# Properties

 Label 392.361 Modulus $392$ Conductor $7$ Order $3$ Real no Primitive no Minimal no Parity even

# Related objects

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

sage: H = DirichletGroup(392, base_ring=CyclotomicField(6))

sage: M = H._module

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

pari: [g,chi] = znchar(Mod(361,392))

## Basic properties

 Modulus: $$392$$ Conductor: $$7$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$3$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{7}(4,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: no Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 392.i

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: $$\Q(\sqrt{-3})$$ Fixed field: $$\Q(\zeta_{7})^+$$

## Values on generators

$$(295,197,297)$$ → $$(1,1,e\left(\frac{2}{3}\right))$$

## Values

 $$-1$$ $$1$$ $$3$$ $$5$$ $$9$$ $$11$$ $$13$$ $$15$$ $$17$$ $$19$$ $$23$$ $$25$$ $$1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$
 value at e.g. 2

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 392 }(361,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{392}(361,\cdot)) = \sum_{r\in \Z/392\Z} \chi_{392}(361,r) e\left(\frac{r}{196}\right) = 0.0$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 392 }(361,·),\chi_{ 392 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{392}(361,\cdot),\chi_{392}(1,\cdot)) = \sum_{r\in \Z/392\Z} \chi_{392}(361,r) \chi_{392}(1,1-r) = 0$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 392 }(361,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{392}(361,·)) = \sum_{r \in \Z/392\Z} \chi_{392}(361,r) e\left(\frac{1 r + 2 r^{-1}}{392}\right) = 0.0$$