# Properties

 Label 68.3 Modulus $68$ Conductor $68$ Order $16$ 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(68)

sage: M = H._module

sage: chi = DirichletCharacter(H, M([8,1]))

pari: [g,chi] = znchar(Mod(3,68))

## Basic properties

 Modulus: $$68$$ Conductor: $$68$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$16$$ 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 68.i

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

$$(35,37)$$ → $$(-1,e\left(\frac{1}{16}\right))$$

## Values

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

## Related number fields

 Field of values: $$\Q(\zeta_{16})$$ Fixed field: $$\Q(\zeta_{68})^+$$

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 68 }(3,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{68}(3,\cdot)) = \sum_{r\in \Z/68\Z} \chi_{68}(3,r) e\left(\frac{r}{34}\right) = -0.0$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 68 }(3,·),\chi_{ 68 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{68}(3,\cdot),\chi_{68}(1,\cdot)) = \sum_{r\in \Z/68\Z} \chi_{68}(3,r) \chi_{68}(1,1-r) = 0$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 68 }(3,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{68}(3,·)) = \sum_{r \in \Z/68\Z} \chi_{68}(3,r) e\left(\frac{1 r + 2 r^{-1}}{68}\right) = 9.8144240156+-4.0652675341i$$