# Properties

 Label 764.639 Modulus $764$ Conductor $764$ Order $38$ Real no Primitive yes Minimal yes Parity even

# Related objects

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

sage: H = DirichletGroup(764, base_ring=CyclotomicField(38))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([19,11]))

pari: [g,chi] = znchar(Mod(639,764))

## Basic properties

 Modulus: $$764$$ Conductor: $$764$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$38$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: yes sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 764.l

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

$$(383,401)$$ → $$(-1,e\left(\frac{11}{38}\right))$$

## Values

 $$-1$$ $$1$$ $$3$$ $$5$$ $$7$$ $$9$$ $$11$$ $$13$$ $$15$$ $$17$$ $$19$$ $$21$$ $$1$$ $$1$$ $$e\left(\frac{3}{38}\right)$$ $$e\left(\frac{9}{19}\right)$$ $$1$$ $$e\left(\frac{3}{19}\right)$$ $$e\left(\frac{2}{19}\right)$$ $$e\left(\frac{8}{19}\right)$$ $$e\left(\frac{21}{38}\right)$$ $$e\left(\frac{7}{19}\right)$$ $$e\left(\frac{15}{19}\right)$$ $$e\left(\frac{3}{38}\right)$$
 value at e.g. 2

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 764 }(639,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{764}(639,\cdot)) = \sum_{r\in \Z/764\Z} \chi_{764}(639,r) e\left(\frac{r}{382}\right) = 0.0$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 764 }(639,·),\chi_{ 764 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{764}(639,\cdot),\chi_{764}(1,\cdot)) = \sum_{r\in \Z/764\Z} \chi_{764}(639,r) \chi_{764}(1,1-r) = 0$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 764 }(639,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{764}(639,·)) = \sum_{r \in \Z/764\Z} \chi_{764}(639,r) e\left(\frac{1 r + 2 r^{-1}}{764}\right) = -12.0692877642+13.1107414772i$$