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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(15730, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([0,2,11]))
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pari:[g,chi] = znchar(Mod(7171,15730))
\(\chi_{15730}(21,\cdot)\)
\(\chi_{15730}(681,\cdot)\)
\(\chi_{15730}(2111,\cdot)\)
\(\chi_{15730}(2881,\cdot)\)
\(\chi_{15730}(3541,\cdot)\)
\(\chi_{15730}(4311,\cdot)\)
\(\chi_{15730}(4971,\cdot)\)
\(\chi_{15730}(5741,\cdot)\)
\(\chi_{15730}(6401,\cdot)\)
\(\chi_{15730}(7171,\cdot)\)
\(\chi_{15730}(7831,\cdot)\)
\(\chi_{15730}(8601,\cdot)\)
\(\chi_{15730}(9261,\cdot)\)
\(\chi_{15730}(10031,\cdot)\)
\(\chi_{15730}(10691,\cdot)\)
\(\chi_{15730}(11461,\cdot)\)
\(\chi_{15730}(12121,\cdot)\)
\(\chi_{15730}(12891,\cdot)\)
\(\chi_{15730}(14321,\cdot)\)
\(\chi_{15730}(14981,\cdot)\)
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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 ]
\((3147,3511,1211)\) → \((1,e\left(\frac{1}{22}\right),i)\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) | \(31\) |
| \( \chi_{ 15730 }(7171, a) \) |
\(1\) | \(1\) | \(1\) | \(e\left(\frac{3}{44}\right)\) | \(1\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(1\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{7}{44}\right)\) |
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sage:chi.jacobi_sum(n)
