# Properties

 Label 966.25 Modulus $966$ Conductor $161$ Order $33$ Real no Primitive no Minimal yes Parity even

# Learn more about

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

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

sage: M = H._module

sage: chi = DirichletCharacter(H, M([0,44,6]))

pari: [g,chi] = znchar(Mod(25,966))

## Basic properties

 Modulus: $$966$$ Conductor: $$161$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$33$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{161}(25,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 966.y

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

$$(323,829,925)$$ → $$(1,e\left(\frac{2}{3}\right),e\left(\frac{1}{11}\right))$$

## Values

 $$-1$$ $$1$$ $$5$$ $$11$$ $$13$$ $$17$$ $$19$$ $$25$$ $$29$$ $$31$$ $$37$$ $$41$$ $$1$$ $$1$$ $$e\left(\frac{14}{33}\right)$$ $$e\left(\frac{16}{33}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{10}{33}\right)$$ $$e\left(\frac{23}{33}\right)$$ $$e\left(\frac{28}{33}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{7}{33}\right)$$ $$e\left(\frac{8}{33}\right)$$ $$e\left(\frac{1}{11}\right)$$
 value at e.g. 2

## Related number fields

 Field of values: $$\Q(\zeta_{33})$$ Fixed field: 33.33.277966181338944111003326058293667039541136678070715028736001.1

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 966 }(25,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{966}(25,\cdot)) = \sum_{r\in \Z/966\Z} \chi_{966}(25,r) e\left(\frac{r}{483}\right) = 12.6570660577+-0.8936883188i$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 966 }(25,·),\chi_{ 966 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{966}(25,\cdot),\chi_{966}(1,\cdot)) = \sum_{r\in \Z/966\Z} \chi_{966}(25,r) \chi_{966}(1,1-r) = 0$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 966 }(25,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{966}(25,·)) = \sum_{r \in \Z/966\Z} \chi_{966}(25,r) e\left(\frac{1 r + 2 r^{-1}}{966}\right) = -0.2002903336+4.2046115453i$$