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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1452, base_ring=CyclotomicField(110))
M = H._module
chi = DirichletCharacter(H, M([0,55,37]))
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pari:[g,chi] = znchar(Mod(557,1452))
\(\chi_{1452}(17,\cdot)\)
\(\chi_{1452}(29,\cdot)\)
\(\chi_{1452}(41,\cdot)\)
\(\chi_{1452}(101,\cdot)\)
\(\chi_{1452}(149,\cdot)\)
\(\chi_{1452}(173,\cdot)\)
\(\chi_{1452}(281,\cdot)\)
\(\chi_{1452}(293,\cdot)\)
\(\chi_{1452}(305,\cdot)\)
\(\chi_{1452}(365,\cdot)\)
\(\chi_{1452}(413,\cdot)\)
\(\chi_{1452}(425,\cdot)\)
\(\chi_{1452}(437,\cdot)\)
\(\chi_{1452}(497,\cdot)\)
\(\chi_{1452}(545,\cdot)\)
\(\chi_{1452}(557,\cdot)\)
\(\chi_{1452}(569,\cdot)\)
\(\chi_{1452}(629,\cdot)\)
\(\chi_{1452}(677,\cdot)\)
\(\chi_{1452}(689,\cdot)\)
\(\chi_{1452}(701,\cdot)\)
\(\chi_{1452}(761,\cdot)\)
\(\chi_{1452}(809,\cdot)\)
\(\chi_{1452}(821,\cdot)\)
\(\chi_{1452}(833,\cdot)\)
\(\chi_{1452}(893,\cdot)\)
\(\chi_{1452}(953,\cdot)\)
\(\chi_{1452}(1025,\cdot)\)
\(\chi_{1452}(1073,\cdot)\)
\(\chi_{1452}(1085,\cdot)\)
...
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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 ]
\((727,485,1333)\) → \((1,-1,e\left(\frac{37}{110}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 1452 }(557, a) \) |
\(1\) | \(1\) | \(e\left(\frac{43}{110}\right)\) | \(e\left(\frac{39}{110}\right)\) | \(e\left(\frac{107}{110}\right)\) | \(e\left(\frac{54}{55}\right)\) | \(e\left(\frac{101}{110}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{43}{55}\right)\) | \(e\left(\frac{12}{55}\right)\) | \(e\left(\frac{51}{55}\right)\) | \(e\left(\frac{41}{55}\right)\) |
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sage:chi.jacobi_sum(n)
