# Properties

 Label 547.27 Modulus $547$ Conductor $547$ Order $14$ Real no Primitive yes Minimal yes Parity odd

# Learn more about

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

sage: H = DirichletGroup(547)

sage: M = H._module

sage: chi = DirichletCharacter(H, M([5]))

pari: [g,chi] = znchar(Mod(27,547))

## Basic properties

 Modulus: $$547$$ Conductor: $$547$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$14$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: yes sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: odd sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 547.g

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

$$2$$ → $$e\left(\frac{5}{14}\right)$$

## Values

 $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$ $$8$$ $$9$$ $$10$$ $$11$$ $$-1$$ $$1$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$1$$
 value at e.g. 2

## Related number fields

 Field of values: $$\Q(\zeta_{7})$$ Fixed field: 14.0.392496006513281583932595840500243827.1

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 547 }(27,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{547}(27,\cdot)) = \sum_{r\in \Z/547\Z} \chi_{547}(27,r) e\left(\frac{2r}{547}\right) = 22.7597328296+5.3846598338i$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 547 }(27,·),\chi_{ 547 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{547}(27,\cdot),\chi_{547}(1,\cdot)) = \sum_{r\in \Z/547\Z} \chi_{547}(27,r) \chi_{547}(1,1-r) = -1$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 547 }(27,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{547}(27,·)) = \sum_{r \in \Z/547\Z} \chi_{547}(27,r) e\left(\frac{1 r + 2 r^{-1}}{547}\right) = 2.8226281343+-1.3593060678i$$