![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1309, base_ring=CyclotomicField(80))
M = H._module
chi = DirichletCharacter(H, M([0,56,65]))
![Copy content]()
pari:[g,chi] = znchar(Mod(29,1309))
\(\chi_{1309}(29,\cdot)\)
\(\chi_{1309}(57,\cdot)\)
\(\chi_{1309}(211,\cdot)\)
\(\chi_{1309}(260,\cdot)\)
\(\chi_{1309}(316,\cdot)\)
\(\chi_{1309}(337,\cdot)\)
\(\chi_{1309}(414,\cdot)\)
\(\chi_{1309}(435,\cdot)\)
\(\chi_{1309}(470,\cdot)\)
\(\chi_{1309}(547,\cdot)\)
\(\chi_{1309}(568,\cdot)\)
\(\chi_{1309}(589,\cdot)\)
\(\chi_{1309}(624,\cdot)\)
\(\chi_{1309}(666,\cdot)\)
\(\chi_{1309}(673,\cdot)\)
\(\chi_{1309}(743,\cdot)\)
\(\chi_{1309}(827,\cdot)\)
\(\chi_{1309}(855,\cdot)\)
\(\chi_{1309}(904,\cdot)\)
\(\chi_{1309}(932,\cdot)\)
\(\chi_{1309}(974,\cdot)\)
\(\chi_{1309}(981,\cdot)\)
\(\chi_{1309}(1009,\cdot)\)
\(\chi_{1309}(1030,\cdot)\)
\(\chi_{1309}(1051,\cdot)\)
\(\chi_{1309}(1128,\cdot)\)
\(\chi_{1309}(1163,\cdot)\)
\(\chi_{1309}(1184,\cdot)\)
\(\chi_{1309}(1212,\cdot)\)
\(\chi_{1309}(1261,\cdot)\)
...
![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 ]
\((1123,596,309)\) → \((1,e\left(\frac{7}{10}\right),e\left(\frac{13}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(12\) | \(13\) |
| \( \chi_{ 1309 }(29, a) \) |
\(1\) | \(1\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{33}{80}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{69}{80}\right)\) | \(e\left(\frac{39}{80}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{19}{20}\right)\) |
![Copy content]()
sage:chi.jacobi_sum(n)
