Properties

 Conductor 71 Order 70 Real No Primitive Yes Parity Odd Orbit Label 71.h

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(71)
sage: chi = H
pari: [g,chi] = znchar(Mod(33,71))

Basic properties

 sage: chi.conductor() pari: znconreyconductor(g,chi) Conductor = 71 sage: chi.multiplicative_order() pari: charorder(g,chi) Order = 70 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 = 71.h Orbit index = 8

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

$$7$$ → $$e\left(\frac{57}{70}\right)$$

Values

 -1 1 2 3 4 5 6 7 8 9 10 11 $$-1$$ $$1$$ $$e\left(\frac{31}{35}\right)$$ $$e\left(\frac{6}{35}\right)$$ $$e\left(\frac{27}{35}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{2}{35}\right)$$ $$e\left(\frac{57}{70}\right)$$ $$e\left(\frac{23}{35}\right)$$ $$e\left(\frac{12}{35}\right)$$ $$e\left(\frac{24}{35}\right)$$ $$e\left(\frac{17}{70}\right)$$
value at  e.g. 2

Related number fields

 Field of values $$\Q(\zeta_{35})$$

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
$$\tau_{ a }( \chi_{ 71 }(33,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{71}(33,\cdot)) = \sum_{r\in \Z/71\Z} \chi_{71}(33,r) e\left(\frac{2r}{71}\right) = 3.9313624879+7.4528108113i$$

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
$$J(\chi_{ 71 }(33,·),\chi_{ 71 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{71}(33,\cdot),\chi_{71}(1,\cdot)) = \sum_{r\in \Z/71\Z} \chi_{71}(33,r) \chi_{71}(1,1-r) = -1$$

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
$$K(a,b,\chi_{ 71 }(33,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{71}(33,·)) = \sum_{r \in \Z/71\Z} \chi_{71}(33,r) e\left(\frac{1 r + 2 r^{-1}}{71}\right) = -2.8585794824+-7.6166577925i$$