# Properties

 Conductor 8 Order 2 Real Yes Primitive No Parity Even Orbit Label 120.k

# Related objects

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

sage: H = DirichletGroup_conrey(120)

sage: chi = H[61]

pari: [g,chi] = znchar(Mod(61,120))

## Basic properties

 sage: chi.conductor()  pari: znconreyconductor(g,chi) Conductor = 8 sage: chi.multiplicative_order()  pari: charorder(g,chi) Order = 2 Real = Yes sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization] Primitive = No sage: chi.is_odd()  pari: zncharisodd(g,chi) Parity = Even Orbit label = 120.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 ]

## Inducingprimitive character

$$\chi_{8}(5,\cdot)$$ = $$\displaystyle\left(\frac{8}{\bullet}\right)$$

## Values on generators

$$(31,61,41,97)$$ → $$(1,-1,1,1)$$

## Values

 -1 1 7 11 13 17 19 23 29 31 37 41 $$1$$ $$1$$ $$1$$ $$-1$$ $$-1$$ $$1$$ $$-1$$ $$1$$ $$-1$$ $$1$$ $$-1$$ $$1$$
value at  e.g. 2

## Related number fields

 Field of values $$\Q$$

## Gauss sum

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

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 120 }(61,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{120}(61,\cdot)) = \sum_{r\in \Z/120\Z} \chi_{120}(61,r) e\left(\frac{r}{60}\right) = -0.0$$

## Jacobi sum

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

$$J(\chi_{ 120 }(61,·),\chi_{ 120 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{120}(61,\cdot),\chi_{120}(1,\cdot)) = \sum_{r\in \Z/120\Z} \chi_{120}(61,r) \chi_{120}(1,1-r) = 0$$

## Kloosterman sum

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

$$K(a,b,\chi_{ 120 }(61,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{120}(61,·)) = \sum_{r \in \Z/120\Z} \chi_{120}(61,r) e\left(\frac{1 r + 2 r^{-1}}{120}\right) = 18.3059648902$$