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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([125,243]))
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pari:[g,chi] = znchar(Mod(1439,18225))
\(\chi_{18225}(44,\cdot)\)
\(\chi_{18225}(89,\cdot)\)
\(\chi_{18225}(179,\cdot)\)
\(\chi_{18225}(314,\cdot)\)
\(\chi_{18225}(359,\cdot)\)
\(\chi_{18225}(494,\cdot)\)
\(\chi_{18225}(584,\cdot)\)
\(\chi_{18225}(629,\cdot)\)
\(\chi_{18225}(719,\cdot)\)
\(\chi_{18225}(764,\cdot)\)
\(\chi_{18225}(854,\cdot)\)
\(\chi_{18225}(989,\cdot)\)
\(\chi_{18225}(1034,\cdot)\)
\(\chi_{18225}(1169,\cdot)\)
\(\chi_{18225}(1259,\cdot)\)
\(\chi_{18225}(1304,\cdot)\)
\(\chi_{18225}(1394,\cdot)\)
\(\chi_{18225}(1439,\cdot)\)
\(\chi_{18225}(1529,\cdot)\)
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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 ]
\((4376,13852)\) → \((e\left(\frac{25}{162}\right),e\left(\frac{3}{10}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 18225 }(1439, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{184}{405}\right)\) | \(e\left(\frac{368}{405}\right)\) | \(e\left(\frac{49}{162}\right)\) | \(e\left(\frac{49}{135}\right)\) | \(e\left(\frac{383}{810}\right)\) | \(e\left(\frac{757}{810}\right)\) | \(e\left(\frac{613}{810}\right)\) | \(e\left(\frac{331}{405}\right)\) | \(e\left(\frac{134}{135}\right)\) | \(e\left(\frac{64}{135}\right)\) |
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sage:chi.jacobi_sum(n)
