# Properties

 Label 13.4 Modulus $13$ Conductor $13$ Order $6$ Real no Primitive yes Minimal yes Parity even

# Related objects

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

sage: H = DirichletGroup(13, base_ring=CyclotomicField(6))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([1]))

pari: [g,chi] = znchar(Mod(4,13))

## Basic properties

 Modulus: $$13$$ Conductor: $$13$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$6$$ 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 13.e

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

$$2$$ → $$e\left(\frac{1}{6}\right)$$

## Values

 $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$ $$8$$ $$9$$ $$10$$ $$11$$ $$1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$-1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$-1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$
 value at e.g. 2

## Related number fields

 Field of values: $$\Q(\sqrt{-3})$$ Fixed field: $$\Q(\zeta_{13})^+$$

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 13 }(4,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{13}(4,\cdot)) = \sum_{r\in \Z/13\Z} \chi_{13}(4,r) e\left(\frac{2r}{13}\right) = 3.1453598372+-1.7625865921i$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 13 }(4,·),\chi_{ 13 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{13}(4,\cdot),\chi_{13}(1,\cdot)) = \sum_{r\in \Z/13\Z} \chi_{13}(4,r) \chi_{13}(1,1-r) = -1$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 13 }(4,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{13}(4,·)) = \sum_{r \in \Z/13\Z} \chi_{13}(4,r) e\left(\frac{1 r + 2 r^{-1}}{13}\right) = 2.6071422853+1.5052343002i$$