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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2312, base_ring=CyclotomicField(272))
M = H._module
chi = DirichletCharacter(H, M([0,0,15]))
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pari:[g,chi] = znchar(Mod(57,2312))
\(\chi_{2312}(41,\cdot)\)
\(\chi_{2312}(57,\cdot)\)
\(\chi_{2312}(73,\cdot)\)
\(\chi_{2312}(97,\cdot)\)
\(\chi_{2312}(105,\cdot)\)
\(\chi_{2312}(113,\cdot)\)
\(\chi_{2312}(129,\cdot)\)
\(\chi_{2312}(177,\cdot)\)
\(\chi_{2312}(193,\cdot)\)
\(\chi_{2312}(201,\cdot)\)
\(\chi_{2312}(209,\cdot)\)
\(\chi_{2312}(233,\cdot)\)
\(\chi_{2312}(241,\cdot)\)
\(\chi_{2312}(265,\cdot)\)
\(\chi_{2312}(313,\cdot)\)
\(\chi_{2312}(337,\cdot)\)
\(\chi_{2312}(345,\cdot)\)
\(\chi_{2312}(369,\cdot)\)
\(\chi_{2312}(377,\cdot)\)
\(\chi_{2312}(385,\cdot)\)
\(\chi_{2312}(401,\cdot)\)
\(\chi_{2312}(449,\cdot)\)
\(\chi_{2312}(465,\cdot)\)
\(\chi_{2312}(473,\cdot)\)
\(\chi_{2312}(481,\cdot)\)
\(\chi_{2312}(505,\cdot)\)
\(\chi_{2312}(521,\cdot)\)
\(\chi_{2312}(537,\cdot)\)
\(\chi_{2312}(585,\cdot)\)
\(\chi_{2312}(601,\cdot)\)
...
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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 ]
\((1735,1157,1737)\) → \((1,1,e\left(\frac{15}{272}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(19\) | \(21\) | \(23\) |
| \( \chi_{ 2312 }(57, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{15}{272}\right)\) | \(e\left(\frac{171}{272}\right)\) | \(e\left(\frac{149}{272}\right)\) | \(e\left(\frac{15}{136}\right)\) | \(e\left(\frac{73}{272}\right)\) | \(e\left(\frac{55}{68}\right)\) | \(e\left(\frac{93}{136}\right)\) | \(e\left(\frac{105}{136}\right)\) | \(e\left(\frac{41}{68}\right)\) | \(e\left(\frac{81}{272}\right)\) |
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sage:chi.jacobi_sum(n)
