# Properties

 Label 344.169 Modulus $344$ Conductor $43$ Order $21$ Real no Primitive no Minimal yes Parity even

# Related objects

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

sage: H = DirichletGroup(344)

sage: M = H._module

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

pari: [g,chi] = znchar(Mod(169,344))

## Basic properties

 Modulus: $$344$$ Conductor: $$43$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$21$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{43}(40,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 344.y

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

$$(87,173,89)$$ → $$(1,1,e\left(\frac{11}{21}\right))$$

## Values

 $$-1$$ $$1$$ $$3$$ $$5$$ $$7$$ $$9$$ $$11$$ $$13$$ $$15$$ $$17$$ $$19$$ $$21$$ $$1$$ $$1$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{6}{7}\right)$$
 value at e.g. 2

## Related number fields

 Field of values: $$\Q(\zeta_{21})$$ Fixed field: $$\Q(\zeta_{43})^+$$

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 344 }(169,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{344}(169,\cdot)) = \sum_{r\in \Z/344\Z} \chi_{344}(169,r) e\left(\frac{r}{172}\right) = -0.0$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 344 }(169,·),\chi_{ 344 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{344}(169,\cdot),\chi_{344}(1,\cdot)) = \sum_{r\in \Z/344\Z} \chi_{344}(169,r) \chi_{344}(1,1-r) = 0$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 344 }(169,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{344}(169,·)) = \sum_{r \in \Z/344\Z} \chi_{344}(169,r) e\left(\frac{1 r + 2 r^{-1}}{344}\right) = 0.0$$