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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2312, base_ring=CyclotomicField(136))
M = H._module
chi = DirichletCharacter(H, M([0,0,1]))
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pari:[g,chi] = znchar(Mod(9,2312))
\(\chi_{2312}(9,\cdot)\)
\(\chi_{2312}(25,\cdot)\)
\(\chi_{2312}(49,\cdot)\)
\(\chi_{2312}(121,\cdot)\)
\(\chi_{2312}(145,\cdot)\)
\(\chi_{2312}(161,\cdot)\)
\(\chi_{2312}(185,\cdot)\)
\(\chi_{2312}(257,\cdot)\)
\(\chi_{2312}(281,\cdot)\)
\(\chi_{2312}(297,\cdot)\)
\(\chi_{2312}(321,\cdot)\)
\(\chi_{2312}(393,\cdot)\)
\(\chi_{2312}(417,\cdot)\)
\(\chi_{2312}(433,\cdot)\)
\(\chi_{2312}(457,\cdot)\)
\(\chi_{2312}(529,\cdot)\)
\(\chi_{2312}(553,\cdot)\)
\(\chi_{2312}(569,\cdot)\)
\(\chi_{2312}(593,\cdot)\)
\(\chi_{2312}(665,\cdot)\)
\(\chi_{2312}(689,\cdot)\)
\(\chi_{2312}(705,\cdot)\)
\(\chi_{2312}(729,\cdot)\)
\(\chi_{2312}(801,\cdot)\)
\(\chi_{2312}(825,\cdot)\)
\(\chi_{2312}(841,\cdot)\)
\(\chi_{2312}(865,\cdot)\)
\(\chi_{2312}(937,\cdot)\)
\(\chi_{2312}(961,\cdot)\)
\(\chi_{2312}(1073,\cdot)\)
...
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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 ]
\((1735,1157,1737)\) → \((1,1,e\left(\frac{1}{136}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(19\) | \(21\) | \(23\) |
| \( \chi_{ 2312 }(9, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{136}\right)\) | \(e\left(\frac{93}{136}\right)\) | \(e\left(\frac{19}{136}\right)\) | \(e\left(\frac{1}{68}\right)\) | \(e\left(\frac{23}{136}\right)\) | \(e\left(\frac{15}{34}\right)\) | \(e\left(\frac{47}{68}\right)\) | \(e\left(\frac{7}{68}\right)\) | \(e\left(\frac{5}{34}\right)\) | \(e\left(\frac{87}{136}\right)\) |
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sage:chi.jacobi_sum(n)
