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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(40310, base_ring=CyclotomicField(276))
M = H._module
chi = DirichletCharacter(H, M([0,69,64]))
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pari:[g,chi] = znchar(Mod(41,40310))
\(\chi_{40310}(41,\cdot)\)
\(\chi_{40310}(1201,\cdot)\)
\(\chi_{40310}(1781,\cdot)\)
\(\chi_{40310}(1931,\cdot)\)
\(\chi_{40310}(2221,\cdot)\)
\(\chi_{40310}(2361,\cdot)\)
\(\chi_{40310}(2511,\cdot)\)
\(\chi_{40310}(3521,\cdot)\)
\(\chi_{40310}(3961,\cdot)\)
\(\chi_{40310}(4251,\cdot)\)
\(\chi_{40310}(5121,\cdot)\)
\(\chi_{40310}(5261,\cdot)\)
\(\chi_{40310}(6571,\cdot)\)
\(\chi_{40310}(7001,\cdot)\)
\(\chi_{40310}(7731,\cdot)\)
\(\chi_{40310}(8161,\cdot)\)
\(\chi_{40310}(8601,\cdot)\)
\(\chi_{40310}(9181,\cdot)\)
\(\chi_{40310}(9611,\cdot)\)
\(\chi_{40310}(9761,\cdot)\)
\(\chi_{40310}(10631,\cdot)\)
\(\chi_{40310}(12081,\cdot)\)
\(\chi_{40310}(12801,\cdot)\)
\(\chi_{40310}(12951,\cdot)\)
\(\chi_{40310}(13091,\cdot)\)
\(\chi_{40310}(13381,\cdot)\)
\(\chi_{40310}(13671,\cdot)\)
\(\chi_{40310}(15271,\cdot)\)
\(\chi_{40310}(15411,\cdot)\)
\(\chi_{40310}(15851,\cdot)\)
...
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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 ]
\((24187,19461,16821)\) → \((1,i,e\left(\frac{16}{69}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 40310 }(41, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{209}{276}\right)\) | \(e\left(\frac{41}{69}\right)\) | \(e\left(\frac{71}{138}\right)\) | \(e\left(\frac{241}{276}\right)\) | \(e\left(\frac{47}{138}\right)\) | \(e\left(\frac{17}{276}\right)\) | \(e\left(\frac{109}{276}\right)\) | \(e\left(\frac{97}{276}\right)\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{25}{92}\right)\) |
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sage:chi.jacobi_sum(n)
