Properties

 Label 784.37 Modulus $784$ Conductor $784$ Order $84$ Real no Primitive yes Minimal yes Parity even

Related objects

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

sage: H = DirichletGroup(784, base_ring=CyclotomicField(84))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([0,21,64]))

pari: [g,chi] = znchar(Mod(37,784))

Basic properties

 Modulus: $$784$$ Conductor: $$784$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$84$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: yes sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

Galois orbit 784.bt

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

$$(687,197,689)$$ → $$(1,i,e\left(\frac{16}{21}\right))$$

Values

 $$-1$$ $$1$$ $$3$$ $$5$$ $$9$$ $$11$$ $$13$$ $$15$$ $$17$$ $$19$$ $$23$$ $$25$$ $$1$$ $$1$$ $$e\left(\frac{43}{84}\right)$$ $$e\left(\frac{29}{84}\right)$$ $$e\left(\frac{1}{42}\right)$$ $$e\left(\frac{61}{84}\right)$$ $$e\left(\frac{25}{28}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{19}{42}\right)$$ $$e\left(\frac{29}{42}\right)$$
 value at e.g. 2

Related number fields

 Field of values: $\Q(\zeta_{84})$ Fixed field: Number field defined by a degree 84 polynomial

Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 784 }(37,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{784}(37,\cdot)) = \sum_{r\in \Z/784\Z} \chi_{784}(37,r) e\left(\frac{r}{392}\right) = 0.0$$

Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 784 }(37,·),\chi_{ 784 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{784}(37,\cdot),\chi_{784}(1,\cdot)) = \sum_{r\in \Z/784\Z} \chi_{784}(37,r) \chi_{784}(1,1-r) = 0$$

Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 784 }(37,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{784}(37,·)) = \sum_{r \in \Z/784\Z} \chi_{784}(37,r) e\left(\frac{1 r + 2 r^{-1}}{784}\right) = -8.464095805+-5.5417636576i$$