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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1734, base_ring=CyclotomicField(136))
M = H._module
chi = DirichletCharacter(H, M([0,7]))
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pari:[g,chi] = znchar(Mod(19,1734))
\(\chi_{1734}(19,\cdot)\)
\(\chi_{1734}(25,\cdot)\)
\(\chi_{1734}(43,\cdot)\)
\(\chi_{1734}(49,\cdot)\)
\(\chi_{1734}(121,\cdot)\)
\(\chi_{1734}(127,\cdot)\)
\(\chi_{1734}(145,\cdot)\)
\(\chi_{1734}(151,\cdot)\)
\(\chi_{1734}(223,\cdot)\)
\(\chi_{1734}(229,\cdot)\)
\(\chi_{1734}(247,\cdot)\)
\(\chi_{1734}(253,\cdot)\)
\(\chi_{1734}(325,\cdot)\)
\(\chi_{1734}(331,\cdot)\)
\(\chi_{1734}(349,\cdot)\)
\(\chi_{1734}(355,\cdot)\)
\(\chi_{1734}(427,\cdot)\)
\(\chi_{1734}(433,\cdot)\)
\(\chi_{1734}(451,\cdot)\)
\(\chi_{1734}(457,\cdot)\)
\(\chi_{1734}(529,\cdot)\)
\(\chi_{1734}(535,\cdot)\)
\(\chi_{1734}(553,\cdot)\)
\(\chi_{1734}(559,\cdot)\)
\(\chi_{1734}(631,\cdot)\)
\(\chi_{1734}(637,\cdot)\)
\(\chi_{1734}(655,\cdot)\)
\(\chi_{1734}(661,\cdot)\)
\(\chi_{1734}(739,\cdot)\)
\(\chi_{1734}(763,\cdot)\)
...
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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 ]
\((1157,1159)\) → \((1,e\left(\frac{7}{136}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 1734 }(19, a) \) |
\(1\) | \(1\) | \(e\left(\frac{107}{136}\right)\) | \(e\left(\frac{133}{136}\right)\) | \(e\left(\frac{25}{136}\right)\) | \(e\left(\frac{3}{34}\right)\) | \(e\left(\frac{49}{68}\right)\) | \(e\left(\frac{65}{136}\right)\) | \(e\left(\frac{39}{68}\right)\) | \(e\left(\frac{59}{136}\right)\) | \(e\left(\frac{63}{136}\right)\) | \(e\left(\frac{13}{17}\right)\) |
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sage:chi.jacobi_sum(n)
