# Properties

 Label 709.4 Modulus $709$ Conductor $709$ Order $354$ Real no Primitive yes Minimal yes Parity even

# Learn more about

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

sage: H = DirichletGroup(709)

sage: M = H._module

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

pari: [g,chi] = znchar(Mod(4,709))

## Basic properties

 Modulus: $$709$$ Conductor: $$709$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$354$$ 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 709.k

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

$$2$$ → $$e\left(\frac{1}{354}\right)$$

## Values

 $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$ $$8$$ $$9$$ $$10$$ $$11$$ $$1$$ $$1$$ $$e\left(\frac{1}{354}\right)$$ $$e\left(\frac{86}{177}\right)$$ $$e\left(\frac{1}{177}\right)$$ $$e\left(\frac{161}{177}\right)$$ $$e\left(\frac{173}{354}\right)$$ $$e\left(\frac{26}{177}\right)$$ $$e\left(\frac{1}{118}\right)$$ $$e\left(\frac{172}{177}\right)$$ $$e\left(\frac{323}{354}\right)$$ $$e\left(\frac{158}{177}\right)$$
 value at e.g. 2

## Related number fields

 Field of values: $\Q(\zeta_{177})$ Fixed field: Number field defined by a degree 354 polynomial

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 709 }(4,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{709}(4,\cdot)) = \sum_{r\in \Z/709\Z} \chi_{709}(4,r) e\left(\frac{2r}{709}\right) = 26.0619288437+-5.4567265778i$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 709 }(4,·),\chi_{ 709 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{709}(4,\cdot),\chi_{709}(1,\cdot)) = \sum_{r\in \Z/709\Z} \chi_{709}(4,r) \chi_{709}(1,1-r) = -1$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 709 }(4,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{709}(4,·)) = \sum_{r \in \Z/709\Z} \chi_{709}(4,r) e\left(\frac{1 r + 2 r^{-1}}{709}\right) = 21.225504929+0.1883718672i$$