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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7350, base_ring=CyclotomicField(70))
M = H._module
chi = DirichletCharacter(H, M([0,7,50]))
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pari:[g,chi] = znchar(Mod(6679,7350))
\(\chi_{7350}(169,\cdot)\)
\(\chi_{7350}(379,\cdot)\)
\(\chi_{7350}(1009,\cdot)\)
\(\chi_{7350}(1219,\cdot)\)
\(\chi_{7350}(1429,\cdot)\)
\(\chi_{7350}(1639,\cdot)\)
\(\chi_{7350}(2269,\cdot)\)
\(\chi_{7350}(2479,\cdot)\)
\(\chi_{7350}(2689,\cdot)\)
\(\chi_{7350}(3109,\cdot)\)
\(\chi_{7350}(3319,\cdot)\)
\(\chi_{7350}(3739,\cdot)\)
\(\chi_{7350}(4159,\cdot)\)
\(\chi_{7350}(4369,\cdot)\)
\(\chi_{7350}(4579,\cdot)\)
\(\chi_{7350}(4789,\cdot)\)
\(\chi_{7350}(5209,\cdot)\)
\(\chi_{7350}(5419,\cdot)\)
\(\chi_{7350}(5629,\cdot)\)
\(\chi_{7350}(5839,\cdot)\)
\(\chi_{7350}(6259,\cdot)\)
\(\chi_{7350}(6679,\cdot)\)
\(\chi_{7350}(6889,\cdot)\)
\(\chi_{7350}(7309,\cdot)\)
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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 ]
\((4901,1177,2551)\) → \((1,e\left(\frac{1}{10}\right),e\left(\frac{5}{7}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 7350 }(6679, a) \) |
\(1\) | \(1\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{33}{70}\right)\) | \(e\left(\frac{11}{70}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{17}{70}\right)\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{53}{70}\right)\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{11}{14}\right)\) |
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sage:chi.jacobi_sum(n)
