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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2415, base_ring=CyclotomicField(132))
M = H._module
chi = DirichletCharacter(H, M([0,33,110,36]))
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pari:[g,chi] = znchar(Mod(1342,2415))
\(\chi_{2415}(52,\cdot)\)
\(\chi_{2415}(73,\cdot)\)
\(\chi_{2415}(82,\cdot)\)
\(\chi_{2415}(187,\cdot)\)
\(\chi_{2415}(262,\cdot)\)
\(\chi_{2415}(292,\cdot)\)
\(\chi_{2415}(397,\cdot)\)
\(\chi_{2415}(418,\cdot)\)
\(\chi_{2415}(472,\cdot)\)
\(\chi_{2415}(577,\cdot)\)
\(\chi_{2415}(607,\cdot)\)
\(\chi_{2415}(703,\cdot)\)
\(\chi_{2415}(808,\cdot)\)
\(\chi_{2415}(817,\cdot)\)
\(\chi_{2415}(892,\cdot)\)
\(\chi_{2415}(913,\cdot)\)
\(\chi_{2415}(922,\cdot)\)
\(\chi_{2415}(997,\cdot)\)
\(\chi_{2415}(1018,\cdot)\)
\(\chi_{2415}(1048,\cdot)\)
\(\chi_{2415}(1153,\cdot)\)
\(\chi_{2415}(1228,\cdot)\)
\(\chi_{2415}(1237,\cdot)\)
\(\chi_{2415}(1258,\cdot)\)
\(\chi_{2415}(1342,\cdot)\)
\(\chi_{2415}(1363,\cdot)\)
\(\chi_{2415}(1438,\cdot)\)
\(\chi_{2415}(1522,\cdot)\)
\(\chi_{2415}(1543,\cdot)\)
\(\chi_{2415}(1573,\cdot)\)
...
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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 ]
\((806,967,346,1891)\) → \((1,i,e\left(\frac{5}{6}\right),e\left(\frac{3}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) | \(22\) | \(26\) |
| \( \chi_{ 2415 }(1342, a) \) |
\(1\) | \(1\) | \(e\left(\frac{61}{132}\right)\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{131}{132}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(i\) | \(e\left(\frac{35}{66}\right)\) |
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sage:chi.jacobi_sum(n)
