# Properties

 Label 667.244 Modulus $667$ Conductor $667$ Order $44$ Real no Primitive yes Minimal yes Parity even

# Learn more

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

sage: H = DirichletGroup(667, base_ring=CyclotomicField(44))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([42,11]))

pari: [g,chi] = znchar(Mod(244,667))

## Basic properties

 Modulus: $$667$$ Conductor: $$667$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$44$$ 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 667.q

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

$$(465,553)$$ → $$(e\left(\frac{21}{22}\right),i)$$

## Values

 $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$ $$8$$ $$9$$ $$10$$ $$11$$ $$1$$ $$1$$ $$e\left(\frac{7}{44}\right)$$ $$e\left(\frac{23}{44}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{21}{44}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{27}{44}\right)$$ $$e\left(\frac{37}{44}\right)$$
 value at e.g. 2

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 667 }(244,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{667}(244,\cdot)) = \sum_{r\in \Z/667\Z} \chi_{667}(244,r) e\left(\frac{2r}{667}\right) = 25.4415399785+4.441626225i$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 667 }(244,·),\chi_{ 667 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{667}(244,\cdot),\chi_{667}(1,\cdot)) = \sum_{r\in \Z/667\Z} \chi_{667}(244,r) \chi_{667}(1,1-r) = 1$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 667 }(244,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{667}(244,·)) = \sum_{r \in \Z/667\Z} \chi_{667}(244,r) e\left(\frac{1 r + 2 r^{-1}}{667}\right) = 13.6616486722+7.459824561i$$