# Properties

 Label 517.157 Modulus $517$ Conductor $517$ Order $115$ Real no Primitive yes Minimal yes Parity even

# Related objects

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

sage: H = DirichletGroup(517, base_ring=CyclotomicField(230))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([184,130]))

pari: [g,chi] = znchar(Mod(157,517))

## Basic properties

 Modulus: $$517$$ Conductor: $$517$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$115$$ 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 517.m

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: $\Q(\zeta_{115})$ Fixed field: Number field defined by a degree 115 polynomial (not computed)

## Values on generators

$$(189,287)$$ → $$(e\left(\frac{4}{5}\right),e\left(\frac{13}{23}\right))$$

## Values

 $$a$$ $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$ $$8$$ $$9$$ $$10$$ $$12$$ $$\chi_{ 517 }(157, a)$$ $$1$$ $$1$$ $$e\left(\frac{112}{115}\right)$$ $$e\left(\frac{81}{115}\right)$$ $$e\left(\frac{109}{115}\right)$$ $$e\left(\frac{88}{115}\right)$$ $$e\left(\frac{78}{115}\right)$$ $$e\left(\frac{79}{115}\right)$$ $$e\left(\frac{106}{115}\right)$$ $$e\left(\frac{47}{115}\right)$$ $$e\left(\frac{17}{23}\right)$$ $$e\left(\frac{15}{23}\right)$$
sage: chi.jacobi_sum(n)

$$\chi_{ 517 }(157,a) \;$$ at $$\;a =$$ e.g. 2

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 517 }(157,·) )\;$$ at $$\;a =$$ e.g. 2

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 517 }(157,·),\chi_{ 517 }(n,·)) \;$$ for $$\; n =$$ e.g. 1

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 517 }(157,·)) \;$$ at $$\; a,b =$$ e.g. 1,2