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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1815, base_ring=CyclotomicField(110))
M = H._module
chi = DirichletCharacter(H, M([0,55,53]))
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pari:[g,chi] = znchar(Mod(514,1815))
\(\chi_{1815}(19,\cdot)\)
\(\chi_{1815}(79,\cdot)\)
\(\chi_{1815}(139,\cdot)\)
\(\chi_{1815}(184,\cdot)\)
\(\chi_{1815}(244,\cdot)\)
\(\chi_{1815}(259,\cdot)\)
\(\chi_{1815}(304,\cdot)\)
\(\chi_{1815}(349,\cdot)\)
\(\chi_{1815}(409,\cdot)\)
\(\chi_{1815}(424,\cdot)\)
\(\chi_{1815}(469,\cdot)\)
\(\chi_{1815}(514,\cdot)\)
\(\chi_{1815}(574,\cdot)\)
\(\chi_{1815}(589,\cdot)\)
\(\chi_{1815}(634,\cdot)\)
\(\chi_{1815}(679,\cdot)\)
\(\chi_{1815}(739,\cdot)\)
\(\chi_{1815}(754,\cdot)\)
\(\chi_{1815}(799,\cdot)\)
\(\chi_{1815}(904,\cdot)\)
\(\chi_{1815}(919,\cdot)\)
\(\chi_{1815}(964,\cdot)\)
\(\chi_{1815}(1009,\cdot)\)
\(\chi_{1815}(1069,\cdot)\)
\(\chi_{1815}(1084,\cdot)\)
\(\chi_{1815}(1174,\cdot)\)
\(\chi_{1815}(1234,\cdot)\)
\(\chi_{1815}(1249,\cdot)\)
\(\chi_{1815}(1294,\cdot)\)
\(\chi_{1815}(1339,\cdot)\)
...
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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 ]
\((1211,727,1696)\) → \((1,-1,e\left(\frac{53}{110}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) | \(23\) |
| \( \chi_{ 1815 }(514, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{54}{55}\right)\) | \(e\left(\frac{53}{55}\right)\) | \(e\left(\frac{48}{55}\right)\) | \(e\left(\frac{52}{55}\right)\) | \(e\left(\frac{9}{55}\right)\) | \(e\left(\frac{47}{55}\right)\) | \(e\left(\frac{51}{55}\right)\) | \(e\left(\frac{6}{55}\right)\) | \(e\left(\frac{109}{110}\right)\) | \(e\left(\frac{5}{22}\right)\) |
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sage:chi.jacobi_sum(n)
