# Properties

 Modulus 61 Conductor 61 Order 60 Real no Primitive yes Minimal yes Parity odd Orbit label 61.l

# Learn more about

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

sage: H = DirichletGroup(61)

sage: M = H._module

sage: chi = DirichletCharacter(H, M([53]))

pari: [g,chi] = znchar(Mod(51,61))

## Basic properties

 sage: chi.conductor()  pari: znconreyconductor(g,chi) Modulus = 61 Conductor = 61 sage: chi.multiplicative_order()  pari: charorder(g,chi) Order = 60 Real = no sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization] Primitive = yes Minimal = yes sage: chi.is_odd()  pari: zncharisodd(g,chi) Parity = odd Orbit label = 61.l Orbit index = 12

## Galois orbit

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

$$2$$ → $$e\left(\frac{53}{60}\right)$$

## Values

 -1 1 2 3 4 5 6 7 8 9 10 11 $$-1$$ $$1$$ $$e\left(\frac{53}{60}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{11}{60}\right)$$ $$e\left(\frac{17}{60}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{19}{60}\right)$$ $$i$$
value at  e.g. 2

## Related number fields

 Field of values $$\Q(\zeta_{60})$$

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 61 }(51,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{61}(51,\cdot)) = \sum_{r\in \Z/61\Z} \chi_{61}(51,r) e\left(\frac{2r}{61}\right) = -3.3746873492+-7.0435420986i$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 61 }(51,·),\chi_{ 61 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{61}(51,\cdot),\chi_{61}(1,\cdot)) = \sum_{r\in \Z/61\Z} \chi_{61}(51,r) \chi_{61}(1,1-r) = -1$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 61 }(51,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{61}(51,·)) = \sum_{r \in \Z/61\Z} \chi_{61}(51,r) e\left(\frac{1 r + 2 r^{-1}}{61}\right) = 3.4387920001+8.9583594352i$$