Properties

 Modulus 90 Conductor 45 Order 12 Real no Primitive no Minimal yes Parity even Orbit label 90.l

Related objects

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

sage: H = DirichletGroup(90)

sage: M = H._module

sage: chi = DirichletCharacter(H, M([10,9]))

pari: [g,chi] = znchar(Mod(23,90))

Basic properties

 sage: chi.conductor()  pari: znconreyconductor(g,chi) Modulus = 90 Conductor = 45 sage: chi.multiplicative_order()  pari: charorder(g,chi) Order = 12 Real = no sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization] Primitive = no Minimal = yes sage: chi.is_odd()  pari: zncharisodd(g,chi) Parity = even Orbit label = 90.l Orbit index = 12

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

$$(11,37)$$ → $$(e\left(\frac{5}{6}\right),-i)$$

Values

 -1 1 7 11 13 17 19 23 29 31 37 41 $$1$$ $$1$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$i$$ $$-1$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$-i$$ $$e\left(\frac{1}{6}\right)$$
value at  e.g. 2

Related number fields

 Field of values $$\Q(\zeta_{12})$$

Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 90 }(23,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{90}(23,\cdot)) = \sum_{r\in \Z/90\Z} \chi_{90}(23,r) e\left(\frac{r}{45}\right) = 6.2320540123+-2.4822374559i$$

Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 90 }(23,·),\chi_{ 90 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{90}(23,\cdot),\chi_{90}(1,\cdot)) = \sum_{r\in \Z/90\Z} \chi_{90}(23,r) \chi_{90}(1,1-r) = 0$$

Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 90 }(23,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{90}(23,·)) = \sum_{r \in \Z/90\Z} \chi_{90}(23,r) e\left(\frac{1 r + 2 r^{-1}}{90}\right) = 2.0886650724+-7.7950041702i$$