Properties

 Label 64.63 Modulus $64$ Conductor $4$ Order $2$ Real yes Primitive no Minimal no Parity odd

Related objects

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

sage: H = DirichletGroup(64, base_ring=CyclotomicField(2))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([1,0]))

pari: [g,chi] = znchar(Mod(63,64))

Basic properties

 Modulus: $$64$$ Conductor: $$4$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$2$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: yes Primitive: no, induced from $$\chi_{4}(3,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: no Parity: odd sage: chi.is_odd()  pari: zncharisodd(g,chi)

Galois orbit 64.c

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

Values on generators

$$(63,5)$$ → $$(-1,1)$$

Values

 $$-1$$ $$1$$ $$3$$ $$5$$ $$7$$ $$9$$ $$11$$ $$13$$ $$15$$ $$17$$ $$19$$ $$21$$ $$-1$$ $$1$$ $$-1$$ $$1$$ $$-1$$ $$1$$ $$-1$$ $$1$$ $$-1$$ $$1$$ $$-1$$ $$1$$
 value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 64 }(63,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{64}(63,\cdot)) = \sum_{r\in \Z/64\Z} \chi_{64}(63,r) e\left(\frac{r}{32}\right) = 0.0$$

Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 64 }(63,·),\chi_{ 64 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{64}(63,\cdot),\chi_{64}(1,\cdot)) = \sum_{r\in \Z/64\Z} \chi_{64}(63,r) \chi_{64}(1,1-r) = 0$$

Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 64 }(63,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{64}(63,·)) = \sum_{r \in \Z/64\Z} \chi_{64}(63,r) e\left(\frac{1 r + 2 r^{-1}}{64}\right) = -0.0$$