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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2268, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([0,37,45]))
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pari:[g,chi] = znchar(Mod(677,2268))
\(\chi_{2268}(173,\cdot)\)
\(\chi_{2268}(185,\cdot)\)
\(\chi_{2268}(425,\cdot)\)
\(\chi_{2268}(437,\cdot)\)
\(\chi_{2268}(677,\cdot)\)
\(\chi_{2268}(689,\cdot)\)
\(\chi_{2268}(929,\cdot)\)
\(\chi_{2268}(941,\cdot)\)
\(\chi_{2268}(1181,\cdot)\)
\(\chi_{2268}(1193,\cdot)\)
\(\chi_{2268}(1433,\cdot)\)
\(\chi_{2268}(1445,\cdot)\)
\(\chi_{2268}(1685,\cdot)\)
\(\chi_{2268}(1697,\cdot)\)
\(\chi_{2268}(1937,\cdot)\)
\(\chi_{2268}(1949,\cdot)\)
\(\chi_{2268}(2189,\cdot)\)
\(\chi_{2268}(2201,\cdot)\)
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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 ]
\((1135,1541,325)\) → \((1,e\left(\frac{37}{54}\right),e\left(\frac{5}{6}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 2268 }(677, a) \) |
\(1\) | \(1\) | \(e\left(\frac{25}{27}\right)\) | \(e\left(\frac{13}{54}\right)\) | \(e\left(\frac{53}{54}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{11}{54}\right)\) | \(e\left(\frac{23}{27}\right)\) | \(e\left(\frac{19}{54}\right)\) | \(e\left(\frac{29}{54}\right)\) | \(e\left(\frac{4}{9}\right)\) |
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sage:chi.jacobi_sum(n)
