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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(15730, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([11,42,22]))
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pari:[g,chi] = znchar(Mod(3717,15730))
\(\chi_{15730}(857,\cdot)\)
\(\chi_{15730}(1143,\cdot)\)
\(\chi_{15730}(2287,\cdot)\)
\(\chi_{15730}(2573,\cdot)\)
\(\chi_{15730}(3717,\cdot)\)
\(\chi_{15730}(4003,\cdot)\)
\(\chi_{15730}(5147,\cdot)\)
\(\chi_{15730}(5433,\cdot)\)
\(\chi_{15730}(6577,\cdot)\)
\(\chi_{15730}(6863,\cdot)\)
\(\chi_{15730}(8007,\cdot)\)
\(\chi_{15730}(8293,\cdot)\)
\(\chi_{15730}(9723,\cdot)\)
\(\chi_{15730}(10867,\cdot)\)
\(\chi_{15730}(11153,\cdot)\)
\(\chi_{15730}(12297,\cdot)\)
\(\chi_{15730}(13727,\cdot)\)
\(\chi_{15730}(14013,\cdot)\)
\(\chi_{15730}(15157,\cdot)\)
\(\chi_{15730}(15443,\cdot)\)
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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 ]
\((3147,3511,1211)\) → \((i,e\left(\frac{21}{22}\right),-1)\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) | \(31\) |
| \( \chi_{ 15730 }(3717, a) \) |
\(1\) | \(1\) | \(-i\) | \(e\left(\frac{19}{44}\right)\) | \(-1\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{25}{44}\right)\) | \(i\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{13}{22}\right)\) |
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sage:chi.jacobi_sum(n)
