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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5184, base_ring=CyclotomicField(216))
M = H._module
chi = DirichletCharacter(H, M([108,81,76]))
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pari:[g,chi] = znchar(Mod(1271,5184))
\(\chi_{5184}(23,\cdot)\)
\(\chi_{5184}(119,\cdot)\)
\(\chi_{5184}(167,\cdot)\)
\(\chi_{5184}(263,\cdot)\)
\(\chi_{5184}(311,\cdot)\)
\(\chi_{5184}(407,\cdot)\)
\(\chi_{5184}(455,\cdot)\)
\(\chi_{5184}(551,\cdot)\)
\(\chi_{5184}(599,\cdot)\)
\(\chi_{5184}(695,\cdot)\)
\(\chi_{5184}(743,\cdot)\)
\(\chi_{5184}(839,\cdot)\)
\(\chi_{5184}(887,\cdot)\)
\(\chi_{5184}(983,\cdot)\)
\(\chi_{5184}(1031,\cdot)\)
\(\chi_{5184}(1127,\cdot)\)
\(\chi_{5184}(1175,\cdot)\)
\(\chi_{5184}(1271,\cdot)\)
\(\chi_{5184}(1319,\cdot)\)
\(\chi_{5184}(1415,\cdot)\)
\(\chi_{5184}(1463,\cdot)\)
\(\chi_{5184}(1559,\cdot)\)
\(\chi_{5184}(1607,\cdot)\)
\(\chi_{5184}(1703,\cdot)\)
\(\chi_{5184}(1751,\cdot)\)
\(\chi_{5184}(1847,\cdot)\)
\(\chi_{5184}(1895,\cdot)\)
\(\chi_{5184}(1991,\cdot)\)
\(\chi_{5184}(2039,\cdot)\)
\(\chi_{5184}(2135,\cdot)\)
...
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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 ]
\((2431,325,1217)\) → \((-1,e\left(\frac{3}{8}\right),e\left(\frac{19}{54}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 5184 }(1271, a) \) |
\(1\) | \(1\) | \(e\left(\frac{101}{216}\right)\) | \(e\left(\frac{95}{108}\right)\) | \(e\left(\frac{205}{216}\right)\) | \(e\left(\frac{95}{216}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{1}{72}\right)\) | \(e\left(\frac{67}{108}\right)\) | \(e\left(\frac{101}{108}\right)\) | \(e\left(\frac{31}{216}\right)\) | \(e\left(\frac{29}{54}\right)\) |
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sage:chi.jacobi_sum(n)
