# Properties

 Label 2.1 Modulus $2$ Conductor $1$ Order $1$ Real yes Primitive no Minimal yes Parity even

# Related objects

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

sage: H = DirichletGroup(2, base_ring=CyclotomicField(1))

sage: M = H._module

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

pari: [g,chi] = znchar(Mod(1,2))

## Basic properties

 Modulus: $$2$$ Conductor: $$1$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$1$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: yes Primitive: no, induced from $$\chi_{1}(1,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 2.a

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: $$\Q$$ Fixed field: $$\Q$$

## Values

 $$-1$$ $$1$$
 value at e.g. 2

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 2 }(1,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{2}(1,\cdot)) = \sum_{r\in \Z/2\Z} \chi_{2}(1,r) e\left(\frac{r}{1}\right) = 1.0$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 2 }(1,·),\chi_{ 2 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{2}(1,\cdot),\chi_{2}(1,\cdot)) = \sum_{r\in \Z/2\Z} \chi_{2}(1,r) \chi_{2}(1,1-r) = 0$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 2 }(1,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{2}(1,·)) = \sum_{r \in \Z/2\Z} \chi_{2}(1,r) e\left(\frac{1 r + 2 r^{-1}}{2}\right) = -1.0$$