# Properties

 Conductor 45 Order 12 Real no Primitive no Minimal yes Parity odd Orbit label 315.by

# 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(315)

sage: chi = H[43]

pari: [g,chi] = znchar(Mod(43,315))

## 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 = 315.by Orbit index = 51

## 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

$$(281,127,136)$$ → $$(e\left(\frac{2}{3}\right),-i,1)$$

## Values

 -1 1 2 4 8 11 13 16 17 19 22 23 $$-1$$ $$1$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$i$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$-i$$ $$-1$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{7}{12}\right)$$
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)

$$\tau_{ a }( \chi_{ 315 }(43,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{315}(43,\cdot)) = \sum_{r\in \Z/315\Z} \chi_{315}(43,r) e\left(\frac{2r}{315}\right) = -5.0072399573+4.4640282269i$$

## Jacobi sum

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

$$J(\chi_{ 315 }(43,·),\chi_{ 315 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{315}(43,\cdot),\chi_{315}(1,\cdot)) = \sum_{r\in \Z/315\Z} \chi_{315}(43,r) \chi_{315}(1,1-r) = 0$$

## Kloosterman sum

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

$$K(a,b,\chi_{ 315 }(43,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{315}(43,·)) = \sum_{r \in \Z/315\Z} \chi_{315}(43,r) e\left(\frac{1 r + 2 r^{-1}}{315}\right) = -15.9713192059+4.2795020833i$$