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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(18225, base_ring=CyclotomicField(972))
M = H._module
chi = DirichletCharacter(H, M([80,729]))
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pari:[g,chi] = znchar(Mod(718,18225))
\(\chi_{18225}(7,\cdot)\)
\(\chi_{18225}(43,\cdot)\)
\(\chi_{18225}(157,\cdot)\)
\(\chi_{18225}(193,\cdot)\)
\(\chi_{18225}(232,\cdot)\)
\(\chi_{18225}(268,\cdot)\)
\(\chi_{18225}(382,\cdot)\)
\(\chi_{18225}(418,\cdot)\)
\(\chi_{18225}(457,\cdot)\)
\(\chi_{18225}(493,\cdot)\)
\(\chi_{18225}(607,\cdot)\)
\(\chi_{18225}(643,\cdot)\)
\(\chi_{18225}(682,\cdot)\)
\(\chi_{18225}(718,\cdot)\)
\(\chi_{18225}(832,\cdot)\)
\(\chi_{18225}(868,\cdot)\)
\(\chi_{18225}(907,\cdot)\)
\(\chi_{18225}(943,\cdot)\)
\(\chi_{18225}(1057,\cdot)\)
\(\chi_{18225}(1093,\cdot)\)
\(\chi_{18225}(1132,\cdot)\)
\(\chi_{18225}(1168,\cdot)\)
\(\chi_{18225}(1282,\cdot)\)
\(\chi_{18225}(1318,\cdot)\)
\(\chi_{18225}(1357,\cdot)\)
\(\chi_{18225}(1393,\cdot)\)
\(\chi_{18225}(1507,\cdot)\)
\(\chi_{18225}(1543,\cdot)\)
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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 ]
\((4376,13852)\) → \((e\left(\frac{20}{243}\right),-i)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 18225 }(718, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{809}{972}\right)\) | \(e\left(\frac{323}{486}\right)\) | \(e\left(\frac{173}{972}\right)\) | \(e\left(\frac{161}{324}\right)\) | \(e\left(\frac{71}{243}\right)\) | \(e\left(\frac{559}{972}\right)\) | \(e\left(\frac{5}{486}\right)\) | \(e\left(\frac{80}{243}\right)\) | \(e\left(\frac{151}{324}\right)\) | \(e\left(\frac{109}{162}\right)\) |
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sage:chi.jacobi_sum(n)
