# Properties

 Modulus 74 Conductor 37 Order 12 Real no Primitive no Minimal yes Parity odd Orbit label 74.g

# Learn more about

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

sage: H = DirichletGroup(74)

sage: M = H._module

sage: chi = DirichletCharacter(H, M([5]))

pari: [g,chi] = znchar(Mod(23,74))

## Basic properties

 sage: chi.conductor()  pari: znconreyconductor(g,chi) Modulus = 74 Conductor = 37 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 = odd Orbit label = 74.g Orbit index = 7

## 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

$$39$$ → $$e\left(\frac{5}{12}\right)$$

## Values

 -1 1 3 5 7 9 11 13 15 17 19 21 $$-1$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$-1$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$
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)

$$\tau_{ a }( \chi_{ 74 }(23,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{74}(23,\cdot)) = \sum_{r\in \Z/74\Z} \chi_{74}(23,r) e\left(\frac{r}{37}\right) = 2.8706234267+-5.3627904249i$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 74 }(23,·),\chi_{ 74 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{74}(23,\cdot),\chi_{74}(1,\cdot)) = \sum_{r\in \Z/74\Z} \chi_{74}(23,r) \chi_{74}(1,1-r) = 0$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 74 }(23,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{74}(23,·)) = \sum_{r \in \Z/74\Z} \chi_{74}(23,r) e\left(\frac{1 r + 2 r^{-1}}{74}\right) = -5.0628744969+1.3565931328i$$