# Properties

 Label 690.31 Modulus $690$ Conductor $23$ Order $11$ Real no Primitive no Minimal yes Parity even

# Related objects

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

sage: H = DirichletGroup(690, base_ring=CyclotomicField(22))

sage: M = H._module

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

pari: [g,chi] = znchar(Mod(31,690))

## Basic properties

 Modulus: $$690$$ Conductor: $$23$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$11$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{23}(8,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 690.m

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

$$(461,277,511)$$ → $$(1,1,e\left(\frac{3}{11}\right))$$

## Values

 $$-1$$ $$1$$ $$7$$ $$11$$ $$13$$ $$17$$ $$19$$ $$29$$ $$31$$ $$37$$ $$41$$ $$43$$ $$1$$ $$1$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{4}{11}\right)$$
 value at e.g. 2

## Related number fields

 Field of values: $$\Q(\zeta_{11})$$ Fixed field: $$\Q(\zeta_{23})^+$$

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 690 }(31,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{690}(31,\cdot)) = \sum_{r\in \Z/690\Z} \chi_{690}(31,r) e\left(\frac{r}{345}\right) = 2.1513119095+-4.2862404351i$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 690 }(31,·),\chi_{ 690 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{690}(31,\cdot),\chi_{690}(1,\cdot)) = \sum_{r\in \Z/690\Z} \chi_{690}(31,r) \chi_{690}(1,1-r) = 0$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 690 }(31,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{690}(31,·)) = \sum_{r \in \Z/690\Z} \chi_{690}(31,r) e\left(\frac{1 r + 2 r^{-1}}{690}\right) = 0.5945101353+-4.1349088151i$$