Properties

 Label 416.179 Modulus $416$ Conductor $416$ Order $24$ Real no Primitive yes Minimal yes Parity odd

Related objects

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

sage: H = DirichletGroup(416, base_ring=CyclotomicField(24))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([12,21,20]))

pari: [g,chi] = znchar(Mod(179,416))

Basic properties

 Modulus: $$416$$ Conductor: $$416$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$24$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: yes sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: odd sage: chi.is_odd()  pari: zncharisodd(g,chi)

Galois orbit 416.cb

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

$$(287,261,353)$$ → $$(-1,e\left(\frac{7}{8}\right),e\left(\frac{5}{6}\right))$$

Values

 $$-1$$ $$1$$ $$3$$ $$5$$ $$7$$ $$9$$ $$11$$ $$15$$ $$17$$ $$19$$ $$21$$ $$23$$ $$-1$$ $$1$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{17}{24}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{19}{24}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{1}{12}\right)$$
 value at e.g. 2

Related number fields

 Field of values: $$\Q(\zeta_{24})$$ Fixed field: 24.0.188216044816745326913150945765287080795084676923392.1

Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 416 }(179,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{416}(179,\cdot)) = \sum_{r\in \Z/416\Z} \chi_{416}(179,r) e\left(\frac{r}{208}\right) = 0.0$$

Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 416 }(179,·),\chi_{ 416 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{416}(179,\cdot),\chi_{416}(1,\cdot)) = \sum_{r\in \Z/416\Z} \chi_{416}(179,r) \chi_{416}(1,1-r) = 0$$

Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 416 }(179,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{416}(179,·)) = \sum_{r \in \Z/416\Z} \chi_{416}(179,r) e\left(\frac{1 r + 2 r^{-1}}{416}\right) = 21.3427286066+1.3988763388i$$