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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(15730, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,54,44]))
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pari:[g,chi] = znchar(Mod(10539,15730))
\(\chi_{15730}(419,\cdot)\)
\(\chi_{15730}(529,\cdot)\)
\(\chi_{15730}(1849,\cdot)\)
\(\chi_{15730}(1959,\cdot)\)
\(\chi_{15730}(3279,\cdot)\)
\(\chi_{15730}(4709,\cdot)\)
\(\chi_{15730}(4819,\cdot)\)
\(\chi_{15730}(6139,\cdot)\)
\(\chi_{15730}(6249,\cdot)\)
\(\chi_{15730}(7569,\cdot)\)
\(\chi_{15730}(7679,\cdot)\)
\(\chi_{15730}(8999,\cdot)\)
\(\chi_{15730}(9109,\cdot)\)
\(\chi_{15730}(10429,\cdot)\)
\(\chi_{15730}(10539,\cdot)\)
\(\chi_{15730}(11969,\cdot)\)
\(\chi_{15730}(13289,\cdot)\)
\(\chi_{15730}(13399,\cdot)\)
\(\chi_{15730}(14719,\cdot)\)
\(\chi_{15730}(14829,\cdot)\)
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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 ]
\((3147,3511,1211)\) → \((-1,e\left(\frac{9}{11}\right),e\left(\frac{2}{3}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) | \(31\) |
| \( \chi_{ 15730 }(10539, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(-1\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{4}{11}\right)\) |
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sage:chi.jacobi_sum(n)
