# Properties

 Label 65.48 Modulus $65$ Conductor $65$ Order $12$ Real no Primitive yes Minimal yes Parity odd

# Related objects

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

sage: H = DirichletGroup(65, base_ring=CyclotomicField(12))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([9,8]))

pari: [g,chi] = znchar(Mod(48,65))

## Basic properties

 Modulus: $$65$$ Conductor: $$65$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$12$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: yes sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: odd sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 65.q

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: $$\Q(\zeta_{12})$$ Fixed field: 12.0.1593224064453125.1

## Values on generators

$$(27,41)$$ → $$(-i,e\left(\frac{2}{3}\right))$$

## Values

 $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$6$$ $$7$$ $$8$$ $$9$$ $$11$$ $$12$$ $$14$$ $$-1$$ $$1$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$i$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$-i$$ $$-1$$
 value at e.g. 2

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 65 }(48,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{65}(48,\cdot)) = \sum_{r\in \Z/65\Z} \chi_{65}(48,r) e\left(\frac{2r}{65}\right) = 1.8019785066+-7.8582996546i$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 65 }(48,·),\chi_{ 65 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{65}(48,\cdot),\chi_{65}(1,\cdot)) = \sum_{r\in \Z/65\Z} \chi_{65}(48,r) \chi_{65}(1,1-r) = 1$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 65 }(48,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{65}(48,·)) = \sum_{r \in \Z/65\Z} \chi_{65}(48,r) e\left(\frac{1 r + 2 r^{-1}}{65}\right) = 6.3006884211+-1.6882643742i$$