# Properties

 Label 105.41 Modulus $105$ Conductor $21$ Order $2$ Real yes Primitive no Minimal yes Parity even

# Learn more about

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

sage: H = DirichletGroup(105, base_ring=CyclotomicField(2))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([1,0,1]))

pari: [g,chi] = znchar(Mod(41,105))

## Basic properties

 Modulus: $$105$$ Conductor: $$21$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$2$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: yes Primitive: no, induced from $$\chi_{21}(20,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 105.b

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: $$\Q$$ Fixed field: $$\Q(\sqrt{21})$$

## Values on generators

$$(71,22,31)$$ → $$(-1,1,-1)$$

## Values

 $$-1$$ $$1$$ $$2$$ $$4$$ $$8$$ $$11$$ $$13$$ $$16$$ $$17$$ $$19$$ $$22$$ $$23$$ $$1$$ $$1$$ $$-1$$ $$1$$ $$-1$$ $$-1$$ $$-1$$ $$1$$ $$1$$ $$-1$$ $$1$$ $$-1$$
 value at e.g. 2

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 105 }(41,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{105}(41,\cdot)) = \sum_{r\in \Z/105\Z} \chi_{105}(41,r) e\left(\frac{2r}{105}\right) = 4.582575695$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 105 }(41,·),\chi_{ 105 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{105}(41,\cdot),\chi_{105}(1,\cdot)) = \sum_{r\in \Z/105\Z} \chi_{105}(41,r) \chi_{105}(1,1-r) = 3$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 105 }(41,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{105}(41,·)) = \sum_{r \in \Z/105\Z} \chi_{105}(41,r) e\left(\frac{1 r + 2 r^{-1}}{105}\right) = -0.0$$