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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(18225, base_ring=CyclotomicField(972))
M = H._module
chi = DirichletCharacter(H, M([10,243]))
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pari:[g,chi] = znchar(Mod(32,18225))
\(\chi_{18225}(32,\cdot)\)
\(\chi_{18225}(68,\cdot)\)
\(\chi_{18225}(182,\cdot)\)
\(\chi_{18225}(218,\cdot)\)
\(\chi_{18225}(257,\cdot)\)
\(\chi_{18225}(293,\cdot)\)
\(\chi_{18225}(407,\cdot)\)
\(\chi_{18225}(443,\cdot)\)
\(\chi_{18225}(482,\cdot)\)
\(\chi_{18225}(518,\cdot)\)
\(\chi_{18225}(632,\cdot)\)
\(\chi_{18225}(668,\cdot)\)
\(\chi_{18225}(707,\cdot)\)
\(\chi_{18225}(743,\cdot)\)
\(\chi_{18225}(857,\cdot)\)
\(\chi_{18225}(893,\cdot)\)
\(\chi_{18225}(932,\cdot)\)
\(\chi_{18225}(968,\cdot)\)
\(\chi_{18225}(1082,\cdot)\)
\(\chi_{18225}(1118,\cdot)\)
\(\chi_{18225}(1157,\cdot)\)
\(\chi_{18225}(1193,\cdot)\)
\(\chi_{18225}(1307,\cdot)\)
\(\chi_{18225}(1343,\cdot)\)
\(\chi_{18225}(1382,\cdot)\)
\(\chi_{18225}(1418,\cdot)\)
\(\chi_{18225}(1532,\cdot)\)
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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 ]
\((4376,13852)\) → \((e\left(\frac{5}{486}\right),i)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 18225 }(32, a) \) |
\(1\) | \(1\) | \(e\left(\frac{253}{972}\right)\) | \(e\left(\frac{253}{486}\right)\) | \(e\left(\frac{295}{972}\right)\) | \(e\left(\frac{253}{324}\right)\) | \(e\left(\frac{443}{486}\right)\) | \(e\left(\frac{161}{972}\right)\) | \(e\left(\frac{137}{243}\right)\) | \(e\left(\frac{10}{243}\right)\) | \(e\left(\frac{191}{324}\right)\) | \(e\left(\frac{125}{162}\right)\) |
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sage:chi.jacobi_sum(n)
