# Properties

 Label 196.135 Modulus $196$ Conductor $196$ Order $42$ Real no Primitive yes Minimal yes Parity odd

# Related objects

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

sage: H = DirichletGroup(196, base_ring=CyclotomicField(42))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([21,32]))

pari: [g,chi] = znchar(Mod(135,196))

## Basic properties

 Modulus: $$196$$ Conductor: $$196$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$42$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: yes sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: odd sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 196.o

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

$$(99,101)$$ → $$(-1,e\left(\frac{16}{21}\right))$$

## Values

 $$-1$$ $$1$$ $$3$$ $$5$$ $$9$$ $$11$$ $$13$$ $$15$$ $$17$$ $$19$$ $$23$$ $$25$$ $$-1$$ $$1$$ $$e\left(\frac{11}{42}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{41}{42}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{19}{42}\right)$$ $$e\left(\frac{4}{21}\right)$$
 value at e.g. 2

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 196 }(135,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{196}(135,\cdot)) = \sum_{r\in \Z/196\Z} \chi_{196}(135,r) e\left(\frac{r}{98}\right) = 0.0$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 196 }(135,·),\chi_{ 196 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{196}(135,\cdot),\chi_{196}(1,\cdot)) = \sum_{r\in \Z/196\Z} \chi_{196}(135,r) \chi_{196}(1,1-r) = 0$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 196 }(135,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{196}(135,·)) = \sum_{r \in \Z/196\Z} \chi_{196}(135,r) e\left(\frac{1 r + 2 r^{-1}}{196}\right) = -0.0$$