# Properties

 Label 15.11 Modulus $15$ Conductor $3$ Order $2$ Real yes Primitive no Minimal yes Parity odd

# Learn more about

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

sage: H = DirichletGroup(15, base_ring=CyclotomicField(2))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([1,0]))

pari: [g,chi] = znchar(Mod(11,15))

## Basic properties

 Modulus: $$15$$ Conductor: $$3$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$2$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: yes Primitive: no, induced from $$\chi_{3}(2,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: odd sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 15.c

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

$$(11,7)$$ → $$(-1,1)$$

## Values

 $$-1$$ $$1$$ $$2$$ $$4$$ $$7$$ $$8$$ $$11$$ $$13$$ $$-1$$ $$1$$ $$-1$$ $$1$$ $$1$$ $$-1$$ $$-1$$ $$1$$
 value at e.g. 2

## Related number fields

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

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 15 }(11,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{15}(11,\cdot)) = \sum_{r\in \Z/15\Z} \chi_{15}(11,r) e\left(\frac{2r}{15}\right) = -1.7320508076i$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 15 }(11,·),\chi_{ 15 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{15}(11,\cdot),\chi_{15}(1,\cdot)) = \sum_{r\in \Z/15\Z} \chi_{15}(11,r) \chi_{15}(1,1-r) = -3$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 15 }(11,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{15}(11,·)) = \sum_{r \in \Z/15\Z} \chi_{15}(11,r) e\left(\frac{1 r + 2 r^{-1}}{15}\right) = 0.0$$