Properties

 Label 264.197 Modulus $264$ Conductor $264$ Order $2$ Real yes Primitive yes Minimal yes Parity even

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

sage: H = DirichletGroup(264, base_ring=CyclotomicField(2))

sage: M = H._module

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

pari: [g,chi] = znchar(Mod(197,264))

Kronecker symbol representation

sage: kronecker_character(264)

pari: znchartokronecker(g,chi)

$$\displaystyle\left(\frac{264}{\bullet}\right)$$

Basic properties

 Modulus: $$264$$ Conductor: $$264$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$2$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: yes 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 264.m

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: $$\Q(\sqrt{66})$$

Values on generators

$$(199,133,89,145)$$ → $$(1,-1,-1,-1)$$

Values

 $$-1$$ $$1$$ $$5$$ $$7$$ $$13$$ $$17$$ $$19$$ $$23$$ $$25$$ $$29$$ $$31$$ $$35$$ $$1$$ $$1$$ $$1$$ $$-1$$ $$1$$ $$1$$ $$1$$ $$-1$$ $$1$$ $$-1$$ $$1$$ $$-1$$
 value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 264 }(197,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{264}(197,\cdot)) = \sum_{r\in \Z/264\Z} \chi_{264}(197,r) e\left(\frac{r}{132}\right) = 0.0$$

Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 264 }(197,·),\chi_{ 264 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{264}(197,\cdot),\chi_{264}(1,\cdot)) = \sum_{r\in \Z/264\Z} \chi_{264}(197,r) \chi_{264}(1,1-r) = 0$$

Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 264 }(197,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{264}(197,·)) = \sum_{r \in \Z/264\Z} \chi_{264}(197,r) e\left(\frac{1 r + 2 r^{-1}}{264}\right) = 0.0$$