# Properties

 Label 421.27 Modulus $421$ Conductor $421$ Order $35$ Real no Primitive yes Minimal yes Parity even

# Learn more about

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

sage: H = DirichletGroup(421, base_ring=CyclotomicField(70))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([62]))

pari: [g,chi] = znchar(Mod(27,421))

## Basic properties

 Modulus: $$421$$ Conductor: $$421$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$35$$ 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 421.p

sage: chi.galois_orbit()

pari: order = charorder(g,chi)

pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]

## Related number fields

 Field of values: $\Q(\zeta_{35})$ Fixed field: 35.35.168109113671617086350535469888345045006842268262463039332288137975520602419531747373726681.1

## Values on generators

$$2$$ → $$e\left(\frac{31}{35}\right)$$

## Values

 $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$ $$8$$ $$9$$ $$10$$ $$11$$ $$1$$ $$1$$ $$e\left(\frac{31}{35}\right)$$ $$e\left(\frac{29}{35}\right)$$ $$e\left(\frac{27}{35}\right)$$ $$e\left(\frac{8}{35}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{6}{35}\right)$$ $$e\left(\frac{23}{35}\right)$$ $$e\left(\frac{23}{35}\right)$$ $$e\left(\frac{4}{35}\right)$$ $$e\left(\frac{1}{35}\right)$$
 value at e.g. 2

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 421 }(27,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{421}(27,\cdot)) = \sum_{r\in \Z/421\Z} \chi_{421}(27,r) e\left(\frac{2r}{421}\right) = 14.2957823731+-14.7183764846i$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 421 }(27,·),\chi_{ 421 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{421}(27,\cdot),\chi_{421}(1,\cdot)) = \sum_{r\in \Z/421\Z} \chi_{421}(27,r) \chi_{421}(1,1-r) = -1$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 421 }(27,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{421}(27,·)) = \sum_{r \in \Z/421\Z} \chi_{421}(27,r) e\left(\frac{1 r + 2 r^{-1}}{421}\right) = 10.4709381998+-3.9298088368i$$