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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(18225, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([23,0]))
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pari:[g,chi] = znchar(Mod(16901,18225))
\(\chi_{18225}(26,\cdot)\)
\(\chi_{18225}(701,\cdot)\)
\(\chi_{18225}(2051,\cdot)\)
\(\chi_{18225}(2726,\cdot)\)
\(\chi_{18225}(4076,\cdot)\)
\(\chi_{18225}(4751,\cdot)\)
\(\chi_{18225}(6101,\cdot)\)
\(\chi_{18225}(6776,\cdot)\)
\(\chi_{18225}(8126,\cdot)\)
\(\chi_{18225}(8801,\cdot)\)
\(\chi_{18225}(10151,\cdot)\)
\(\chi_{18225}(10826,\cdot)\)
\(\chi_{18225}(12176,\cdot)\)
\(\chi_{18225}(12851,\cdot)\)
\(\chi_{18225}(14201,\cdot)\)
\(\chi_{18225}(14876,\cdot)\)
\(\chi_{18225}(16226,\cdot)\)
\(\chi_{18225}(16901,\cdot)\)
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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 ]
\((4376,13852)\) → \((e\left(\frac{23}{54}\right),1)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 18225 }(16901, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{23}{54}\right)\) | \(e\left(\frac{23}{27}\right)\) | \(e\left(\frac{22}{27}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{29}{54}\right)\) | \(e\left(\frac{11}{27}\right)\) | \(e\left(\frac{13}{54}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{4}{9}\right)\) |
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sage:chi.jacobi_sum(n)
