Properties

 Label 240.13 Modulus $240$ Conductor $80$ Order $4$ Real no Primitive no Minimal yes Parity odd

Related objects

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

sage: H = DirichletGroup(240, base_ring=CyclotomicField(4))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([0,3,0,3]))

pari: [g,chi] = znchar(Mod(13,240))

Basic properties

 Modulus: $$240$$ Conductor: $$80$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$4$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{80}(13,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: odd sage: chi.is_odd()  pari: zncharisodd(g,chi)

Galois orbit 240.ba

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: $$\Q(\sqrt{-1})$$ Fixed field: 4.0.256000.4

Values on generators

$$(31,181,161,97)$$ → $$(1,-i,1,-i)$$

Values

 $$a$$ $$-1$$ $$1$$ $$7$$ $$11$$ $$13$$ $$17$$ $$19$$ $$23$$ $$29$$ $$31$$ $$37$$ $$41$$ $$\chi_{ 240 }(13, a)$$ $$-1$$ $$1$$ $$i$$ $$-i$$ $$-1$$ $$-i$$ $$-i$$ $$-i$$ $$-i$$ $$1$$ $$-1$$ $$-1$$
sage: chi.jacobi_sum(n)

$$\chi_{ 240 }(13,a) \;$$ at $$\;a =$$ e.g. 2

Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 240 }(13,·) )\;$$ at $$\;a =$$ e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 240 }(13,·),\chi_{ 240 }(n,·)) \;$$ for $$\; n =$$ e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 240 }(13,·)) \;$$ at $$\; a,b =$$ e.g. 1,2