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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7803, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([16,9]))
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pari:[g,chi] = znchar(Mod(6898,7803))
\(\chi_{7803}(616,\cdot)\)
\(\chi_{7803}(1483,\cdot)\)
\(\chi_{7803}(1696,\cdot)\)
\(\chi_{7803}(2563,\cdot)\)
\(\chi_{7803}(3217,\cdot)\)
\(\chi_{7803}(4084,\cdot)\)
\(\chi_{7803}(4297,\cdot)\)
\(\chi_{7803}(5164,\cdot)\)
\(\chi_{7803}(5818,\cdot)\)
\(\chi_{7803}(6685,\cdot)\)
\(\chi_{7803}(6898,\cdot)\)
\(\chi_{7803}(7765,\cdot)\)
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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 ]
\((2891,2026)\) → \((e\left(\frac{4}{9}\right),i)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 7803 }(6898, a) \) |
\(1\) | \(1\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{7}{9}\right)\) |
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sage:chi.jacobi_sum(n)
