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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7865, base_ring=CyclotomicField(220))
M = H._module
chi = DirichletCharacter(H, M([55,102,0]))
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pari:[g,chi] = znchar(Mod(2367,7865))
\(\chi_{7865}(183,\cdot)\)
\(\chi_{7865}(222,\cdot)\)
\(\chi_{7865}(248,\cdot)\)
\(\chi_{7865}(508,\cdot)\)
\(\chi_{7865}(547,\cdot)\)
\(\chi_{7865}(612,\cdot)\)
\(\chi_{7865}(677,\cdot)\)
\(\chi_{7865}(833,\cdot)\)
\(\chi_{7865}(898,\cdot)\)
\(\chi_{7865}(937,\cdot)\)
\(\chi_{7865}(963,\cdot)\)
\(\chi_{7865}(1223,\cdot)\)
\(\chi_{7865}(1262,\cdot)\)
\(\chi_{7865}(1327,\cdot)\)
\(\chi_{7865}(1392,\cdot)\)
\(\chi_{7865}(1548,\cdot)\)
\(\chi_{7865}(1652,\cdot)\)
\(\chi_{7865}(1678,\cdot)\)
\(\chi_{7865}(1938,\cdot)\)
\(\chi_{7865}(1977,\cdot)\)
\(\chi_{7865}(2042,\cdot)\)
\(\chi_{7865}(2107,\cdot)\)
\(\chi_{7865}(2263,\cdot)\)
\(\chi_{7865}(2328,\cdot)\)
\(\chi_{7865}(2367,\cdot)\)
\(\chi_{7865}(2692,\cdot)\)
\(\chi_{7865}(2757,\cdot)\)
\(\chi_{7865}(2822,\cdot)\)
\(\chi_{7865}(2978,\cdot)\)
\(\chi_{7865}(3043,\cdot)\)
...
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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 ]
\((3147,3511,1211)\) → \((i,e\left(\frac{51}{110}\right),1)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(14\) | \(16\) |
| \( \chi_{ 7865 }(2367, a) \) |
\(1\) | \(1\) | \(e\left(\frac{157}{220}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{47}{110}\right)\) | \(e\left(\frac{29}{110}\right)\) | \(e\left(\frac{109}{220}\right)\) | \(e\left(\frac{31}{220}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{23}{110}\right)\) | \(e\left(\frac{47}{55}\right)\) |
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sage:chi.jacobi_sum(n)
