Properties

 Modulus 97 Conductor 97 Order 32 Real no Primitive yes Minimal yes Parity odd Orbit label 97.j

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

sage: H = DirichletGroup(97)

sage: M = H._module

sage: chi = DirichletCharacter(H, M([21]))

pari: [g,chi] = znchar(Mod(51,97))

Basic properties

 sage: chi.conductor()  pari: znconreyconductor(g,chi) Modulus = 97 Conductor = 97 sage: chi.multiplicative_order()  pari: charorder(g,chi) Order = 32 Real = no sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization] Primitive = yes Minimal = yes sage: chi.is_odd()  pari: zncharisodd(g,chi) Parity = odd Orbit label = 97.j Orbit index = 10

Galois orbit

sage: chi.galois_orbit()

pari: order = charorder(g,chi)

pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]

Values on generators

$$5$$ → $$e\left(\frac{21}{32}\right)$$

Values

 -1 1 2 3 4 5 6 7 8 9 10 11 $$-1$$ $$1$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{21}{32}\right)$$ $$i$$ $$e\left(\frac{11}{32}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{31}{32}\right)$$ $$e\left(\frac{7}{16}\right)$$
value at  e.g. 2

Related number fields

 Field of values $$\Q(\zeta_{32})$$

Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 97 }(51,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{97}(51,\cdot)) = \sum_{r\in \Z/97\Z} \chi_{97}(51,r) e\left(\frac{2r}{97}\right) = -9.6223956189+2.0998815569i$$

Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 97 }(51,·),\chi_{ 97 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{97}(51,\cdot),\chi_{97}(1,\cdot)) = \sum_{r\in \Z/97\Z} \chi_{97}(51,r) \chi_{97}(1,1-r) = -1$$

Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 97 }(51,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{97}(51,·)) = \sum_{r \in \Z/97\Z} \chi_{97}(51,r) e\left(\frac{1 r + 2 r^{-1}}{97}\right) = -2.4648301587+1.6469468581i$$