sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7260, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([11,11,0,6]))
pari:[g,chi] = znchar(Mod(3191,7260))
\(\chi_{7260}(551,\cdot)\)
\(\chi_{7260}(1871,\cdot)\)
\(\chi_{7260}(2531,\cdot)\)
\(\chi_{7260}(3191,\cdot)\)
\(\chi_{7260}(3851,\cdot)\)
\(\chi_{7260}(4511,\cdot)\)
\(\chi_{7260}(5171,\cdot)\)
\(\chi_{7260}(5831,\cdot)\)
\(\chi_{7260}(6491,\cdot)\)
\(\chi_{7260}(7151,\cdot)\)
sage:chi.galois_orbit()
pari:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
\((3631,4841,4357,7141)\) → \((-1,-1,1,e\left(\frac{3}{11}\right))\)
\(a\) |
\(-1\) | \(1\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
\( \chi_{ 7260 }(3191, a) \) |
\(1\) | \(1\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{7}{22}\right)\) |
sage:chi.jacobi_sum(n)