Properties

 Label 9.4 Modulus $9$ Conductor $9$ Order $3$ 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(9)

sage: M = H._module

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

pari: [g,chi] = znchar(Mod(4,9))

Basic properties

 Modulus: $$9$$ Conductor: $$9$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$3$$ 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 9.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

$$2$$ → $$e\left(\frac{1}{3}\right)$$

Values

 $$-1$$ $$1$$ $$2$$ $$4$$ $$5$$ $$7$$ $$1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$
 value at e.g. 2

Related number fields

 Field of values: $$\Q(\sqrt{-3})$$ Fixed field: $$\Q(\zeta_{9})^+$$

Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 9 }(4,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{9}(4,\cdot)) = \sum_{r\in \Z/9\Z} \chi_{9}(4,r) e\left(\frac{2r}{9}\right) = 0.520944533+-2.954423259i$$

Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 9 }(4,·),\chi_{ 9 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{9}(4,\cdot),\chi_{9}(1,\cdot)) = \sum_{r\in \Z/9\Z} \chi_{9}(4,r) \chi_{9}(1,1-r) = 0$$

Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 9 }(4,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{9}(4,·)) = \sum_{r \in \Z/9\Z} \chi_{9}(4,r) e\left(\frac{1 r + 2 r^{-1}}{9}\right) = -1.5+-2.5980762114i$$