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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2700, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,10,9]))
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pari:[g,chi] = znchar(Mod(2057,2700))
\(\chi_{2700}(257,\cdot)\)
\(\chi_{2700}(293,\cdot)\)
\(\chi_{2700}(857,\cdot)\)
\(\chi_{2700}(893,\cdot)\)
\(\chi_{2700}(1157,\cdot)\)
\(\chi_{2700}(1193,\cdot)\)
\(\chi_{2700}(1757,\cdot)\)
\(\chi_{2700}(1793,\cdot)\)
\(\chi_{2700}(2057,\cdot)\)
\(\chi_{2700}(2093,\cdot)\)
\(\chi_{2700}(2657,\cdot)\)
\(\chi_{2700}(2693,\cdot)\)
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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 ]
\((1351,1001,2377)\) → \((1,e\left(\frac{5}{18}\right),i)\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 2700 }(2057, a) \) |
\(1\) | \(1\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{13}{18}\right)\) |
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sage:chi.jacobi_sum(n)
