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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([0,37,0]))
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pari:[g,chi] = znchar(Mod(29,2268))
\(\chi_{2268}(29,\cdot)\)
\(\chi_{2268}(113,\cdot)\)
\(\chi_{2268}(281,\cdot)\)
\(\chi_{2268}(365,\cdot)\)
\(\chi_{2268}(533,\cdot)\)
\(\chi_{2268}(617,\cdot)\)
\(\chi_{2268}(785,\cdot)\)
\(\chi_{2268}(869,\cdot)\)
\(\chi_{2268}(1037,\cdot)\)
\(\chi_{2268}(1121,\cdot)\)
\(\chi_{2268}(1289,\cdot)\)
\(\chi_{2268}(1373,\cdot)\)
\(\chi_{2268}(1541,\cdot)\)
\(\chi_{2268}(1625,\cdot)\)
\(\chi_{2268}(1793,\cdot)\)
\(\chi_{2268}(1877,\cdot)\)
\(\chi_{2268}(2045,\cdot)\)
\(\chi_{2268}(2129,\cdot)\)
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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 ]
\((1135,1541,325)\) → \((1,e\left(\frac{37}{54}\right),1)\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 2268 }(29, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{41}{54}\right)\) | \(e\left(\frac{49}{54}\right)\) | \(e\left(\frac{13}{27}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{29}{54}\right)\) | \(e\left(\frac{14}{27}\right)\) | \(e\left(\frac{19}{54}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{7}{9}\right)\) |
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sage:chi.jacobi_sum(n)
