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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(44100, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([15,10,18,25]))
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pari:[g,chi] = znchar(Mod(2371,44100))
\(\chi_{44100}(2371,\cdot)\)
\(\chi_{44100}(5911,\cdot)\)
\(\chi_{44100}(11191,\cdot)\)
\(\chi_{44100}(14731,\cdot)\)
\(\chi_{44100}(20011,\cdot)\)
\(\chi_{44100}(28831,\cdot)\)
\(\chi_{44100}(32371,\cdot)\)
\(\chi_{44100}(41191,\cdot)\)
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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 ]
\((22051,34301,15877,9901)\) → \((-1,e\left(\frac{1}{3}\right),e\left(\frac{3}{5}\right),e\left(\frac{5}{6}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 44100 }(2371, a) \) |
\(1\) | \(1\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{5}{6}\right)\) |
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sage:chi.jacobi_sum(n)
