Properties

 Modulus 137 Conductor 137 Order 136 Real no Primitive yes Minimal yes Parity odd Orbit label 137.h

Related objects

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

sage: H = DirichletGroup(137)

sage: M = H._module

sage: chi = DirichletCharacter(H, M([91]))

pari: [g,chi] = znchar(Mod(29,137))

Basic properties

 sage: chi.conductor()  pari: znconreyconductor(g,chi) Modulus = 137 Conductor = 137 sage: chi.multiplicative_order()  pari: charorder(g,chi) Order = 136 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 = 137.h Orbit index = 8

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

$$3$$ → $$e\left(\frac{91}{136}\right)$$

Values

 -1 1 2 3 4 5 6 7 8 9 10 11 $$-1$$ $$1$$ $$e\left(\frac{47}{68}\right)$$ $$e\left(\frac{91}{136}\right)$$ $$e\left(\frac{13}{34}\right)$$ $$e\left(\frac{25}{136}\right)$$ $$e\left(\frac{49}{136}\right)$$ $$e\left(\frac{7}{68}\right)$$ $$e\left(\frac{5}{68}\right)$$ $$e\left(\frac{23}{68}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{43}{68}\right)$$
value at  e.g. 2

Related number fields

 Field of values $$\Q(\zeta_{136})$$

Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 137 }(29,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{137}(29,\cdot)) = \sum_{r\in \Z/137\Z} \chi_{137}(29,r) e\left(\frac{2r}{137}\right) = 10.8954650306+4.2765455413i$$

Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 137 }(29,·),\chi_{ 137 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{137}(29,\cdot),\chi_{137}(1,\cdot)) = \sum_{r\in \Z/137\Z} \chi_{137}(29,r) \chi_{137}(1,1-r) = -1$$

Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 137 }(29,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{137}(29,·)) = \sum_{r \in \Z/137\Z} \chi_{137}(29,r) e\left(\frac{1 r + 2 r^{-1}}{137}\right) = -7.0935650162+-4.8592050548i$$