# Properties

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

# 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(45)

sage: chi = H[13]

pari: [g,chi] = znchar(Mod(13,45))

## 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 = yes Minimal = yes sage: chi.is_odd()  pari: zncharisodd(g,chi) Parity = odd Orbit label = 45.k Orbit index = 11

## 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

$$(11,37)$$ → $$(e\left(\frac{1}{3}\right),-i)$$

## Values

 -1 1 2 4 7 8 11 13 14 16 17 19 $$-1$$ $$1$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$i$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$-i$$ $$-1$$
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)

$$\tau_{ a }( \chi_{ 45 }(13,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{45}(13,\cdot)) = \sum_{r\in \Z/45\Z} \chi_{45}(13,r) e\left(\frac{2r}{45}\right) = -4.1559983645+5.2657077012i$$

## Jacobi sum

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

$$J(\chi_{ 45 }(13,·),\chi_{ 45 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{45}(13,\cdot),\chi_{45}(1,\cdot)) = \sum_{r\in \Z/45\Z} \chi_{45}(13,r) \chi_{45}(1,1-r) = 0$$

## Kloosterman sum

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

$$K(a,b,\chi_{ 45 }(13,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{45}(13,·)) = \sum_{r \in \Z/45\Z} \chi_{45}(13,r) e\left(\frac{1 r + 2 r^{-1}}{45}\right) = 1.2908660059+-4.8175775196i$$