# Properties

 Label 392.205 Modulus $392$ Conductor $392$ Order $42$ Real no Primitive yes Minimal yes Parity even

# Related objects

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(392, base_ring=CyclotomicField(42))

M = H._module

chi = DirichletCharacter(H, M([0,21,2]))

pari: [g,chi] = znchar(Mod(205,392))

## Basic properties

 Modulus: $$392$$ Conductor: $$392$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$42$$ 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 392.z

sage: chi.galois_orbit()

order = charorder(g,chi)

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

## Values on generators

$$(295,197,297)$$ → $$(1,-1,e\left(\frac{1}{21}\right))$$

## First values

 $$a$$ $$-1$$ $$1$$ $$3$$ $$5$$ $$9$$ $$11$$ $$13$$ $$15$$ $$17$$ $$19$$ $$23$$ $$25$$ $$\chi_{ 392 }(205, a)$$ $$1$$ $$1$$ $$e\left(\frac{23}{42}\right)$$ $$e\left(\frac{37}{42}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{17}{42}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{16}{21}\right)$$
sage: chi.jacobi_sum(n)

$$\chi_{ 392 }(205,a) \;$$ at $$\;a =$$ e.g. 2

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 392 }(205,·) )\;$$ at $$\;a =$$ e.g. 2

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 392 }(205,·),\chi_{ 392 }(n,·)) \;$$ for $$\; n =$$ e.g. 1

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 392 }(205,·)) \;$$ at $$\; a,b =$$ e.g. 1,2