left(\frac{1 r + 2 r^{-1}}{27}\right) = -0.0 \)