# Properties

 Modulus 320 Conductor 320 Order 16 Real no Primitive yes Minimal yes Parity even Orbit label 320.bj

# Related objects

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

sage: H = DirichletGroup(320)

sage: M = H._module

sage: chi = DirichletCharacter(H, M([8,9,4]))

pari: [g,chi] = znchar(Mod(27,320))

## Basic properties

 sage: chi.conductor()  pari: znconreyconductor(g,chi) Modulus = 320 Conductor = 320 sage: chi.multiplicative_order()  pari: charorder(g,chi) Order = 16 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 = even Orbit label = 320.bj Orbit index = 36

## 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

$$(191,261,257)$$ → $$(-1,e\left(\frac{9}{16}\right),i)$$

## Values

 -1 1 3 7 9 11 13 17 19 21 23 27 $$1$$ $$1$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$1$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{13}{16}\right)$$
value at  e.g. 2

## Related number fields

 Field of values $$\Q(\zeta_{16})$$

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 320 }(27,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{320}(27,\cdot)) = \sum_{r\in \Z/320\Z} \chi_{320}(27,r) e\left(\frac{r}{160}\right) = -0.0$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 320 }(27,·),\chi_{ 320 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{320}(27,\cdot),\chi_{320}(1,\cdot)) = \sum_{r\in \Z/320\Z} \chi_{320}(27,r) \chi_{320}(1,1-r) = 0$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 320 }(27,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{320}(27,·)) = \sum_{r \in \Z/320\Z} \chi_{320}(27,r) e\left(\frac{1 r + 2 r^{-1}}{320}\right) = -1.3036340443+13.236018575i$$