# Properties

 Label 469.156 Modulus $469$ Conductor $469$ Order $33$ 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(469, base_ring=CyclotomicField(66))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([22,60]))

pari: [g,chi] = znchar(Mod(156,469))

## Basic properties

 Modulus: $$469$$ Conductor: $$469$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$33$$ 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 469.ba

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

$$(269,337)$$ → $$(e\left(\frac{1}{3}\right),e\left(\frac{10}{11}\right))$$

## Values

 $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$8$$ $$9$$ $$10$$ $$11$$ $$12$$ $$1$$ $$1$$ $$e\left(\frac{19}{33}\right)$$ $$e\left(\frac{26}{33}\right)$$ $$e\left(\frac{5}{33}\right)$$ $$e\left(\frac{10}{33}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{19}{33}\right)$$ $$e\left(\frac{29}{33}\right)$$ $$e\left(\frac{32}{33}\right)$$ $$e\left(\frac{31}{33}\right)$$
 value at e.g. 2

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 469 }(156,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{469}(156,\cdot)) = \sum_{r\in \Z/469\Z} \chi_{469}(156,r) e\left(\frac{2r}{469}\right) = -13.3525587095+17.0501957733i$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 469 }(156,·),\chi_{ 469 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{469}(156,\cdot),\chi_{469}(1,\cdot)) = \sum_{r\in \Z/469\Z} \chi_{469}(156,r) \chi_{469}(1,1-r) = 1$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 469 }(156,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{469}(156,·)) = \sum_{r \in \Z/469\Z} \chi_{469}(156,r) e\left(\frac{1 r + 2 r^{-1}}{469}\right) = 4.1175043509+-16.9725841024i$$