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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1452, base_ring=CyclotomicField(110))
M = H._module
chi = DirichletCharacter(H, M([55,0,12]))
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pari:[g,chi] = znchar(Mod(103,1452))
\(\chi_{1452}(31,\cdot)\)
\(\chi_{1452}(91,\cdot)\)
\(\chi_{1452}(103,\cdot)\)
\(\chi_{1452}(115,\cdot)\)
\(\chi_{1452}(163,\cdot)\)
\(\chi_{1452}(223,\cdot)\)
\(\chi_{1452}(235,\cdot)\)
\(\chi_{1452}(247,\cdot)\)
\(\chi_{1452}(295,\cdot)\)
\(\chi_{1452}(355,\cdot)\)
\(\chi_{1452}(367,\cdot)\)
\(\chi_{1452}(379,\cdot)\)
\(\chi_{1452}(427,\cdot)\)
\(\chi_{1452}(499,\cdot)\)
\(\chi_{1452}(559,\cdot)\)
\(\chi_{1452}(619,\cdot)\)
\(\chi_{1452}(631,\cdot)\)
\(\chi_{1452}(643,\cdot)\)
\(\chi_{1452}(691,\cdot)\)
\(\chi_{1452}(751,\cdot)\)
\(\chi_{1452}(763,\cdot)\)
\(\chi_{1452}(775,\cdot)\)
\(\chi_{1452}(823,\cdot)\)
\(\chi_{1452}(883,\cdot)\)
\(\chi_{1452}(895,\cdot)\)
\(\chi_{1452}(907,\cdot)\)
\(\chi_{1452}(955,\cdot)\)
\(\chi_{1452}(1015,\cdot)\)
\(\chi_{1452}(1027,\cdot)\)
\(\chi_{1452}(1039,\cdot)\)
...
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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 ]
\((727,485,1333)\) → \((-1,1,e\left(\frac{6}{55}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 1452 }(103, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{4}{55}\right)\) | \(e\left(\frac{29}{110}\right)\) | \(e\left(\frac{1}{55}\right)\) | \(e\left(\frac{19}{55}\right)\) | \(e\left(\frac{61}{110}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{8}{55}\right)\) | \(e\left(\frac{47}{55}\right)\) | \(e\left(\frac{97}{110}\right)\) | \(e\left(\frac{37}{110}\right)\) |
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sage:chi.jacobi_sum(n)
