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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4600, base_ring=CyclotomicField(110))
M = H._module
chi = DirichletCharacter(H, M([55,0,11,105]))
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pari:[g,chi] = znchar(Mod(1279,4600))
\(\chi_{4600}(79,\cdot)\)
\(\chi_{4600}(159,\cdot)\)
\(\chi_{4600}(319,\cdot)\)
\(\chi_{4600}(359,\cdot)\)
\(\chi_{4600}(479,\cdot)\)
\(\chi_{4600}(559,\cdot)\)
\(\chi_{4600}(839,\cdot)\)
\(\chi_{4600}(879,\cdot)\)
\(\chi_{4600}(1079,\cdot)\)
\(\chi_{4600}(1119,\cdot)\)
\(\chi_{4600}(1239,\cdot)\)
\(\chi_{4600}(1279,\cdot)\)
\(\chi_{4600}(1479,\cdot)\)
\(\chi_{4600}(1719,\cdot)\)
\(\chi_{4600}(1759,\cdot)\)
\(\chi_{4600}(1919,\cdot)\)
\(\chi_{4600}(2039,\cdot)\)
\(\chi_{4600}(2159,\cdot)\)
\(\chi_{4600}(2319,\cdot)\)
\(\chi_{4600}(2639,\cdot)\)
\(\chi_{4600}(2679,\cdot)\)
\(\chi_{4600}(2719,\cdot)\)
\(\chi_{4600}(2839,\cdot)\)
\(\chi_{4600}(2919,\cdot)\)
\(\chi_{4600}(2959,\cdot)\)
\(\chi_{4600}(3079,\cdot)\)
\(\chi_{4600}(3119,\cdot)\)
\(\chi_{4600}(3239,\cdot)\)
\(\chi_{4600}(3319,\cdot)\)
\(\chi_{4600}(3559,\cdot)\)
...
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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 ]
\((1151,2301,2577,1201)\) → \((-1,1,e\left(\frac{1}{10}\right),e\left(\frac{21}{22}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(27\) | \(29\) |
| \( \chi_{ 4600 }(1279, a) \) |
\(1\) | \(1\) | \(e\left(\frac{26}{55}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{52}{55}\right)\) | \(e\left(\frac{38}{55}\right)\) | \(e\left(\frac{29}{110}\right)\) | \(e\left(\frac{54}{55}\right)\) | \(e\left(\frac{34}{55}\right)\) | \(e\left(\frac{67}{110}\right)\) | \(e\left(\frac{23}{55}\right)\) | \(e\left(\frac{21}{55}\right)\) |
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sage:chi.jacobi_sum(n)
