# Properties

 Label 216.215 Modulus $216$ Conductor $12$ Order $2$ Real yes Primitive no Minimal no Parity even

# Related objects

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

sage: H = DirichletGroup(216)

sage: M = H._module

sage: chi = DirichletCharacter(H, M([1,0,1]))

pari: [g,chi] = znchar(Mod(215,216))

## Basic properties

 Modulus: $$216$$ Conductor: $$12$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$2$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: yes Primitive: no, induced from $$\chi_{12}(11,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: no Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 216.c

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

$$(55,109,137)$$ → $$(-1,1,-1)$$

## Values

 $$-1$$ $$1$$ $$5$$ $$7$$ $$11$$ $$13$$ $$17$$ $$19$$ $$23$$ $$25$$ $$29$$ $$31$$ $$1$$ $$1$$ $$-1$$ $$-1$$ $$1$$ $$1$$ $$-1$$ $$-1$$ $$1$$ $$1$$ $$-1$$ $$-1$$
 value at e.g. 2

## Related number fields

 Field of values: $$\Q$$ Fixed field: $$\Q(\sqrt{3})$$

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 216 }(215,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{216}(215,\cdot)) = \sum_{r\in \Z/216\Z} \chi_{216}(215,r) e\left(\frac{r}{108}\right) = -0.0$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 216 }(215,·),\chi_{ 216 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{216}(215,\cdot),\chi_{216}(1,\cdot)) = \sum_{r\in \Z/216\Z} \chi_{216}(215,r) \chi_{216}(1,1-r) = 0$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 216 }(215,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{216}(215,·)) = \sum_{r \in \Z/216\Z} \chi_{216}(215,r) e\left(\frac{1 r + 2 r^{-1}}{216}\right) = -0.0$$