# Properties

 Label 547.29 Modulus $547$ Conductor $547$ Order $91$ 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(547)

sage: M = H._module

sage: chi = DirichletCharacter(H, M([53]))

pari: [g,chi] = znchar(Mod(29,547))

## Basic properties

 Modulus: $$547$$ Conductor: $$547$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$91$$ 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.m

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

$$2$$ → $$e\left(\frac{53}{91}\right)$$

## Values

 $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$ $$8$$ $$9$$ $$10$$ $$11$$ $$1$$ $$1$$ $$e\left(\frac{53}{91}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{15}{91}\right)$$ $$e\left(\frac{23}{91}\right)$$ $$e\left(\frac{40}{91}\right)$$ $$e\left(\frac{90}{91}\right)$$ $$e\left(\frac{68}{91}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{76}{91}\right)$$ $$e\left(\frac{10}{13}\right)$$
 value at e.g. 2

## Related number fields

 Field of values: $\Q(\zeta_{91})$ Fixed field: Number field defined by a degree 91 polynomial

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 547 }(29,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{547}(29,\cdot)) = \sum_{r\in \Z/547\Z} \chi_{547}(29,r) e\left(\frac{2r}{547}\right) = 12.4915688446+19.7727263623i$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 547 }(29,·),\chi_{ 547 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{547}(29,\cdot),\chi_{547}(1,\cdot)) = \sum_{r\in \Z/547\Z} \chi_{547}(29,r) \chi_{547}(1,1-r) = -1$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 547 }(29,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{547}(29,·)) = \sum_{r \in \Z/547\Z} \chi_{547}(29,r) e\left(\frac{1 r + 2 r^{-1}}{547}\right) = 2.5885830232+-9.7731026379i$$