# Properties

 Label 429.25 Modulus $429$ Conductor $143$ Order $10$ Real no Primitive no 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(10))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([0,8,5]))

pari: [g,chi] = znchar(Mod(25,429))

## Basic properties

 Modulus: $$429$$ Conductor: $$143$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$10$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{143}(25,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 429.bb

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

$$(287,79,67)$$ → $$(1,e\left(\frac{4}{5}\right),-1)$$

## Values

 $$-1$$ $$1$$ $$2$$ $$4$$ $$5$$ $$7$$ $$8$$ $$10$$ $$14$$ $$16$$ $$17$$ $$19$$ $$1$$ $$1$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$1$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{9}{10}\right)$$
 value at e.g. 2

## Related number fields

 Field of values: $$\Q(\zeta_{5})$$ Fixed field: 10.10.79589952003133.1

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 429 }(25,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{429}(25,\cdot)) = \sum_{r\in \Z/429\Z} \chi_{429}(25,r) e\left(\frac{2r}{429}\right) = 3.4239066438+-11.4576115877i$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 429 }(25,·),\chi_{ 429 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{429}(25,\cdot),\chi_{429}(1,\cdot)) = \sum_{r\in \Z/429\Z} \chi_{429}(25,r) \chi_{429}(1,1-r) = 1$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 429 }(25,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{429}(25,·)) = \sum_{r \in \Z/429\Z} \chi_{429}(25,r) e\left(\frac{1 r + 2 r^{-1}}{429}\right) = -0.0$$