# Properties

 Label 210.73 Modulus $210$ Conductor $35$ Order $12$ 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(210)

sage: M = H._module

sage: chi = DirichletCharacter(H, M([0,9,2]))

pari: [g,chi] = znchar(Mod(73,210))

## Basic properties

 Modulus: $$210$$ Conductor: $$35$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$12$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{35}(3,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 210.u

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

$$(71,127,31)$$ → $$(1,-i,e\left(\frac{1}{6}\right))$$

## Values

 $$-1$$ $$1$$ $$11$$ $$13$$ $$17$$ $$19$$ $$23$$ $$29$$ $$31$$ $$37$$ $$41$$ $$43$$ $$1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$-i$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$-1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$-1$$ $$i$$
 value at e.g. 2

## Related number fields

 Field of values: $$\Q(\zeta_{12})$$ Fixed field: $$\Q(\zeta_{35})^+$$

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 210 }(73,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{210}(73,\cdot)) = \sum_{r\in \Z/210\Z} \chi_{210}(73,r) e\left(\frac{r}{105}\right) = 4.8136853347+3.439248973i$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 210 }(73,·),\chi_{ 210 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{210}(73,\cdot),\chi_{210}(1,\cdot)) = \sum_{r\in \Z/210\Z} \chi_{210}(73,r) \chi_{210}(1,1-r) = 0$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 210 }(73,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{210}(73,·)) = \sum_{r \in \Z/210\Z} \chi_{210}(73,r) e\left(\frac{1 r + 2 r^{-1}}{210}\right) = 5.623255447+1.5067467559i$$