# Properties

 Label 507.2 Modulus $507$ Conductor $507$ Order $156$ 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(507, base_ring=CyclotomicField(156))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([78,1]))

pari: [g,chi] = znchar(Mod(2,507))

## Basic properties

 Modulus: $$507$$ Conductor: $$507$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$156$$ 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 507.x

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: $\Q(\zeta_{156})$ Fixed field: Number field defined by a degree 156 polynomial (not computed)

## Values on generators

$$(170,340)$$ → $$(-1,e\left(\frac{1}{156}\right))$$

## Values

 $$-1$$ $$1$$ $$2$$ $$4$$ $$5$$ $$7$$ $$8$$ $$10$$ $$11$$ $$14$$ $$16$$ $$17$$ $$1$$ $$1$$ $$e\left(\frac{79}{156}\right)$$ $$e\left(\frac{1}{78}\right)$$ $$e\left(\frac{29}{52}\right)$$ $$e\left(\frac{107}{156}\right)$$ $$e\left(\frac{27}{52}\right)$$ $$e\left(\frac{5}{78}\right)$$ $$e\left(\frac{25}{156}\right)$$ $$e\left(\frac{5}{26}\right)$$ $$e\left(\frac{1}{39}\right)$$ $$e\left(\frac{17}{39}\right)$$
 value at e.g. 2

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 507 }(2,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{507}(2,\cdot)) = \sum_{r\in \Z/507\Z} \chi_{507}(2,r) e\left(\frac{2r}{507}\right) = -20.0656575505+-10.2161336652i$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 507 }(2,·),\chi_{ 507 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{507}(2,\cdot),\chi_{507}(1,\cdot)) = \sum_{r\in \Z/507\Z} \chi_{507}(2,r) \chi_{507}(1,1-r) = 0$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 507 }(2,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{507}(2,·)) = \sum_{r \in \Z/507\Z} \chi_{507}(2,r) e\left(\frac{1 r + 2 r^{-1}}{507}\right) = -0.0$$