# Properties

 Label 211.203 Modulus $211$ Conductor $211$ Order $35$ 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(211, base_ring=CyclotomicField(70))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([36]))

pari: [g,chi] = znchar(Mod(203,211))

## Basic properties

 Modulus: $$211$$ Conductor: $$211$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$35$$ 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 211.l

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: $\Q(\zeta_{35})$ Fixed field: 35.35.10607266494966666158512469468409065269845427817856998775062284008873509080731241.1

## Values on generators

$$2$$ → $$e\left(\frac{18}{35}\right)$$

## Values

 $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$ $$8$$ $$9$$ $$10$$ $$11$$ $$1$$ $$1$$ $$e\left(\frac{18}{35}\right)$$ $$e\left(\frac{4}{35}\right)$$ $$e\left(\frac{1}{35}\right)$$ $$e\left(\frac{31}{35}\right)$$ $$e\left(\frac{22}{35}\right)$$ $$e\left(\frac{17}{35}\right)$$ $$e\left(\frac{19}{35}\right)$$ $$e\left(\frac{8}{35}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{11}{35}\right)$$
 value at e.g. 2

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 211 }(203,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{211}(203,\cdot)) = \sum_{r\in \Z/211\Z} \chi_{211}(203,r) e\left(\frac{2r}{211}\right) = -9.412851114+-11.0633735318i$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 211 }(203,·),\chi_{ 211 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{211}(203,\cdot),\chi_{211}(1,\cdot)) = \sum_{r\in \Z/211\Z} \chi_{211}(203,r) \chi_{211}(1,1-r) = -1$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 211 }(203,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{211}(203,·)) = \sum_{r \in \Z/211\Z} \chi_{211}(203,r) e\left(\frac{1 r + 2 r^{-1}}{211}\right) = -0.2546649561+5.6705558319i$$