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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1225, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([39,50]))
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pari:[g,chi] = znchar(Mod(117,1225))
\(\chi_{1225}(117,\cdot)\)
\(\chi_{1225}(178,\cdot)\)
\(\chi_{1225}(227,\cdot)\)
\(\chi_{1225}(313,\cdot)\)
\(\chi_{1225}(362,\cdot)\)
\(\chi_{1225}(423,\cdot)\)
\(\chi_{1225}(472,\cdot)\)
\(\chi_{1225}(558,\cdot)\)
\(\chi_{1225}(717,\cdot)\)
\(\chi_{1225}(803,\cdot)\)
\(\chi_{1225}(852,\cdot)\)
\(\chi_{1225}(913,\cdot)\)
\(\chi_{1225}(962,\cdot)\)
\(\chi_{1225}(1048,\cdot)\)
\(\chi_{1225}(1097,\cdot)\)
\(\chi_{1225}(1158,\cdot)\)
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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 ]
\((1177,101)\) → \((e\left(\frac{13}{20}\right),e\left(\frac{5}{6}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
| \( \chi_{ 1225 }(117, a) \) |
\(1\) | \(1\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{4}{15}\right)\) |
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sage:chi.jacobi_sum(n)
