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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6069, base_ring=CyclotomicField(408))
M = H._module
chi = DirichletCharacter(H, M([0,136,315]))
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pari:[g,chi] = znchar(Mod(247,6069))
\(\chi_{6069}(25,\cdot)\)
\(\chi_{6069}(100,\cdot)\)
\(\chi_{6069}(121,\cdot)\)
\(\chi_{6069}(151,\cdot)\)
\(\chi_{6069}(172,\cdot)\)
\(\chi_{6069}(247,\cdot)\)
\(\chi_{6069}(298,\cdot)\)
\(\chi_{6069}(331,\cdot)\)
\(\chi_{6069}(382,\cdot)\)
\(\chi_{6069}(457,\cdot)\)
\(\chi_{6069}(478,\cdot)\)
\(\chi_{6069}(508,\cdot)\)
\(\chi_{6069}(529,\cdot)\)
\(\chi_{6069}(604,\cdot)\)
\(\chi_{6069}(655,\cdot)\)
\(\chi_{6069}(739,\cdot)\)
\(\chi_{6069}(814,\cdot)\)
\(\chi_{6069}(835,\cdot)\)
\(\chi_{6069}(865,\cdot)\)
\(\chi_{6069}(886,\cdot)\)
\(\chi_{6069}(961,\cdot)\)
\(\chi_{6069}(1012,\cdot)\)
\(\chi_{6069}(1045,\cdot)\)
\(\chi_{6069}(1096,\cdot)\)
\(\chi_{6069}(1171,\cdot)\)
\(\chi_{6069}(1192,\cdot)\)
\(\chi_{6069}(1222,\cdot)\)
\(\chi_{6069}(1243,\cdot)\)
\(\chi_{6069}(1318,\cdot)\)
\(\chi_{6069}(1369,\cdot)\)
...
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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 ]
\((2024,4336,3760)\) → \((1,e\left(\frac{1}{3}\right),e\left(\frac{105}{136}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(19\) | \(20\) |
| \( \chi_{ 6069 }(247, a) \) |
\(1\) | \(1\) | \(e\left(\frac{73}{204}\right)\) | \(e\left(\frac{73}{102}\right)\) | \(e\left(\frac{191}{408}\right)\) | \(e\left(\frac{5}{68}\right)\) | \(e\left(\frac{337}{408}\right)\) | \(e\left(\frac{37}{408}\right)\) | \(e\left(\frac{11}{34}\right)\) | \(e\left(\frac{22}{51}\right)\) | \(e\left(\frac{97}{204}\right)\) | \(e\left(\frac{25}{136}\right)\) |
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sage:chi.jacobi_sum(n)
