# Properties

 Label 10.7 Modulus $10$ Conductor $5$ Order $4$ Real no Primitive no Minimal yes Parity odd

# Related objects

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

sage: H = DirichletGroup(10, base_ring=CyclotomicField(4))

sage: M = H._module

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

pari: [g,chi] = znchar(Mod(7,10))

## Basic properties

 Modulus: $$10$$ Conductor: $$5$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$4$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{5}(2,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: odd sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 10.c

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

$$7$$ → $$i$$

## Values

 $$-1$$ $$1$$ $$3$$ $$7$$ $$-1$$ $$1$$ $$-i$$ $$i$$
 value at e.g. 2

## Related number fields

 Field of values: $$\Q(\sqrt{-1})$$ Fixed field: $$\Q(\zeta_{5})$$

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 10 }(7,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{10}(7,\cdot)) = \sum_{r\in \Z/10\Z} \chi_{10}(7,r) e\left(\frac{r}{5}\right) = -1.1755705046+1.9021130326i$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 10 }(7,·),\chi_{ 10 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{10}(7,\cdot),\chi_{10}(1,\cdot)) = \sum_{r\in \Z/10\Z} \chi_{10}(7,r) \chi_{10}(1,1-r) = 0$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 10 }(7,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{10}(7,·)) = \sum_{r \in \Z/10\Z} \chi_{10}(7,r) e\left(\frac{1 r + 2 r^{-1}}{10}\right) = -1.9021130326+1.9021130326i$$