# Properties

 Label 648.13 Modulus $648$ Conductor $648$ Order $54$ Real no Primitive yes Minimal yes 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([0,27,8]))

pari: [g,chi] = znchar(Mod(13,648))

## Basic properties

 Modulus: $$648$$ Conductor: $$648$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$54$$ 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 648.bd

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

$$(487,325,569)$$ → $$(1,-1,e\left(\frac{4}{27}\right))$$

## Values

 $$-1$$ $$1$$ $$5$$ $$7$$ $$11$$ $$13$$ $$17$$ $$19$$ $$23$$ $$25$$ $$29$$ $$31$$ $$1$$ $$1$$ $$e\left(\frac{49}{54}\right)$$ $$e\left(\frac{10}{27}\right)$$ $$e\left(\frac{23}{54}\right)$$ $$e\left(\frac{37}{54}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{17}{27}\right)$$ $$e\left(\frac{22}{27}\right)$$ $$e\left(\frac{53}{54}\right)$$ $$e\left(\frac{26}{27}\right)$$
 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)

$$\tau_{ a }( \chi_{ 648 }(13,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{648}(13,\cdot)) = \sum_{r\in \Z/648\Z} \chi_{648}(13,r) e\left(\frac{r}{324}\right) = 0.0$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 648 }(13,·),\chi_{ 648 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{648}(13,\cdot),\chi_{648}(1,\cdot)) = \sum_{r\in \Z/648\Z} \chi_{648}(13,r) \chi_{648}(1,1-r) = 0$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 648 }(13,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{648}(13,·)) = \sum_{r \in \Z/648\Z} \chi_{648}(13,r) e\left(\frac{1 r + 2 r^{-1}}{648}\right) = -0.0$$