sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3150, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([50,39,10]))
pari:[g,chi] = znchar(Mod(1067,3150))
\(\chi_{3150}(47,\cdot)\)
\(\chi_{3150}(173,\cdot)\)
\(\chi_{3150}(437,\cdot)\)
\(\chi_{3150}(563,\cdot)\)
\(\chi_{3150}(677,\cdot)\)
\(\chi_{3150}(803,\cdot)\)
\(\chi_{3150}(1067,\cdot)\)
\(\chi_{3150}(1433,\cdot)\)
\(\chi_{3150}(1697,\cdot)\)
\(\chi_{3150}(1823,\cdot)\)
\(\chi_{3150}(1937,\cdot)\)
\(\chi_{3150}(2063,\cdot)\)
\(\chi_{3150}(2327,\cdot)\)
\(\chi_{3150}(2453,\cdot)\)
\(\chi_{3150}(2567,\cdot)\)
\(\chi_{3150}(3083,\cdot)\)
sage:chi.galois_orbit()
pari:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
\((2801,127,451)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{13}{20}\right),e\left(\frac{1}{6}\right))\)
\(a\) |
\(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
\( \chi_{ 3150 }(1067, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{1}{12}\right)\) |
sage:chi.jacobi_sum(n)