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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([70,729]))
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pari:[g,chi] = znchar(Mod(68,18225))
\(\chi_{18225}(32,\cdot)\)
\(\chi_{18225}(68,\cdot)\)
\(\chi_{18225}(182,\cdot)\)
\(\chi_{18225}(218,\cdot)\)
\(\chi_{18225}(257,\cdot)\)
\(\chi_{18225}(293,\cdot)\)
\(\chi_{18225}(407,\cdot)\)
\(\chi_{18225}(443,\cdot)\)
\(\chi_{18225}(482,\cdot)\)
\(\chi_{18225}(518,\cdot)\)
\(\chi_{18225}(632,\cdot)\)
\(\chi_{18225}(668,\cdot)\)
\(\chi_{18225}(707,\cdot)\)
\(\chi_{18225}(743,\cdot)\)
\(\chi_{18225}(857,\cdot)\)
\(\chi_{18225}(893,\cdot)\)
\(\chi_{18225}(932,\cdot)\)
\(\chi_{18225}(968,\cdot)\)
\(\chi_{18225}(1082,\cdot)\)
\(\chi_{18225}(1118,\cdot)\)
\(\chi_{18225}(1157,\cdot)\)
\(\chi_{18225}(1193,\cdot)\)
\(\chi_{18225}(1307,\cdot)\)
\(\chi_{18225}(1343,\cdot)\)
\(\chi_{18225}(1382,\cdot)\)
\(\chi_{18225}(1418,\cdot)\)
\(\chi_{18225}(1532,\cdot)\)
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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 ]
\((4376,13852)\) → \((e\left(\frac{35}{486}\right),-i)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 18225 }(68, a) \) |
\(1\) | \(1\) | \(e\left(\frac{799}{972}\right)\) | \(e\left(\frac{313}{486}\right)\) | \(e\left(\frac{121}{972}\right)\) | \(e\left(\frac{151}{324}\right)\) | \(e\left(\frac{185}{486}\right)\) | \(e\left(\frac{155}{972}\right)\) | \(e\left(\frac{230}{243}\right)\) | \(e\left(\frac{70}{243}\right)\) | \(e\left(\frac{41}{324}\right)\) | \(e\left(\frac{65}{162}\right)\) |
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sage:chi.jacobi_sum(n)
