Properties

 Label 837.17 Modulus $837$ Conductor $279$ Order $30$ Real no Primitive no Minimal no Parity even

Related objects

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

sage: H = DirichletGroup(837, base_ring=CyclotomicField(30))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([25,7]))

pari: [g,chi] = znchar(Mod(17,837))

Basic properties

 Modulus: $$837$$ Conductor: $$279$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$30$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{279}(203,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: no Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

Galois orbit 837.bt

sage: chi.galois_orbit()

pari: order = charorder(g,chi)

pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]

Related number fields

 Field of values: $$\Q(\zeta_{15})$$ Fixed field: 30.30.52474170681294939938779612124082598308717540593134452231209594253.2

Values on generators

$$(218,406)$$ → $$(e\left(\frac{5}{6}\right),e\left(\frac{7}{30}\right))$$

Values

 $$-1$$ $$1$$ $$2$$ $$4$$ $$5$$ $$7$$ $$8$$ $$10$$ $$11$$ $$13$$ $$14$$ $$16$$ $$1$$ $$1$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{11}{15}\right)$$
 value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 837 }(17,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{837}(17,\cdot)) = \sum_{r\in \Z/837\Z} \chi_{837}(17,r) e\left(\frac{2r}{837}\right) = 0.0$$

Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 837 }(17,·),\chi_{ 837 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{837}(17,\cdot),\chi_{837}(1,\cdot)) = \sum_{r\in \Z/837\Z} \chi_{837}(17,r) \chi_{837}(1,1-r) = 0$$

Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 837 }(17,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{837}(17,·)) = \sum_{r \in \Z/837\Z} \chi_{837}(17,r) e\left(\frac{1 r + 2 r^{-1}}{837}\right) = -0.0$$