# Properties

 Label 784.55 Modulus $784$ Conductor $392$ Order $14$ 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(784, base_ring=CyclotomicField(14))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([7,7,9]))

pari: [g,chi] = znchar(Mod(55,784))

## Basic properties

 Modulus: $$784$$ Conductor: $$392$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$14$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{392}(251,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: no Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 784.bc

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

$$(687,197,689)$$ → $$(-1,-1,e\left(\frac{9}{14}\right))$$

## Values

 $$-1$$ $$1$$ $$3$$ $$5$$ $$9$$ $$11$$ $$13$$ $$15$$ $$17$$ $$19$$ $$23$$ $$25$$ $$1$$ $$1$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$-1$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{2}{7}\right)$$
 value at e.g. 2

## Related number fields

 Field of values: $$\Q(\zeta_{7})$$ Fixed field: 14.14.2812424737865523319657201664.1

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 784 }(55,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{784}(55,\cdot)) = \sum_{r\in \Z/784\Z} \chi_{784}(55,r) e\left(\frac{r}{392}\right) = 27.5476235488+-28.445183016i$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 784 }(55,·),\chi_{ 784 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{784}(55,\cdot),\chi_{784}(1,\cdot)) = \sum_{r\in \Z/784\Z} \chi_{784}(55,r) \chi_{784}(1,1-r) = 0$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 784 }(55,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{784}(55,·)) = \sum_{r \in \Z/784\Z} \chi_{784}(55,r) e\left(\frac{1 r + 2 r^{-1}}{784}\right) = 0.0$$