# Properties

 Label 328.245 Modulus $328$ Conductor $328$ Order $2$ Real yes Primitive yes Minimal yes Parity even

# Related objects

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

sage: H = DirichletGroup(328, base_ring=CyclotomicField(2))

sage: M = H._module

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

pari: [g,chi] = znchar(Mod(245,328))

## Kronecker symbol representation

sage: kronecker_character(328)

pari: znchartokronecker(g,chi)

$$\displaystyle\left(\frac{328}{\bullet}\right)$$

## Basic properties

 Modulus: $$328$$ Conductor: $$328$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$2$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: yes 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 328.g

sage: chi.galois_orbit()

pari: order = charorder(g,chi)

pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]

## Related number fields

 Field of values: $$\Q$$ Fixed field: $$\Q(\sqrt{82})$$

## Values on generators

$$(247,165,129)$$ → $$(1,-1,-1)$$

## Values

 $$-1$$ $$1$$ $$3$$ $$5$$ $$7$$ $$9$$ $$11$$ $$13$$ $$15$$ $$17$$ $$19$$ $$21$$ $$1$$ $$1$$ $$1$$ $$-1$$ $$-1$$ $$1$$ $$1$$ $$1$$ $$-1$$ $$-1$$ $$1$$ $$-1$$
 value at e.g. 2

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 328 }(245,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{328}(245,\cdot)) = \sum_{r\in \Z/328\Z} \chi_{328}(245,r) e\left(\frac{r}{164}\right) = 0.0$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 328 }(245,·),\chi_{ 328 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{328}(245,\cdot),\chi_{328}(1,\cdot)) = \sum_{r\in \Z/328\Z} \chi_{328}(245,r) \chi_{328}(1,1-r) = 0$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 328 }(245,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{328}(245,·)) = \sum_{r \in \Z/328\Z} \chi_{328}(245,r) e\left(\frac{1 r + 2 r^{-1}}{328}\right) = -21.9584183073$$