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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(18225, base_ring=CyclotomicField(810))
M = H._module
chi = DirichletCharacter(H, M([610,486]))
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pari:[g,chi] = znchar(Mod(1171,18225))
\(\chi_{18225}(46,\cdot)\)
\(\chi_{18225}(91,\cdot)\)
\(\chi_{18225}(181,\cdot)\)
\(\chi_{18225}(316,\cdot)\)
\(\chi_{18225}(361,\cdot)\)
\(\chi_{18225}(496,\cdot)\)
\(\chi_{18225}(586,\cdot)\)
\(\chi_{18225}(631,\cdot)\)
\(\chi_{18225}(721,\cdot)\)
\(\chi_{18225}(766,\cdot)\)
\(\chi_{18225}(856,\cdot)\)
\(\chi_{18225}(991,\cdot)\)
\(\chi_{18225}(1036,\cdot)\)
\(\chi_{18225}(1171,\cdot)\)
\(\chi_{18225}(1261,\cdot)\)
\(\chi_{18225}(1306,\cdot)\)
\(\chi_{18225}(1396,\cdot)\)
\(\chi_{18225}(1441,\cdot)\)
\(\chi_{18225}(1531,\cdot)\)
\(\chi_{18225}(1666,\cdot)\)
\(\chi_{18225}(1711,\cdot)\)
\(\chi_{18225}(1846,\cdot)\)
\(\chi_{18225}(1936,\cdot)\)
\(\chi_{18225}(1981,\cdot)\)
\(\chi_{18225}(2071,\cdot)\)
\(\chi_{18225}(2116,\cdot)\)
\(\chi_{18225}(2206,\cdot)\)
\(\chi_{18225}(2341,\cdot)\)
\(\chi_{18225}(2386,\cdot)\)
\(\chi_{18225}(2521,\cdot)\)
...
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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 ]
\((4376,13852)\) → \((e\left(\frac{61}{81}\right),e\left(\frac{3}{5}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 18225 }(1171, a) \) |
\(1\) | \(1\) | \(e\left(\frac{143}{405}\right)\) | \(e\left(\frac{286}{405}\right)\) | \(e\left(\frac{58}{81}\right)\) | \(e\left(\frac{8}{135}\right)\) | \(e\left(\frac{293}{405}\right)\) | \(e\left(\frac{172}{405}\right)\) | \(e\left(\frac{28}{405}\right)\) | \(e\left(\frac{167}{405}\right)\) | \(e\left(\frac{88}{135}\right)\) | \(e\left(\frac{38}{135}\right)\) |
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sage:chi.jacobi_sum(n)
