Properties

Conductor 32
Order 8
Real No
Primitive No
Parity Odd
Orbit Label 96.m

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(96)
 
sage: chi = H[67]
 
pari: [g,chi] = znchar(Mod(67,96))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 32
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 8
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 = 96.m
Orbit index = 13

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 ]
 

\(\chi_{96}(19,\cdot)\) \(\chi_{96}(43,\cdot)\) \(\chi_{96}(67,\cdot)\) \(\chi_{96}(91,\cdot)\)

Inducing primitive character

\(\chi_{32}(3,\cdot)\)

Values on generators

\((31,37,65)\) → \((-1,e\left(\frac{3}{8}\right),1)\)

Values

-11571113171923252931
\(-1\)\(1\)\(e\left(\frac{3}{8}\right)\)\(i\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{5}{8}\right)\)\(-1\)\(e\left(\frac{1}{8}\right)\)\(-i\)\(-i\)\(e\left(\frac{1}{8}\right)\)\(-1\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{8})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 96 }(67,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{96}(67,\cdot)) = \sum_{r\in \Z/96\Z} \chi_{96}(67,r) e\left(\frac{r}{48}\right) = -0.0 \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 96 }(67,·),\chi_{ 96 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{96}(67,\cdot),\chi_{96}(1,\cdot)) = \sum_{r\in \Z/96\Z} \chi_{96}(67,r) \chi_{96}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 96 }(67,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{96}(67,·)) = \sum_{r \in \Z/96\Z} \chi_{96}(67,r) e\left(\frac{1 r + 2 r^{-1}}{96}\right) = -6.2855596671+9.4070048194i \)