# Properties

 Label 90.67 Modulus $90$ Conductor $45$ Order $12$ Real no Primitive no Minimal yes Parity odd

# 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([4,3]))

pari: [g,chi] = znchar(Mod(67,90))

## Basic properties

 Modulus: $$90$$ Conductor: $$45$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$12$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{45}(22,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: odd sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 90.k

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

$$(11,37)$$ → $$(e\left(\frac{1}{3}\right),i)$$

## Values

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

## Related number fields

 Field of values: $$\Q(\zeta_{12})$$ Fixed field: Number field defined by a degree %d polynomial (not computed)

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 90 }(67,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{90}(67,\cdot)) = \sum_{r\in \Z/90\Z} \chi_{90}(67,r) e\left(\frac{r}{45}\right) = -5.0072399573+-4.4640282269i$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 90 }(67,·),\chi_{ 90 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{90}(67,\cdot),\chi_{90}(1,\cdot)) = \sum_{r\in \Z/90\Z} \chi_{90}(67,r) \chi_{90}(1,1-r) = 0$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 90 }(67,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{90}(67,·)) = \sum_{r \in \Z/90\Z} \chi_{90}(67,r) e\left(\frac{1 r + 2 r^{-1}}{90}\right) = -7.7950041702+-2.0886650724i$$