# Properties

 Label 243.217 Modulus $243$ Conductor $27$ Order $9$ Real no Primitive no Minimal no Parity even

# Related objects

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

sage: H = DirichletGroup(243, base_ring=CyclotomicField(18))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([10]))

pari: [g,chi] = znchar(Mod(217,243))

## Basic properties

 Modulus: $$243$$ Conductor: $$27$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$9$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{27}(25,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: no Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 243.e

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

$$2$$ → $$e\left(\frac{5}{9}\right)$$

## Values

 $$-1$$ $$1$$ $$2$$ $$4$$ $$5$$ $$7$$ $$8$$ $$10$$ $$11$$ $$13$$ $$14$$ $$16$$ $$1$$ $$1$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$
 value at e.g. 2

## Related number fields

 Field of values: $$\Q(\zeta_{9})$$ Fixed field: $$\Q(\zeta_{27})^+$$

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 243 }(217,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{243}(217,\cdot)) = \sum_{r\in \Z/243\Z} \chi_{243}(217,r) e\left(\frac{2r}{243}\right) = 0.0$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 243 }(217,·),\chi_{ 243 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{243}(217,\cdot),\chi_{243}(1,\cdot)) = \sum_{r\in \Z/243\Z} \chi_{243}(217,r) \chi_{243}(1,1-r) = 0$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 243 }(217,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{243}(217,·)) = \sum_{r \in \Z/243\Z} \chi_{243}(217,r) e\left(\frac{1 r + 2 r^{-1}}{243}\right) = 0.0$$