# Properties

 Label 287.218 Modulus $287$ Conductor $41$ Order $40$ Real no Primitive no Minimal yes Parity odd

# Learn more

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

sage: H = DirichletGroup(287, base_ring=CyclotomicField(40))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([0,31]))

pari: [g,chi] = znchar(Mod(218,287))

## Basic properties

 Modulus: $$287$$ Conductor: $$41$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$40$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{41}(13,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: odd sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 287.ba

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: $$\Q(\zeta_{40})$$ Fixed field: $$\Q(\zeta_{41})$$

## Values on generators

$$(206,211)$$ → $$(1,e\left(\frac{31}{40}\right))$$

## Values

 $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$8$$ $$9$$ $$10$$ $$11$$ $$12$$ $$-1$$ $$1$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{31}{40}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$i$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{13}{40}\right)$$ $$e\left(\frac{37}{40}\right)$$
sage: chi.jacobi_sum(n)

$$\chi_{ 287 }(218,a) \;$$ at $$\;a =$$ e.g. 2

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 287 }(218,·) )\;$$ at $$\;a =$$ e.g. 2

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 287 }(218,·),\chi_{ 287 }(n,·)) \;$$ for $$\; n =$$ e.g. 1

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 287 }(218,·)) \;$$ at $$\; a,b =$$ e.g. 1,2