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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(30345, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([0,36,16,33]))
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pari:[g,chi] = znchar(Mod(10768,30345))
\(\chi_{30345}(802,\cdot)\)
\(\chi_{30345}(1948,\cdot)\)
\(\chi_{30345}(2377,\cdot)\)
\(\chi_{30345}(6283,\cdot)\)
\(\chi_{30345}(10768,\cdot)\)
\(\chi_{30345}(12007,\cdot)\)
\(\chi_{30345}(12847,\cdot)\)
\(\chi_{30345}(15103,\cdot)\)
\(\chi_{30345}(16342,\cdot)\)
\(\chi_{30345}(17182,\cdot)\)
\(\chi_{30345}(20848,\cdot)\)
\(\chi_{30345}(22213,\cdot)\)
\(\chi_{30345}(25183,\cdot)\)
\(\chi_{30345}(26548,\cdot)\)
\(\chi_{30345}(26812,\cdot)\)
\(\chi_{30345}(28387,\cdot)\)
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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 ]
\((20231,24277,4336,28036)\) → \((1,-i,e\left(\frac{1}{3}\right),e\left(\frac{11}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(19\) | \(22\) | \(23\) | \(26\) |
| \( \chi_{ 30345 }(10768, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{1}{24}\right)\) |
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sage:chi.jacobi_sum(n)
