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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(18225, base_ring=CyclotomicField(540))
M = H._module
chi = DirichletCharacter(H, M([110,459]))
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pari:[g,chi] = znchar(Mod(1997,18225))
\(\chi_{18225}(53,\cdot)\)
\(\chi_{18225}(188,\cdot)\)
\(\chi_{18225}(377,\cdot)\)
\(\chi_{18225}(458,\cdot)\)
\(\chi_{18225}(512,\cdot)\)
\(\chi_{18225}(863,\cdot)\)
\(\chi_{18225}(917,\cdot)\)
\(\chi_{18225}(998,\cdot)\)
\(\chi_{18225}(1187,\cdot)\)
\(\chi_{18225}(1322,\cdot)\)
\(\chi_{18225}(1403,\cdot)\)
\(\chi_{18225}(1592,\cdot)\)
\(\chi_{18225}(1673,\cdot)\)
\(\chi_{18225}(1727,\cdot)\)
\(\chi_{18225}(1808,\cdot)\)
\(\chi_{18225}(1997,\cdot)\)
\(\chi_{18225}(2078,\cdot)\)
\(\chi_{18225}(2213,\cdot)\)
\(\chi_{18225}(2402,\cdot)\)
\(\chi_{18225}(2483,\cdot)\)
\(\chi_{18225}(2537,\cdot)\)
\(\chi_{18225}(2888,\cdot)\)
\(\chi_{18225}(2942,\cdot)\)
\(\chi_{18225}(3023,\cdot)\)
\(\chi_{18225}(3212,\cdot)\)
\(\chi_{18225}(3347,\cdot)\)
\(\chi_{18225}(3428,\cdot)\)
\(\chi_{18225}(3617,\cdot)\)
\(\chi_{18225}(3698,\cdot)\)
\(\chi_{18225}(3752,\cdot)\)
...
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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 ]
\((4376,13852)\) → \((e\left(\frac{11}{54}\right),e\left(\frac{17}{20}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 18225 }(1997, a) \) |
\(1\) | \(1\) | \(e\left(\frac{29}{540}\right)\) | \(e\left(\frac{29}{270}\right)\) | \(e\left(\frac{55}{108}\right)\) | \(e\left(\frac{29}{180}\right)\) | \(e\left(\frac{67}{270}\right)\) | \(e\left(\frac{421}{540}\right)\) | \(e\left(\frac{76}{135}\right)\) | \(e\left(\frac{29}{135}\right)\) | \(e\left(\frac{139}{180}\right)\) | \(e\left(\frac{7}{90}\right)\) |
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sage:chi.jacobi_sum(n)
