# Properties

 Conductor 45 Order 12 Real no Primitive no Minimal yes Parity odd Orbit label 360.bp

# 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(360)

sage: chi = H[97]

pari: [g,chi] = znchar(Mod(97,360))

## Basic properties

 sage: chi.conductor()  pari: znconreyconductor(g,chi) Conductor = 45 sage: chi.multiplicative_order()  pari: charorder(g,chi) Order = 12 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 = 360.bp Orbit index = 42

## 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

$$(271,181,281,217)$$ → $$(1,1,e\left(\frac{2}{3}\right),i)$$

## Values

 -1 1 7 11 13 17 19 23 29 31 37 41 $$-1$$ $$1$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$i$$ $$-1$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$i$$ $$e\left(\frac{1}{3}\right)$$
value at  e.g. 2

## Related number fields

 Field of values $$\Q(\zeta_{12})$$

## Gauss sum

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

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 360 }(97,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{360}(97,\cdot)) = \sum_{r\in \Z/360\Z} \chi_{360}(97,r) e\left(\frac{r}{180}\right) = -0.0$$

## Jacobi sum

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

$$J(\chi_{ 360 }(97,·),\chi_{ 360 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{360}(97,\cdot),\chi_{360}(1,\cdot)) = \sum_{r\in \Z/360\Z} \chi_{360}(97,r) \chi_{360}(1,1-r) = 0$$

## Kloosterman sum

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

$$K(a,b,\chi_{ 360 }(97,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{360}(97,·)) = \sum_{r \in \Z/360\Z} \chi_{360}(97,r) e\left(\frac{1 r + 2 r^{-1}}{360}\right) = -0.0$$