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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2300, base_ring=CyclotomicField(220))
M = H._module
chi = DirichletCharacter(H, M([0,209,140]))
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pari:[g,chi] = znchar(Mod(13,2300))
\(\chi_{2300}(13,\cdot)\)
\(\chi_{2300}(73,\cdot)\)
\(\chi_{2300}(77,\cdot)\)
\(\chi_{2300}(117,\cdot)\)
\(\chi_{2300}(133,\cdot)\)
\(\chi_{2300}(173,\cdot)\)
\(\chi_{2300}(177,\cdot)\)
\(\chi_{2300}(197,\cdot)\)
\(\chi_{2300}(213,\cdot)\)
\(\chi_{2300}(233,\cdot)\)
\(\chi_{2300}(317,\cdot)\)
\(\chi_{2300}(353,\cdot)\)
\(\chi_{2300}(377,\cdot)\)
\(\chi_{2300}(397,\cdot)\)
\(\chi_{2300}(417,\cdot)\)
\(\chi_{2300}(453,\cdot)\)
\(\chi_{2300}(473,\cdot)\)
\(\chi_{2300}(533,\cdot)\)
\(\chi_{2300}(537,\cdot)\)
\(\chi_{2300}(577,\cdot)\)
\(\chi_{2300}(633,\cdot)\)
\(\chi_{2300}(637,\cdot)\)
\(\chi_{2300}(653,\cdot)\)
\(\chi_{2300}(673,\cdot)\)
\(\chi_{2300}(717,\cdot)\)
\(\chi_{2300}(777,\cdot)\)
\(\chi_{2300}(813,\cdot)\)
\(\chi_{2300}(817,\cdot)\)
\(\chi_{2300}(837,\cdot)\)
\(\chi_{2300}(853,\cdot)\)
...
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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 ]
\((1151,277,1201)\) → \((1,e\left(\frac{19}{20}\right),e\left(\frac{7}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(27\) | \(29\) |
| \( \chi_{ 2300 }(13, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{183}{220}\right)\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{73}{110}\right)\) | \(e\left(\frac{51}{55}\right)\) | \(e\left(\frac{211}{220}\right)\) | \(e\left(\frac{177}{220}\right)\) | \(e\left(\frac{71}{110}\right)\) | \(e\left(\frac{37}{55}\right)\) | \(e\left(\frac{109}{220}\right)\) | \(e\left(\frac{39}{110}\right)\) |
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sage:chi.jacobi_sum(n)
