# Properties

 Label 546.71 Modulus $546$ Conductor $39$ 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(546, base_ring=CyclotomicField(12))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([6,0,5]))

pari: [g,chi] = znchar(Mod(71,546))

## Basic properties

 Modulus: $$546$$ Conductor: $$39$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$12$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{39}(32,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 546.bu

sage: chi.galois_orbit()

pari: order = charorder(g,chi)

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

## Related number fields

 Field of values: $$\Q(\zeta_{12})$$ Fixed field: $$\Q(\zeta_{39})^+$$

## Values on generators

$$(365,157,379)$$ → $$(-1,1,e\left(\frac{5}{12}\right))$$

## Values

 $$-1$$ $$1$$ $$5$$ $$11$$ $$17$$ $$19$$ $$23$$ $$25$$ $$29$$ $$31$$ $$37$$ $$41$$ $$1$$ $$1$$ $$i$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$-1$$ $$e\left(\frac{1}{6}\right)$$ $$-i$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{11}{12}\right)$$
 value at e.g. 2

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 546 }(71,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{546}(71,\cdot)) = \sum_{r\in \Z/546\Z} \chi_{546}(71,r) e\left(\frac{r}{273}\right) = 6.2403428627+-0.2410828828i$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 546 }(71,·),\chi_{ 546 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{546}(71,\cdot),\chi_{546}(1,\cdot)) = \sum_{r\in \Z/546\Z} \chi_{546}(71,r) \chi_{546}(1,1-r) = 0$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 546 }(71,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{546}(71,·)) = \sum_{r \in \Z/546\Z} \chi_{546}(71,r) e\left(\frac{1 r + 2 r^{-1}}{546}\right) = 0.0$$