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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1444, base_ring=CyclotomicField(342))
M = H._module
chi = DirichletCharacter(H, M([0,277]))
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pari:[g,chi] = znchar(Mod(185,1444))
\(\chi_{1444}(13,\cdot)\)
\(\chi_{1444}(21,\cdot)\)
\(\chi_{1444}(29,\cdot)\)
\(\chi_{1444}(33,\cdot)\)
\(\chi_{1444}(41,\cdot)\)
\(\chi_{1444}(53,\cdot)\)
\(\chi_{1444}(89,\cdot)\)
\(\chi_{1444}(97,\cdot)\)
\(\chi_{1444}(105,\cdot)\)
\(\chi_{1444}(109,\cdot)\)
\(\chi_{1444}(117,\cdot)\)
\(\chi_{1444}(129,\cdot)\)
\(\chi_{1444}(165,\cdot)\)
\(\chi_{1444}(173,\cdot)\)
\(\chi_{1444}(181,\cdot)\)
\(\chi_{1444}(185,\cdot)\)
\(\chi_{1444}(193,\cdot)\)
\(\chi_{1444}(205,\cdot)\)
\(\chi_{1444}(241,\cdot)\)
\(\chi_{1444}(249,\cdot)\)
\(\chi_{1444}(257,\cdot)\)
\(\chi_{1444}(261,\cdot)\)
\(\chi_{1444}(269,\cdot)\)
\(\chi_{1444}(281,\cdot)\)
\(\chi_{1444}(317,\cdot)\)
\(\chi_{1444}(325,\cdot)\)
\(\chi_{1444}(337,\cdot)\)
\(\chi_{1444}(345,\cdot)\)
\(\chi_{1444}(357,\cdot)\)
\(\chi_{1444}(393,\cdot)\)
...
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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 ]
\((723,1085)\) → \((1,e\left(\frac{277}{342}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(21\) | \(23\) |
| \( \chi_{ 1444 }(185, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{199}{342}\right)\) | \(e\left(\frac{155}{171}\right)\) | \(e\left(\frac{28}{57}\right)\) | \(e\left(\frac{28}{171}\right)\) | \(e\left(\frac{35}{57}\right)\) | \(e\left(\frac{161}{342}\right)\) | \(e\left(\frac{167}{342}\right)\) | \(e\left(\frac{8}{171}\right)\) | \(e\left(\frac{25}{342}\right)\) | \(e\left(\frac{43}{171}\right)\) |
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sage:chi.jacobi_sum(n)
