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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([68,0,45]))
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pari:[g,chi] = znchar(Mod(535,2312))
\(\chi_{2312}(15,\cdot)\)
\(\chi_{2312}(87,\cdot)\)
\(\chi_{2312}(111,\cdot)\)
\(\chi_{2312}(127,\cdot)\)
\(\chi_{2312}(151,\cdot)\)
\(\chi_{2312}(223,\cdot)\)
\(\chi_{2312}(247,\cdot)\)
\(\chi_{2312}(263,\cdot)\)
\(\chi_{2312}(287,\cdot)\)
\(\chi_{2312}(359,\cdot)\)
\(\chi_{2312}(383,\cdot)\)
\(\chi_{2312}(495,\cdot)\)
\(\chi_{2312}(519,\cdot)\)
\(\chi_{2312}(535,\cdot)\)
\(\chi_{2312}(559,\cdot)\)
\(\chi_{2312}(631,\cdot)\)
\(\chi_{2312}(655,\cdot)\)
\(\chi_{2312}(671,\cdot)\)
\(\chi_{2312}(695,\cdot)\)
\(\chi_{2312}(767,\cdot)\)
\(\chi_{2312}(791,\cdot)\)
\(\chi_{2312}(807,\cdot)\)
\(\chi_{2312}(831,\cdot)\)
\(\chi_{2312}(903,\cdot)\)
\(\chi_{2312}(927,\cdot)\)
\(\chi_{2312}(943,\cdot)\)
\(\chi_{2312}(967,\cdot)\)
\(\chi_{2312}(1039,\cdot)\)
\(\chi_{2312}(1063,\cdot)\)
\(\chi_{2312}(1079,\cdot)\)
...
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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 ]
\((1735,1157,1737)\) → \((-1,1,e\left(\frac{45}{136}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(19\) | \(21\) | \(23\) |
| \( \chi_{ 2312 }(535, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{113}{136}\right)\) | \(e\left(\frac{105}{136}\right)\) | \(e\left(\frac{107}{136}\right)\) | \(e\left(\frac{45}{68}\right)\) | \(e\left(\frac{15}{136}\right)\) | \(e\left(\frac{29}{34}\right)\) | \(e\left(\frac{41}{68}\right)\) | \(e\left(\frac{9}{68}\right)\) | \(e\left(\frac{21}{34}\right)\) | \(e\left(\frac{39}{136}\right)\) |
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sage:chi.jacobi_sum(n)
