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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1225, base_ring=CyclotomicField(84))
M = H._module
chi = DirichletCharacter(H, M([63,50]))
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pari:[g,chi] = znchar(Mod(1193,1225))
\(\chi_{1225}(82,\cdot)\)
\(\chi_{1225}(143,\cdot)\)
\(\chi_{1225}(157,\cdot)\)
\(\chi_{1225}(243,\cdot)\)
\(\chi_{1225}(257,\cdot)\)
\(\chi_{1225}(318,\cdot)\)
\(\chi_{1225}(332,\cdot)\)
\(\chi_{1225}(418,\cdot)\)
\(\chi_{1225}(432,\cdot)\)
\(\chi_{1225}(493,\cdot)\)
\(\chi_{1225}(507,\cdot)\)
\(\chi_{1225}(593,\cdot)\)
\(\chi_{1225}(682,\cdot)\)
\(\chi_{1225}(768,\cdot)\)
\(\chi_{1225}(782,\cdot)\)
\(\chi_{1225}(843,\cdot)\)
\(\chi_{1225}(857,\cdot)\)
\(\chi_{1225}(943,\cdot)\)
\(\chi_{1225}(957,\cdot)\)
\(\chi_{1225}(1018,\cdot)\)
\(\chi_{1225}(1032,\cdot)\)
\(\chi_{1225}(1118,\cdot)\)
\(\chi_{1225}(1132,\cdot)\)
\(\chi_{1225}(1193,\cdot)\)
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sage:chi.galois_orbit()
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pari:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
\((1177,101)\) → \((-i,e\left(\frac{25}{42}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
| \( \chi_{ 1225 }(1193, a) \) |
\(1\) | \(1\) | \(e\left(\frac{19}{84}\right)\) | \(e\left(\frac{71}{84}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{25}{84}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{19}{21}\right)\) |
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sage:chi.jacobi_sum(n)
