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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([0,99,44,30]))
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pari:[g,chi] = znchar(Mod(1423,2415))
\(\chi_{2415}(37,\cdot)\)
\(\chi_{2415}(67,\cdot)\)
\(\chi_{2415}(88,\cdot)\)
\(\chi_{2415}(172,\cdot)\)
\(\chi_{2415}(247,\cdot)\)
\(\chi_{2415}(268,\cdot)\)
\(\chi_{2415}(352,\cdot)\)
\(\chi_{2415}(373,\cdot)\)
\(\chi_{2415}(382,\cdot)\)
\(\chi_{2415}(457,\cdot)\)
\(\chi_{2415}(562,\cdot)\)
\(\chi_{2415}(592,\cdot)\)
\(\chi_{2415}(613,\cdot)\)
\(\chi_{2415}(688,\cdot)\)
\(\chi_{2415}(697,\cdot)\)
\(\chi_{2415}(718,\cdot)\)
\(\chi_{2415}(793,\cdot)\)
\(\chi_{2415}(802,\cdot)\)
\(\chi_{2415}(907,\cdot)\)
\(\chi_{2415}(1003,\cdot)\)
\(\chi_{2415}(1033,\cdot)\)
\(\chi_{2415}(1138,\cdot)\)
\(\chi_{2415}(1192,\cdot)\)
\(\chi_{2415}(1213,\cdot)\)
\(\chi_{2415}(1318,\cdot)\)
\(\chi_{2415}(1348,\cdot)\)
\(\chi_{2415}(1423,\cdot)\)
\(\chi_{2415}(1528,\cdot)\)
\(\chi_{2415}(1537,\cdot)\)
\(\chi_{2415}(1558,\cdot)\)
...
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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 ]
\((806,967,346,1891)\) → \((1,-i,e\left(\frac{1}{3}\right),e\left(\frac{5}{22}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) | \(22\) | \(26\) |
| \( \chi_{ 2415 }(1423, a) \) |
\(1\) | \(1\) | \(e\left(\frac{115}{132}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{27}{44}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{89}{132}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(i\) | \(e\left(\frac{10}{33}\right)\) |
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sage:chi.jacobi_sum(n)
