Properties

Conductor 16
Order 4
Real No
Primitive Yes
Parity Odd
Orbit Label 16.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(16)
 
sage: chi = H[11]
 
pari: [g,chi] = znchar(Mod(11,16))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 16
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 4
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 = 16.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 ]
 

\(\chi_{16}(3,\cdot)\) \(\chi_{16}(11,\cdot)\)

Values on generators

\((15,5)\) → \((-1,i)\)

Values

-1135791113
\(-1\)\(1\)\(i\)\(i\)\(1\)\(-1\)\(-i\)\(-i\)
value at  e.g. 2

Related number fields

Field of values \(\Q(i)\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 16 }(11,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{16}(11,\cdot)) = \sum_{r\in \Z/16\Z} \chi_{16}(11,r) e\left(\frac{r}{8}\right) = -0.0 \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 16 }(11,·),\chi_{ 16 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{16}(11,\cdot),\chi_{16}(1,\cdot)) = \sum_{r\in \Z/16\Z} \chi_{16}(11,r) \chi_{16}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 16 }(11,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{16}(11,·)) = \sum_{r \in \Z/16\Z} \chi_{16}(11,r) e\left(\frac{1 r + 2 r^{-1}}{16}\right) = 1.5307337295+3.69551813i \)