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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9025, base_ring=CyclotomicField(684))
M = H._module
chi = DirichletCharacter(H, M([513,554]))
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pari:[g,chi] = znchar(Mod(1268,9025))
\(\chi_{9025}(32,\cdot)\)
\(\chi_{9025}(143,\cdot)\)
\(\chi_{9025}(193,\cdot)\)
\(\chi_{9025}(243,\cdot)\)
\(\chi_{9025}(257,\cdot)\)
\(\chi_{9025}(268,\cdot)\)
\(\chi_{9025}(318,\cdot)\)
\(\chi_{9025}(357,\cdot)\)
\(\chi_{9025}(382,\cdot)\)
\(\chi_{9025}(393,\cdot)\)
\(\chi_{9025}(432,\cdot)\)
\(\chi_{9025}(507,\cdot)\)
\(\chi_{9025}(618,\cdot)\)
\(\chi_{9025}(718,\cdot)\)
\(\chi_{9025}(732,\cdot)\)
\(\chi_{9025}(743,\cdot)\)
\(\chi_{9025}(782,\cdot)\)
\(\chi_{9025}(793,\cdot)\)
\(\chi_{9025}(832,\cdot)\)
\(\chi_{9025}(857,\cdot)\)
\(\chi_{9025}(868,\cdot)\)
\(\chi_{9025}(907,\cdot)\)
\(\chi_{9025}(982,\cdot)\)
\(\chi_{9025}(1093,\cdot)\)
\(\chi_{9025}(1143,\cdot)\)
\(\chi_{9025}(1193,\cdot)\)
\(\chi_{9025}(1207,\cdot)\)
\(\chi_{9025}(1218,\cdot)\)
\(\chi_{9025}(1257,\cdot)\)
\(\chi_{9025}(1268,\cdot)\)
...
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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 ]
\((5777,3251)\) → \((-i,e\left(\frac{277}{342}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 9025 }(1268, a) \) |
\(1\) | \(1\) | \(e\left(\frac{383}{684}\right)\) | \(e\left(\frac{569}{684}\right)\) | \(e\left(\frac{41}{342}\right)\) | \(e\left(\frac{67}{171}\right)\) | \(e\left(\frac{55}{228}\right)\) | \(e\left(\frac{155}{228}\right)\) | \(e\left(\frac{227}{342}\right)\) | \(e\left(\frac{35}{57}\right)\) | \(e\left(\frac{217}{228}\right)\) | \(e\left(\frac{493}{684}\right)\) |
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sage:chi.jacobi_sum(n)
