Properties

Label 243.28
Modulus $243$
Conductor $27$
Order $9$
Real no
Primitive no
Minimal no
Parity even

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(243, base_ring=CyclotomicField(18))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([8]))
 
pari: [g,chi] = znchar(Mod(28,243))
 

Basic properties

Modulus: \(243\)
Conductor: \(27\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(9\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{27}(13,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 243.e

\(\chi_{243}(28,\cdot)\) \(\chi_{243}(55,\cdot)\) \(\chi_{243}(109,\cdot)\) \(\chi_{243}(136,\cdot)\) \(\chi_{243}(190,\cdot)\) \(\chi_{243}(217,\cdot)\)

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

Values on generators

\(2\) → \(e\left(\frac{4}{9}\right)\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\(1\)\(1\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{7}{9}\right)\)\(e\left(\frac{5}{9}\right)\)\(e\left(\frac{5}{9}\right)\)\(e\left(\frac{7}{9}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{9})\)
Fixed field: \(\Q(\zeta_{27})^+\)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 243 }(28,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{243}(28,\cdot)) = \sum_{r\in \Z/243\Z} \chi_{243}(28,r) e\left(\frac{2r}{243}\right) = 0.0 \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 243 }(28,·),\chi_{ 243 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{243}(28,\cdot),\chi_{243}(1,\cdot)) = \sum_{r\in \Z/243\Z} \chi_{243}(28,r) \chi_{243}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 243 }(28,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{243}(28,·)) = \sum_{r \in \Z/243\Z} \chi_{243}(28,r) e\left(\frac{1 r + 2 r^{-1}}{243}\right) = -0.0 \)