# Properties

 Label 966.61 Modulus $966$ Conductor $161$ Order $66$ Real no Primitive no Minimal yes Parity even

# Learn more

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

H = DirichletGroup(966, base_ring=CyclotomicField(66))

M = H._module

chi = DirichletCharacter(H, M([0,55,51]))

pari: [g,chi] = znchar(Mod(61,966))

## Basic properties

 Modulus: $$966$$ Conductor: $$161$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$66$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{161}(61,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 966.be

sage: chi.galois_orbit()

order = charorder(g,chi)

[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]

## Related number fields

 Field of values: $$\Q(\zeta_{33})$$ Fixed field: Number field defined by a degree 66 polynomial

## Values on generators

$$(323,829,925)$$ → $$(1,e\left(\frac{5}{6}\right),e\left(\frac{17}{22}\right))$$

## First values

 $$a$$ $$-1$$ $$1$$ $$5$$ $$11$$ $$13$$ $$17$$ $$19$$ $$25$$ $$29$$ $$31$$ $$37$$ $$41$$ $$\chi_{ 966 }(61, a)$$ $$1$$ $$1$$ $$e\left(\frac{31}{33}\right)$$ $$e\left(\frac{19}{66}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{8}{33}\right)$$ $$e\left(\frac{25}{33}\right)$$ $$e\left(\frac{29}{33}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{31}{66}\right)$$ $$e\left(\frac{59}{66}\right)$$ $$e\left(\frac{17}{22}\right)$$
sage: chi.jacobi_sum(n)

$$\chi_{ 966 }(61,a) \;$$ at $$\;a =$$ e.g. 2

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 966 }(61,·) )\;$$ at $$\;a =$$ e.g. 2

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 966 }(61,·),\chi_{ 966 }(n,·)) \;$$ for $$\; n =$$ e.g. 1

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 966 }(61,·)) \;$$ at $$\; a,b =$$ e.g. 1,2