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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3400, base_ring=CyclotomicField(80))
M = H._module
chi = DirichletCharacter(H, M([0,0,16,55]))
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pari:[g,chi] = znchar(Mod(41,3400))
\(\chi_{3400}(41,\cdot)\)
\(\chi_{3400}(241,\cdot)\)
\(\chi_{3400}(481,\cdot)\)
\(\chi_{3400}(521,\cdot)\)
\(\chi_{3400}(641,\cdot)\)
\(\chi_{3400}(721,\cdot)\)
\(\chi_{3400}(881,\cdot)\)
\(\chi_{3400}(921,\cdot)\)
\(\chi_{3400}(1081,\cdot)\)
\(\chi_{3400}(1161,\cdot)\)
\(\chi_{3400}(1281,\cdot)\)
\(\chi_{3400}(1321,\cdot)\)
\(\chi_{3400}(1561,\cdot)\)
\(\chi_{3400}(1761,\cdot)\)
\(\chi_{3400}(1841,\cdot)\)
\(\chi_{3400}(1881,\cdot)\)
\(\chi_{3400}(1961,\cdot)\)
\(\chi_{3400}(2081,\cdot)\)
\(\chi_{3400}(2241,\cdot)\)
\(\chi_{3400}(2281,\cdot)\)
\(\chi_{3400}(2441,\cdot)\)
\(\chi_{3400}(2521,\cdot)\)
\(\chi_{3400}(2561,\cdot)\)
\(\chi_{3400}(2641,\cdot)\)
\(\chi_{3400}(2681,\cdot)\)
\(\chi_{3400}(2761,\cdot)\)
\(\chi_{3400}(2921,\cdot)\)
\(\chi_{3400}(2961,\cdot)\)
\(\chi_{3400}(3121,\cdot)\)
\(\chi_{3400}(3241,\cdot)\)
...
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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 ]
\((2551,1701,2177,1601)\) → \((1,1,e\left(\frac{1}{5}\right),e\left(\frac{11}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 3400 }(41, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{7}{80}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{1}{80}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{41}{80}\right)\) | \(e\left(\frac{21}{80}\right)\) | \(e\left(\frac{27}{80}\right)\) |
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sage:chi.jacobi_sum(n)
