# Properties

 Label 648.23 Modulus $648$ Conductor $324$ Order $54$ Real no Primitive no Minimal no Parity even

# Related objects

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

sage: H = DirichletGroup(648, base_ring=CyclotomicField(54))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([27,0,11]))

pari: [g,chi] = znchar(Mod(23,648))

## Basic properties

 Modulus: $$648$$ Conductor: $$324$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$54$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{324}(23,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: no Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 648.be

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

$$(487,325,569)$$ → $$(-1,1,e\left(\frac{11}{54}\right))$$

## Values

 $$-1$$ $$1$$ $$5$$ $$7$$ $$11$$ $$13$$ $$17$$ $$19$$ $$23$$ $$25$$ $$29$$ $$31$$ $$1$$ $$1$$ $$e\left(\frac{37}{54}\right)$$ $$e\left(\frac{41}{54}\right)$$ $$e\left(\frac{4}{27}\right)$$ $$e\left(\frac{17}{27}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{20}{27}\right)$$ $$e\left(\frac{10}{27}\right)$$ $$e\left(\frac{29}{54}\right)$$ $$e\left(\frac{31}{54}\right)$$
 value at e.g. 2

## Related number fields

 Field of values: $$\Q(\zeta_{27})$$ Fixed field: Number field defined by a degree 54 polynomial

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 648 }(23,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{648}(23,\cdot)) = \sum_{r\in \Z/648\Z} \chi_{648}(23,r) e\left(\frac{r}{324}\right) = -17.3920535747+-31.5200963269i$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 648 }(23,·),\chi_{ 648 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{648}(23,\cdot),\chi_{648}(1,\cdot)) = \sum_{r\in \Z/648\Z} \chi_{648}(23,r) \chi_{648}(1,1-r) = 0$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 648 }(23,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{648}(23,·)) = \sum_{r \in \Z/648\Z} \chi_{648}(23,r) e\left(\frac{1 r + 2 r^{-1}}{648}\right) = -0.0$$