# Properties

 Conductor 5 Order 4 Real no Primitive no Minimal yes Parity odd Orbit label 35.g

# Learn more about

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

sage: H = DirichletGroup_conrey(35)

sage: chi = H[22]

pari: [g,chi] = znchar(Mod(22,35))

## Basic properties

 sage: chi.conductor()  pari: znconreyconductor(g,chi) Conductor = 5 sage: chi.multiplicative_order()  pari: charorder(g,chi) Order = 4 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 = 35.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

$$(22,31)$$ → $$(i,1)$$

## Values

 -1 1 2 3 4 6 8 9 11 12 13 16 $$-1$$ $$1$$ $$i$$ $$-i$$ $$-1$$ $$1$$ $$-i$$ $$-1$$ $$1$$ $$i$$ $$-i$$ $$1$$
value at  e.g. 2

## Related number fields

 Field of values $$\Q(i)$$

## Gauss sum

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

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 35 }(22,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{35}(22,\cdot)) = \sum_{r\in \Z/35\Z} \chi_{35}(22,r) e\left(\frac{2r}{35}\right) = 1.1755705046+-1.9021130326i$$

## Jacobi sum

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

$$J(\chi_{ 35 }(22,·),\chi_{ 35 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{35}(22,\cdot),\chi_{35}(1,\cdot)) = \sum_{r\in \Z/35\Z} \chi_{35}(22,r) \chi_{35}(1,1-r) = -5$$

## Kloosterman sum

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

$$K(a,b,\chi_{ 35 }(22,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{35}(22,·)) = \sum_{r \in \Z/35\Z} \chi_{35}(22,r) e\left(\frac{1 r + 2 r^{-1}}{35}\right) = -5.1205291252+5.1205291252i$$