Properties

 Label 405.248 Modulus $405$ Conductor $405$ Order $108$ Real no Primitive yes Minimal yes Parity even

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

sage: H = DirichletGroup(405, base_ring=CyclotomicField(108))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([46,81]))

pari: [g,chi] = znchar(Mod(248,405))

Basic properties

 Modulus: $$405$$ Conductor: $$405$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$108$$ 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 405.x

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: $\Q(\zeta_{108})$ Fixed field: Number field defined by a degree 108 polynomial (not computed)

Values on generators

$$(326,82)$$ → $$(e\left(\frac{23}{54}\right),-i)$$

Values

 $$a$$ $$-1$$ $$1$$ $$2$$ $$4$$ $$7$$ $$8$$ $$11$$ $$13$$ $$14$$ $$16$$ $$17$$ $$19$$ $$\chi_{ 405 }(248, a)$$ $$1$$ $$1$$ $$e\left(\frac{19}{108}\right)$$ $$e\left(\frac{19}{54}\right)$$ $$e\left(\frac{61}{108}\right)$$ $$e\left(\frac{19}{36}\right)$$ $$e\left(\frac{29}{54}\right)$$ $$e\left(\frac{71}{108}\right)$$ $$e\left(\frac{20}{27}\right)$$ $$e\left(\frac{19}{27}\right)$$ $$e\left(\frac{29}{36}\right)$$ $$e\left(\frac{17}{18}\right)$$
sage: chi.jacobi_sum(n)

$$\chi_{ 405 }(248,a) \;$$ at $$\;a =$$ e.g. 2

Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 405 }(248,·) )\;$$ at $$\;a =$$ e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 405 }(248,·),\chi_{ 405 }(n,·)) \;$$ for $$\; n =$$ e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 405 }(248,·)) \;$$ at $$\; a,b =$$ e.g. 1,2