Properties

 Conductor 400 Order 20 Real no Primitive yes Minimal yes Parity odd Orbit label 400.bn

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(400)

sage: chi = H[13]

pari: [g,chi] = znchar(Mod(13,400))

Basic properties

 sage: chi.conductor()  pari: znconreyconductor(g,chi) Conductor = 400 sage: chi.multiplicative_order()  pari: charorder(g,chi) Order = 20 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 = 400.bn Orbit index = 40

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

$$(351,101,177)$$ → $$(1,-i,e\left(\frac{19}{20}\right))$$

Values

 -1 1 3 7 9 11 13 17 19 21 23 27 $$-1$$ $$1$$ $$e\left(\frac{9}{10}\right)$$ $$i$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{7}{10}\right)$$
value at  e.g. 2

Related number fields

 Field of values $$\Q(\zeta_{20})$$

Gauss sum

sage: chi.sage_character().gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 400 }(13,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{400}(13,\cdot)) = \sum_{r\in \Z/400\Z} \chi_{400}(13,r) e\left(\frac{r}{200}\right) = -0.0$$

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)

$$J(\chi_{ 400 }(13,·),\chi_{ 400 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{400}(13,\cdot),\chi_{400}(1,\cdot)) = \sum_{r\in \Z/400\Z} \chi_{400}(13,r) \chi_{400}(1,1-r) = 0$$

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)

$$K(a,b,\chi_{ 400 }(13,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{400}(13,·)) = \sum_{r \in \Z/400\Z} \chi_{400}(13,r) e\left(\frac{1 r + 2 r^{-1}}{400}\right) = -0.0$$