# Properties

 Label 547.66 Modulus $547$ Conductor $547$ Order $273$ 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(547, base_ring=CyclotomicField(546))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([500]))

pari: [g,chi] = znchar(Mod(66,547))

## Basic properties

 Modulus: $$547$$ Conductor: $$547$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$273$$ 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 547.o

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

$$2$$ → $$e\left(\frac{250}{273}\right)$$

## Values

 $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$ $$8$$ $$9$$ $$10$$ $$11$$ $$1$$ $$1$$ $$e\left(\frac{250}{273}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{227}{273}\right)$$ $$e\left(\frac{251}{273}\right)$$ $$e\left(\frac{211}{273}\right)$$ $$e\left(\frac{88}{273}\right)$$ $$e\left(\frac{68}{91}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{76}{91}\right)$$ $$e\left(\frac{4}{39}\right)$$
 value at e.g. 2

## Related number fields

 Field of values: $\Q(\zeta_{273})$ Fixed field: Number field defined by a degree 273 polynomial (not computed)

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 547 }(66,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{547}(66,\cdot)) = \sum_{r\in \Z/547\Z} \chi_{547}(66,r) e\left(\frac{2r}{547}\right) = 6.9710005239+-22.3249893997i$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 547 }(66,·),\chi_{ 547 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{547}(66,\cdot),\chi_{547}(1,\cdot)) = \sum_{r\in \Z/547\Z} \chi_{547}(66,r) \chi_{547}(1,1-r) = -1$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 547 }(66,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{547}(66,·)) = \sum_{r \in \Z/547\Z} \chi_{547}(66,r) e\left(\frac{1 r + 2 r^{-1}}{547}\right) = -6.8655029475+1.8607919304i$$