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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4176, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,21,28,24]))
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pari:[g,chi] = znchar(Mod(25,4176))
\(\chi_{4176}(25,\cdot)\)
\(\chi_{4176}(169,\cdot)\)
\(\chi_{4176}(313,\cdot)\)
\(\chi_{4176}(745,\cdot)\)
\(\chi_{4176}(1321,\cdot)\)
\(\chi_{4176}(1417,\cdot)\)
\(\chi_{4176}(1561,\cdot)\)
\(\chi_{4176}(1705,\cdot)\)
\(\chi_{4176}(2137,\cdot)\)
\(\chi_{4176}(2617,\cdot)\)
\(\chi_{4176}(2713,\cdot)\)
\(\chi_{4176}(4009,\cdot)\)
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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 ]
\((1567,1045,929,4033)\) → \((1,-1,e\left(\frac{2}{3}\right),e\left(\frac{4}{7}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(31\) | \(35\) |
| \( \chi_{ 4176 }(25, a) \) |
\(1\) | \(1\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(1\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{13}{14}\right)\) |
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sage:chi.jacobi_sum(n)
