# Properties

 Modulus 73 Conductor 73 Order 36 Real no Primitive yes Minimal yes Parity even Orbit label 73.k

# Related objects

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

sage: H = DirichletGroup(73)

sage: M = H._module

sage: chi = DirichletCharacter(H, M([35]))

pari: [g,chi] = znchar(Mod(38,73))

## Basic properties

 sage: chi.conductor()  pari: znconreyconductor(g,chi) Modulus = 73 Conductor = 73 sage: chi.multiplicative_order()  pari: charorder(g,chi) Order = 36 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 = even Orbit label = 73.k Orbit index = 11

## 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

$$5$$ → $$e\left(\frac{35}{36}\right)$$

## Values

 -1 1 2 3 4 5 6 7 8 9 10 11 $$1$$ $$1$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{35}{36}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$-i$$ $$e\left(\frac{17}{36}\right)$$
value at  e.g. 2

## Related number fields

 Field of values $$\Q(\zeta_{36})$$

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 73 }(38,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{73}(38,\cdot)) = \sum_{r\in \Z/73\Z} \chi_{73}(38,r) e\left(\frac{2r}{73}\right) = 7.4016336093+-4.2679995213i$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 73 }(38,·),\chi_{ 73 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{73}(38,\cdot),\chi_{73}(1,\cdot)) = \sum_{r\in \Z/73\Z} \chi_{73}(38,r) \chi_{73}(1,1-r) = -1$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 73 }(38,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{73}(38,·)) = \sum_{r \in \Z/73\Z} \chi_{73}(38,r) e\left(\frac{1 r + 2 r^{-1}}{73}\right) = 1.8368823736+-1.5413273222i$$