sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5160, base_ring=CyclotomicField(84))
M = H._module
chi = DirichletCharacter(H, M([0,42,0,21,58]))
pari:[g,chi] = znchar(Mod(3157,5160))
\(\chi_{5160}(157,\cdot)\)
\(\chi_{5160}(277,\cdot)\)
\(\chi_{5160}(373,\cdot)\)
\(\chi_{5160}(493,\cdot)\)
\(\chi_{5160}(757,\cdot)\)
\(\chi_{5160}(1093,\cdot)\)
\(\chi_{5160}(1237,\cdot)\)
\(\chi_{5160}(1453,\cdot)\)
\(\chi_{5160}(2413,\cdot)\)
\(\chi_{5160}(2437,\cdot)\)
\(\chi_{5160}(2557,\cdot)\)
\(\chi_{5160}(2653,\cdot)\)
\(\chi_{5160}(2893,\cdot)\)
\(\chi_{5160}(3013,\cdot)\)
\(\chi_{5160}(3157,\cdot)\)
\(\chi_{5160}(3253,\cdot)\)
\(\chi_{5160}(3373,\cdot)\)
\(\chi_{5160}(3517,\cdot)\)
\(\chi_{5160}(3853,\cdot)\)
\(\chi_{5160}(4333,\cdot)\)
\(\chi_{5160}(4477,\cdot)\)
\(\chi_{5160}(4717,\cdot)\)
\(\chi_{5160}(4957,\cdot)\)
\(\chi_{5160}(5077,\cdot)\)
sage:chi.galois_orbit()
pari:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
\((3871,2581,1721,3097,4561)\) → \((1,-1,1,i,e\left(\frac{29}{42}\right))\)
\(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
\( \chi_{ 5160 }(3157, a) \) |
\(1\) | \(1\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{29}{84}\right)\) | \(e\left(\frac{41}{84}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{67}{84}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{7}\right)\) |
sage:chi.jacobi_sum(n)