Properties

Modulus 162
Conductor 81
Order 27
Real no
Primitive no
Minimal yes
Parity even
Orbit label 162.g

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(162)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([8]))
 
pari: [g,chi] = znchar(Mod(7,162))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Modulus = 162
Conductor = 81
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 27
Real = no
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
 
Primitive = no
Minimal = yes
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = even
Orbit label = 162.g
Orbit index = 7

Galois orbit

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

\(\chi_{162}(7,\cdot)\) \(\chi_{162}(13,\cdot)\) \(\chi_{162}(25,\cdot)\) \(\chi_{162}(31,\cdot)\) \(\chi_{162}(43,\cdot)\) \(\chi_{162}(49,\cdot)\) \(\chi_{162}(61,\cdot)\) \(\chi_{162}(67,\cdot)\) \(\chi_{162}(79,\cdot)\) \(\chi_{162}(85,\cdot)\) \(\chi_{162}(97,\cdot)\) \(\chi_{162}(103,\cdot)\) \(\chi_{162}(115,\cdot)\) \(\chi_{162}(121,\cdot)\) \(\chi_{162}(133,\cdot)\) \(\chi_{162}(139,\cdot)\) \(\chi_{162}(151,\cdot)\) \(\chi_{162}(157,\cdot)\)

Values on generators

\(83\) → \(e\left(\frac{8}{27}\right)\)

Values

-11571113171923252931
\(1\)\(1\)\(e\left(\frac{22}{27}\right)\)\(e\left(\frac{20}{27}\right)\)\(e\left(\frac{23}{27}\right)\)\(e\left(\frac{10}{27}\right)\)\(e\left(\frac{7}{9}\right)\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{7}{27}\right)\)\(e\left(\frac{17}{27}\right)\)\(e\left(\frac{26}{27}\right)\)\(e\left(\frac{25}{27}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{27})\)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 162 }(7,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{162}(7,\cdot)) = \sum_{r\in \Z/162\Z} \chi_{162}(7,r) e\left(\frac{r}{81}\right) = 8.9729364705+-0.697431786i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 162 }(7,·),\chi_{ 162 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{162}(7,\cdot),\chi_{162}(1,\cdot)) = \sum_{r\in \Z/162\Z} \chi_{162}(7,r) \chi_{162}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 162 }(7,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{162}(7,·)) = \sum_{r \in \Z/162\Z} \chi_{162}(7,r) e\left(\frac{1 r + 2 r^{-1}}{162}\right) = -0.0 \)