![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(740, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,27,1]))
![Copy content]()
pari:[g,chi] = znchar(Mod(113,740))
\(\chi_{740}(17,\cdot)\)
\(\chi_{740}(113,\cdot)\)
\(\chi_{740}(257,\cdot)\)
\(\chi_{740}(353,\cdot)\)
\(\chi_{740}(457,\cdot)\)
\(\chi_{740}(513,\cdot)\)
\(\chi_{740}(533,\cdot)\)
\(\chi_{740}(537,\cdot)\)
\(\chi_{740}(573,\cdot)\)
\(\chi_{740}(577,\cdot)\)
\(\chi_{740}(597,\cdot)\)
\(\chi_{740}(653,\cdot)\)
![Copy content]()
sage:chi.galois_orbit()
![Copy content]()
pari:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
\((371,297,261)\) → \((1,-i,e\left(\frac{1}{36}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 740 }(113, a) \) |
\(1\) | \(1\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{11}{12}\right)\) |
![Copy content]()
sage:chi.jacobi_sum(n)
![Copy content]()
sage:chi.gauss_sum(a)
![Copy content]()
pari:znchargauss(g,chi,a)
![Copy content]()
sage:chi.jacobi_sum(n)
![Copy content]()
sage:chi.kloosterman_sum(a,b)
