![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2580, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,0,0,34]))
![Copy content]()
pari:[g,chi] = znchar(Mod(31,2580))
\(\chi_{2580}(31,\cdot)\)
\(\chi_{2580}(271,\cdot)\)
\(\chi_{2580}(511,\cdot)\)
\(\chi_{2580}(1171,\cdot)\)
\(\chi_{2580}(1471,\cdot)\)
\(\chi_{2580}(1651,\cdot)\)
\(\chi_{2580}(1831,\cdot)\)
\(\chi_{2580}(2131,\cdot)\)
\(\chi_{2580}(2251,\cdot)\)
\(\chi_{2580}(2431,\cdot)\)
\(\chi_{2580}(2491,\cdot)\)
\(\chi_{2580}(2551,\cdot)\)
![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 ]
\((1291,1721,517,1981)\) → \((-1,1,1,e\left(\frac{17}{21}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 2580 }(31, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{6}{7}\right)\) |
![Copy content]()
sage:chi.jacobi_sum(n)
