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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8925, base_ring=CyclotomicField(240))
M = H._module
chi = DirichletCharacter(H, M([0,36,80,135]))
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pari:[g,chi] = znchar(Mod(2683,8925))
\(\chi_{8925}(88,\cdot)\)
\(\chi_{8925}(142,\cdot)\)
\(\chi_{8925}(403,\cdot)\)
\(\chi_{8925}(487,\cdot)\)
\(\chi_{8925}(583,\cdot)\)
\(\chi_{8925}(772,\cdot)\)
\(\chi_{8925}(877,\cdot)\)
\(\chi_{8925}(898,\cdot)\)
\(\chi_{8925}(1108,\cdot)\)
\(\chi_{8925}(1348,\cdot)\)
\(\chi_{8925}(1423,\cdot)\)
\(\chi_{8925}(1537,\cdot)\)
\(\chi_{8925}(1642,\cdot)\)
\(\chi_{8925}(1663,\cdot)\)
\(\chi_{8925}(1873,\cdot)\)
\(\chi_{8925}(1927,\cdot)\)
\(\chi_{8925}(2188,\cdot)\)
\(\chi_{8925}(2272,\cdot)\)
\(\chi_{8925}(2662,\cdot)\)
\(\chi_{8925}(2683,\cdot)\)
\(\chi_{8925}(2692,\cdot)\)
\(\chi_{8925}(3133,\cdot)\)
\(\chi_{8925}(3208,\cdot)\)
\(\chi_{8925}(3292,\cdot)\)
\(\chi_{8925}(3322,\cdot)\)
\(\chi_{8925}(3427,\cdot)\)
\(\chi_{8925}(3448,\cdot)\)
\(\chi_{8925}(3658,\cdot)\)
\(\chi_{8925}(3712,\cdot)\)
\(\chi_{8925}(3973,\cdot)\)
...
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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 ]
\((5951,6427,2551,8401)\) → \((1,e\left(\frac{3}{20}\right),e\left(\frac{1}{3}\right),e\left(\frac{9}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(19\) | \(22\) | \(23\) | \(26\) |
| \( \chi_{ 8925 }(2683, a) \) |
\(1\) | \(1\) | \(e\left(\frac{83}{120}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{161}{240}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{29}{120}\right)\) | \(e\left(\frac{29}{80}\right)\) | \(e\left(\frac{181}{240}\right)\) | \(e\left(\frac{19}{24}\right)\) |
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sage:chi.jacobi_sum(n)
