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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(44100, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,0,0,1]))
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pari:[g,chi] = znchar(Mod(9901,44100))
\(\chi_{44100}(3601,\cdot)\)
\(\chi_{44100}(7201,\cdot)\)
\(\chi_{44100}(9901,\cdot)\)
\(\chi_{44100}(13501,\cdot)\)
\(\chi_{44100}(19801,\cdot)\)
\(\chi_{44100}(22501,\cdot)\)
\(\chi_{44100}(26101,\cdot)\)
\(\chi_{44100}(28801,\cdot)\)
\(\chi_{44100}(32401,\cdot)\)
\(\chi_{44100}(35101,\cdot)\)
\(\chi_{44100}(38701,\cdot)\)
\(\chi_{44100}(41401,\cdot)\)
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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 ]
\((22051,34301,15877,9901)\) → \((1,1,1,e\left(\frac{1}{42}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 44100 }(9901, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{1}{7}\right)\) |
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sage:chi.jacobi_sum(n)
