# Properties

 Modulus 80 Conductor 80 Order 4 Real no Primitive yes Minimal yes Parity odd Orbit label 80.i

# Related objects

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter

sage: H = DirichletGroup(80)

sage: M = H._module

sage: chi = DirichletCharacter(H, M([0,3,3]))

pari: [g,chi] = znchar(Mod(13,80))

## Basic properties

 sage: chi.conductor()  pari: znconreyconductor(g,chi) Modulus = 80 Conductor = 80 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 = yes Minimal = yes sage: chi.is_odd()  pari: zncharisodd(g,chi) Parity = odd Orbit label = 80.i Orbit index = 9

## Galois orbit

sage: chi.galois_orbit()

pari: order = charorder(g,chi)

pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]

## Values on generators

$$(31,21,17)$$ → $$(1,-i,-i)$$

## Values

 -1 1 3 7 9 11 13 17 19 21 23 27 $$-1$$ $$1$$ $$-1$$ $$i$$ $$1$$ $$-i$$ $$-1$$ $$-i$$ $$-i$$ $$-i$$ $$-i$$ $$-1$$
value at  e.g. 2

## Related number fields

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

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 80 }(13,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{80}(13,\cdot)) = \sum_{r\in \Z/80\Z} \chi_{80}(13,r) e\left(\frac{r}{40}\right) = -0.0$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 80 }(13,·),\chi_{ 80 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{80}(13,\cdot),\chi_{80}(1,\cdot)) = \sum_{r\in \Z/80\Z} \chi_{80}(13,r) \chi_{80}(1,1-r) = 0$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 80 }(13,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{80}(13,·)) = \sum_{r \in \Z/80\Z} \chi_{80}(13,r) e\left(\frac{1 r + 2 r^{-1}}{80}\right) = 2.5448566901+6.1438275356i$$