# Properties

 Label 407.34 Modulus $407$ Conductor $37$ Order $9$ 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(407, base_ring=CyclotomicField(18))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([0,4]))

pari: [g,chi] = znchar(Mod(34,407))

## Basic properties

 Modulus: $$407$$ Conductor: $$37$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$9$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{37}(34,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 407.l

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: $$\Q(\zeta_{9})$$ Fixed field: 9.9.3512479453921.1

## Values on generators

$$(112,298)$$ → $$(1,e\left(\frac{2}{9}\right))$$

## Values

 $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$ $$8$$ $$9$$ $$10$$ $$12$$ $$1$$ $$1$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$1$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{9}\right)$$
 value at e.g. 2

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 407 }(34,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{407}(34,\cdot)) = \sum_{r\in \Z/407\Z} \chi_{407}(34,r) e\left(\frac{2r}{407}\right) = 5.5662079377+2.4530244994i$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 407 }(34,·),\chi_{ 407 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{407}(34,\cdot),\chi_{407}(1,\cdot)) = \sum_{r\in \Z/407\Z} \chi_{407}(34,r) \chi_{407}(1,1-r) = -9$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 407 }(34,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{407}(34,·)) = \sum_{r \in \Z/407\Z} \chi_{407}(34,r) e\left(\frac{1 r + 2 r^{-1}}{407}\right) = 0.0256808926+0.0215488275i$$