Properties

 Label 405.197 Modulus $405$ Conductor $135$ Order $36$ Real no Primitive no Minimal no Parity even

Related objects

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

sage: H = DirichletGroup(405, base_ring=CyclotomicField(36))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([26,9]))

pari: [g,chi] = znchar(Mod(197,405))

Basic properties

 Modulus: $$405$$ Conductor: $$135$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$36$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{135}(92,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: no Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

Galois orbit 405.r

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: $$\Q(\zeta_{36})$$ Fixed field: $$\Q(\zeta_{135})^+$$

Values on generators

$$(326,82)$$ → $$(e\left(\frac{13}{18}\right),i)$$

Values

 $$a$$ $$-1$$ $$1$$ $$2$$ $$4$$ $$7$$ $$8$$ $$11$$ $$13$$ $$14$$ $$16$$ $$17$$ $$19$$ $$\chi_{ 405 }(197, a)$$ $$1$$ $$1$$ $$e\left(\frac{35}{36}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{29}{36}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{19}{36}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$
sage: chi.jacobi_sum(n)

$$\chi_{ 405 }(197,a) \;$$ at $$\;a =$$ e.g. 2

Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 405 }(197,·) )\;$$ at $$\;a =$$ e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 405 }(197,·),\chi_{ 405 }(n,·)) \;$$ for $$\; n =$$ e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 405 }(197,·)) \;$$ at $$\; a,b =$$ e.g. 1,2