# Properties

 Label 261.47 Modulus $261$ Conductor $261$ Order $84$ 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(261, base_ring=CyclotomicField(84))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([14,33]))

pari: [g,chi] = znchar(Mod(47,261))

## Basic properties

 Modulus: $$261$$ Conductor: $$261$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$84$$ 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 261.x

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

$$(146,118)$$ → $$(e\left(\frac{1}{6}\right),e\left(\frac{11}{28}\right))$$

## Values

 $$-1$$ $$1$$ $$2$$ $$4$$ $$5$$ $$7$$ $$8$$ $$10$$ $$11$$ $$13$$ $$14$$ $$16$$ $$1$$ $$1$$ $$e\left(\frac{47}{84}\right)$$ $$e\left(\frac{5}{42}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{19}{28}\right)$$ $$e\left(\frac{1}{28}\right)$$ $$e\left(\frac{83}{84}\right)$$ $$e\left(\frac{17}{42}\right)$$ $$e\left(\frac{79}{84}\right)$$ $$e\left(\frac{5}{21}\right)$$
 value at e.g. 2

## Related number fields

 Field of values: $\Q(\zeta_{84})$ Fixed field: Number field defined by a degree 84 polynomial

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 261 }(47,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{261}(47,\cdot)) = \sum_{r\in \Z/261\Z} \chi_{261}(47,r) e\left(\frac{2r}{261}\right) = 15.050946346+5.8710317739i$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 261 }(47,·),\chi_{ 261 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{261}(47,\cdot),\chi_{261}(1,\cdot)) = \sum_{r\in \Z/261\Z} \chi_{261}(47,r) \chi_{261}(1,1-r) = 0$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 261 }(47,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{261}(47,·)) = \sum_{r \in \Z/261\Z} \chi_{261}(47,r) e\left(\frac{1 r + 2 r^{-1}}{261}\right) = -3.296131517+17.420474006i$$