# Properties

 Conductor 11 Order 10 Real no Primitive no Minimal yes Parity odd Orbit label 33.g

# Related objects

Show commands for: Pari/GP / SageMath
sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed

sage: H = DirichletGroup_conrey(33)

sage: chi = H[13]

pari: [g,chi] = znchar(Mod(13,33))

## Basic properties

 sage: chi.conductor()  pari: znconreyconductor(g,chi) Conductor = 11 sage: chi.multiplicative_order()  pari: charorder(g,chi) Order = 10 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 = odd Orbit label = 33.g Orbit index = 7

## 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 ]

## Values on generators

$$(23,13)$$ → $$(1,e\left(\frac{1}{10}\right))$$

## Values

 -1 1 2 4 5 7 8 10 13 14 16 17 $$-1$$ $$1$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$-1$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{9}{10}\right)$$
value at  e.g. 2

## Related number fields

 Field of values $$\Q(\zeta_{5})$$

## Gauss sum

sage: chi.sage_character().gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 33 }(13,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{33}(13,\cdot)) = \sum_{r\in \Z/33\Z} \chi_{33}(13,r) e\left(\frac{2r}{33}\right) = -3.3158232425+0.0729124433i$$

## Jacobi sum

sage: chi.sage_character().jacobi_sum(n)

$$J(\chi_{ 33 }(13,·),\chi_{ 33 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{33}(13,\cdot),\chi_{33}(1,\cdot)) = \sum_{r\in \Z/33\Z} \chi_{33}(13,r) \chi_{33}(1,1-r) = -1$$

## Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)

$$K(a,b,\chi_{ 33 }(13,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{33}(13,·)) = \sum_{r \in \Z/33\Z} \chi_{33}(13,r) e\left(\frac{1 r + 2 r^{-1}}{33}\right) = -0.1729543846+0.5322988621i$$