Properties

 Modulus 320 Conductor 64 Order 16 Real no Primitive no Minimal yes Parity even Orbit label 320.be

Learn more about

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

sage: H = DirichletGroup(320)

sage: M = H._module

sage: chi = DirichletCharacter(H, M([0,15,0]))

pari: [g,chi] = znchar(Mod(141,320))

Basic properties

 sage: chi.conductor()  pari: znconreyconductor(g,chi) Modulus = 320 Conductor = 64 sage: chi.multiplicative_order()  pari: charorder(g,chi) Order = 16 Real = no sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization] Primitive = no Minimal = yes sage: chi.is_odd()  pari: zncharisodd(g,chi) Parity = even Orbit label = 320.be Orbit index = 31

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

$$(191,261,257)$$ → $$(1,e\left(\frac{15}{16}\right),1)$$

Values

 -1 1 3 7 9 11 13 17 19 21 23 27 $$1$$ $$1$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$i$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{7}{16}\right)$$
value at  e.g. 2

Related number fields

 Field of values $$\Q(\zeta_{16})$$

Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 320 }(141,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{320}(141,\cdot)) = \sum_{r\in \Z/320\Z} \chi_{320}(141,r) e\left(\frac{r}{160}\right) = -0.0$$

Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 320 }(141,·),\chi_{ 320 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{320}(141,\cdot),\chi_{320}(1,\cdot)) = \sum_{r\in \Z/320\Z} \chi_{320}(141,r) \chi_{320}(1,1-r) = 0$$

Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 320 }(141,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{320}(141,·)) = \sum_{r \in \Z/320\Z} \chi_{320}(141,r) e\left(\frac{1 r + 2 r^{-1}}{320}\right) = 20.0121149952+16.4235183362i$$