# Properties

 Conductor 11 Order 2 Real Yes Primitive Yes Parity Odd

# Related objects

Show commands for: SageMath / Pari/GP
sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
sage: H = DirichletGroup_conrey(11)
sage: chi = H[10]
pari: [g,chi] = znchar(Mod(10,11))

## Kronecker symbol representation

sage: kronecker_character(-11)

$\displaystyle\left(\frac{-11}{\bullet}\right)$

## Values on generators

sage: chi(k) for k in H.gens()
pari: [ chareval(g,chi,x) | x <- g.gen ] \\ value in Q/Z

$2$ → $-1$

## Values

 1 2 3 4 5 6 7 8 9 10 1 $-1$ 1 1 1 $-1$ $-1$ $-1$ 1 $-1$
value at  e.g. 2

## Basic properties

 sage: chi.conductor() pari: znconreyconductor(g,chi) Conductor = 11 sage: chi.multiplicative_order() pari: charorder(g,chi) Order = 2 sage: chi.is_odd() pari: zncharisodd(g,chi) Parity = Odd Real = Yes sage: chi.is_primitive() pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization] Primitive = Yes

## 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 ]

## Related number fields

 Field of values $\Q$

## Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
$\tau_{ a }( \chi_{ 11 }(10,·) )\;$ at $\;a =$ e.g. 2
$\displaystyle \tau_{2}(\chi_{11}(10,\cdot)) = \sum_{r\in \Z/11\Z} \chi_{11}(10,r) e\left(\frac{2r}{11}\right) = -3.3166247904i.$

## Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
$J(\chi_{ 11 }(10,·),\chi_{ 11 }(n,·)) \;$ for $\; n =$ e.g. 1
$\displaystyle J(\chi_{11}(10,\cdot),\chi_{11}(1,\cdot)) = \sum_{r\in \Z/11\Z} \chi_{11}(10,r) \chi_{11}(1,1-r) = -1.$

## Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
$K(a,b,\chi_{ 11 }(10,·)) \;$ at $\; a,b =$ e.g. 1,2
$\displaystyle K(1,2,\chi_{11}(10,·)) = \sum_{r \in \Z/11\Z} \chi_{11}(10,r) e\left(\frac{1 r + 2 r^{-1}}{11}\right) = -0.0.$