![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(30345, base_ring=CyclotomicField(136))
M = H._module
chi = DirichletCharacter(H, M([68,102,0,19]))
![Copy content]()
pari:[g,chi] = znchar(Mod(13343,30345))
\(\chi_{30345}(722,\cdot)\)
\(\chi_{30345}(848,\cdot)\)
\(\chi_{30345}(1352,\cdot)\)
\(\chi_{30345}(1583,\cdot)\)
\(\chi_{30345}(2507,\cdot)\)
\(\chi_{30345}(2633,\cdot)\)
\(\chi_{30345}(3137,\cdot)\)
\(\chi_{30345}(3368,\cdot)\)
\(\chi_{30345}(4292,\cdot)\)
\(\chi_{30345}(4418,\cdot)\)
\(\chi_{30345}(4922,\cdot)\)
\(\chi_{30345}(5153,\cdot)\)
\(\chi_{30345}(6077,\cdot)\)
\(\chi_{30345}(6707,\cdot)\)
\(\chi_{30345}(6938,\cdot)\)
\(\chi_{30345}(7862,\cdot)\)
\(\chi_{30345}(7988,\cdot)\)
\(\chi_{30345}(8492,\cdot)\)
\(\chi_{30345}(8723,\cdot)\)
\(\chi_{30345}(9773,\cdot)\)
\(\chi_{30345}(10277,\cdot)\)
\(\chi_{30345}(10508,\cdot)\)
\(\chi_{30345}(11432,\cdot)\)
\(\chi_{30345}(11558,\cdot)\)
\(\chi_{30345}(12062,\cdot)\)
\(\chi_{30345}(13217,\cdot)\)
\(\chi_{30345}(13343,\cdot)\)
\(\chi_{30345}(13847,\cdot)\)
\(\chi_{30345}(14078,\cdot)\)
\(\chi_{30345}(15002,\cdot)\)
...
![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 ]
\((20231,24277,4336,28036)\) → \((-1,-i,1,e\left(\frac{19}{136}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(19\) | \(22\) | \(23\) | \(26\) |
| \( \chi_{ 30345 }(13343, a) \) |
\(1\) | \(1\) | \(e\left(\frac{27}{34}\right)\) | \(e\left(\frac{10}{17}\right)\) | \(e\left(\frac{13}{34}\right)\) | \(e\left(\frac{97}{136}\right)\) | \(e\left(\frac{43}{68}\right)\) | \(e\left(\frac{3}{17}\right)\) | \(e\left(\frac{31}{68}\right)\) | \(e\left(\frac{69}{136}\right)\) | \(e\left(\frac{123}{136}\right)\) | \(e\left(\frac{29}{68}\right)\) |
![Copy content]()
sage:chi.jacobi_sum(n)
