Properties

 Conductor 61 Order 60 Real No Primitive Yes Parity Odd Orbit Label 61.l

Related objects

Show commands for: SageMath / Pari/GP
sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
sage: H = DirichletGroup_conrey(61)
sage: chi = H[51]
pari: [g,chi] = znchar(Mod(51,61))

Basic properties

 sage: chi.conductor() pari: znconreyconductor(g,chi) 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 sage: chi.is_odd() pari: zncharisodd(g,chi) Parity = Odd Orbit label = 61.l Orbit index = 12

Galois orbit

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