# Properties

 Label 67.26 Modulus $67$ Conductor $67$ Order $33$ 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(67)

sage: M = H._module

sage: chi = DirichletCharacter(H, M([10]))

pari: [g,chi] = znchar(Mod(26,67))

## Basic properties

 Modulus: $$67$$ Conductor: $$67$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$33$$ 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 67.g

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

$$2$$ → $$e\left(\frac{10}{33}\right)$$

## Values

 $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$ $$8$$ $$9$$ $$10$$ $$11$$ $$1$$ $$1$$ $$e\left(\frac{10}{33}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{20}{33}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{4}{33}\right)$$ $$e\left(\frac{32}{33}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{28}{33}\right)$$ $$e\left(\frac{29}{33}\right)$$
 value at e.g. 2

## Related number fields

 Field of values: $$\Q(\zeta_{33})$$ Fixed field: Number field defined by a degree %d polynomial (not computed)

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 67 }(26,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{67}(26,\cdot)) = \sum_{r\in \Z/67\Z} \chi_{67}(26,r) e\left(\frac{2r}{67}\right) = -5.9776700487+-5.5917314661i$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 67 }(26,·),\chi_{ 67 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{67}(26,\cdot),\chi_{67}(1,\cdot)) = \sum_{r\in \Z/67\Z} \chi_{67}(26,r) \chi_{67}(1,1-r) = -1$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 67 }(26,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{67}(26,·)) = \sum_{r \in \Z/67\Z} \chi_{67}(26,r) e\left(\frac{1 r + 2 r^{-1}}{67}\right) = 8.3920418484+11.784973794i$$