Properties

 Label 101.4 Modulus $101$ Conductor $101$ Order $50$ Real no Primitive yes Minimal yes Parity even

Related objects

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

sage: H = DirichletGroup(101)

sage: M = H._module

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

pari: [g,chi] = znchar(Mod(4,101))

Basic properties

 Modulus: $$101$$ Conductor: $$101$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$50$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no 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 101.h

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

$$2$$ → $$e\left(\frac{1}{50}\right)$$

Values

 $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$ $$8$$ $$9$$ $$10$$ $$11$$ $$1$$ $$1$$ $$e\left(\frac{1}{50}\right)$$ $$e\left(\frac{19}{50}\right)$$ $$e\left(\frac{1}{25}\right)$$ $$e\left(\frac{12}{25}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{9}{50}\right)$$ $$e\left(\frac{3}{50}\right)$$ $$e\left(\frac{19}{25}\right)$$ $$-1$$ $$e\left(\frac{13}{50}\right)$$
 value at e.g. 2

Related number fields

 Field of values: $$\Q(\zeta_{25})$$ Fixed field: Number field defined by a degree 50 polynomial

Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 101 }(4,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{101}(4,\cdot)) = \sum_{r\in \Z/101\Z} \chi_{101}(4,r) e\left(\frac{2r}{101}\right) = 2.0403975696+9.8405679591i$$

Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 101 }(4,·),\chi_{ 101 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{101}(4,\cdot),\chi_{101}(1,\cdot)) = \sum_{r\in \Z/101\Z} \chi_{101}(4,r) \chi_{101}(1,1-r) = -1$$

Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 101 }(4,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{101}(4,·)) = \sum_{r \in \Z/101\Z} \chi_{101}(4,r) e\left(\frac{1 r + 2 r^{-1}}{101}\right) = -9.194084781+-0.5784427847i$$