![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8352, base_ring=CyclotomicField(56))
M = H._module
chi = DirichletCharacter(H, M([28,7,28,40]))
![Copy content]()
pari:[g,chi] = znchar(Mod(8027,8352))
\(\chi_{8352}(107,\cdot)\)
\(\chi_{8352}(683,\cdot)\)
\(\chi_{8352}(1475,\cdot)\)
\(\chi_{8352}(1619,\cdot)\)
\(\chi_{8352}(1763,\cdot)\)
\(\chi_{8352}(1979,\cdot)\)
\(\chi_{8352}(2195,\cdot)\)
\(\chi_{8352}(2771,\cdot)\)
\(\chi_{8352}(3563,\cdot)\)
\(\chi_{8352}(3707,\cdot)\)
\(\chi_{8352}(3851,\cdot)\)
\(\chi_{8352}(4067,\cdot)\)
\(\chi_{8352}(4283,\cdot)\)
\(\chi_{8352}(4859,\cdot)\)
\(\chi_{8352}(5651,\cdot)\)
\(\chi_{8352}(5795,\cdot)\)
\(\chi_{8352}(5939,\cdot)\)
\(\chi_{8352}(6155,\cdot)\)
\(\chi_{8352}(6371,\cdot)\)
\(\chi_{8352}(6947,\cdot)\)
\(\chi_{8352}(7739,\cdot)\)
\(\chi_{8352}(7883,\cdot)\)
\(\chi_{8352}(8027,\cdot)\)
\(\chi_{8352}(8243,\cdot)\)
![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 ]
\((1567,5221,929,4033)\) → \((-1,e\left(\frac{1}{8}\right),-1,e\left(\frac{5}{7}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(31\) | \(35\) |
| \( \chi_{ 8352 }(8027, a) \) |
\(1\) | \(1\) | \(e\left(\frac{19}{56}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{27}{56}\right)\) | \(e\left(\frac{41}{56}\right)\) | \(1\) | \(e\left(\frac{45}{56}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{37}{56}\right)\) |
![Copy content]()
sage:chi.jacobi_sum(n)
