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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4000, base_ring=CyclotomicField(100))
M = H._module
chi = DirichletCharacter(H, M([50,25,36]))
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pari:[g,chi] = znchar(Mod(3111,4000))
\(\chi_{4000}(71,\cdot)\)
\(\chi_{4000}(231,\cdot)\)
\(\chi_{4000}(311,\cdot)\)
\(\chi_{4000}(391,\cdot)\)
\(\chi_{4000}(471,\cdot)\)
\(\chi_{4000}(631,\cdot)\)
\(\chi_{4000}(711,\cdot)\)
\(\chi_{4000}(791,\cdot)\)
\(\chi_{4000}(871,\cdot)\)
\(\chi_{4000}(1031,\cdot)\)
\(\chi_{4000}(1111,\cdot)\)
\(\chi_{4000}(1191,\cdot)\)
\(\chi_{4000}(1271,\cdot)\)
\(\chi_{4000}(1431,\cdot)\)
\(\chi_{4000}(1511,\cdot)\)
\(\chi_{4000}(1591,\cdot)\)
\(\chi_{4000}(1671,\cdot)\)
\(\chi_{4000}(1831,\cdot)\)
\(\chi_{4000}(1911,\cdot)\)
\(\chi_{4000}(1991,\cdot)\)
\(\chi_{4000}(2071,\cdot)\)
\(\chi_{4000}(2231,\cdot)\)
\(\chi_{4000}(2311,\cdot)\)
\(\chi_{4000}(2391,\cdot)\)
\(\chi_{4000}(2471,\cdot)\)
\(\chi_{4000}(2631,\cdot)\)
\(\chi_{4000}(2711,\cdot)\)
\(\chi_{4000}(2791,\cdot)\)
\(\chi_{4000}(2871,\cdot)\)
\(\chi_{4000}(3031,\cdot)\)
...
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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 ]
\((2751,2501,1377)\) → \((-1,i,e\left(\frac{9}{25}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 4000 }(3111, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{77}{100}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{27}{50}\right)\) | \(e\left(\frac{11}{100}\right)\) | \(e\left(\frac{79}{100}\right)\) | \(e\left(\frac{7}{25}\right)\) | \(e\left(\frac{73}{100}\right)\) | \(e\left(\frac{37}{100}\right)\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{31}{100}\right)\) |
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sage:chi.jacobi_sum(n)
