# Properties

 Label 429.428 Modulus $429$ Conductor $429$ Order $2$ Real yes Primitive yes Minimal yes Parity even

# Related objects

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

sage: H = DirichletGroup(429, base_ring=CyclotomicField(2))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([1,1,1]))

pari: [g,chi] = znchar(Mod(428,429))

## Kronecker symbol representation

sage: kronecker_character(429)

pari: znchartokronecker(g,chi)

$$\displaystyle\left(\frac{429}{\bullet}\right)$$

## Basic properties

 Modulus: $$429$$ Conductor: $$429$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$2$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: yes 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 429.e

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

$$(287,79,67)$$ → $$(-1,-1,-1)$$

## Values

 $$-1$$ $$1$$ $$2$$ $$4$$ $$5$$ $$7$$ $$8$$ $$10$$ $$14$$ $$16$$ $$17$$ $$19$$ $$1$$ $$1$$ $$-1$$ $$1$$ $$1$$ $$1$$ $$-1$$ $$-1$$ $$-1$$ $$1$$ $$1$$ $$1$$
 value at e.g. 2

## Related number fields

 Field of values: $$\Q$$ Fixed field: $$\Q(\sqrt{429})$$

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 429 }(428,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{429}(428,\cdot)) = \sum_{r\in \Z/429\Z} \chi_{429}(428,r) e\left(\frac{2r}{429}\right) = -20.7123151772$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 429 }(428,·),\chi_{ 429 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{429}(428,\cdot),\chi_{429}(1,\cdot)) = \sum_{r\in \Z/429\Z} \chi_{429}(428,r) \chi_{429}(1,1-r) = -1$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 429 }(428,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{429}(428,·)) = \sum_{r \in \Z/429\Z} \chi_{429}(428,r) e\left(\frac{1 r + 2 r^{-1}}{429}\right) = 0.0$$