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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9800, base_ring=CyclotomicField(70))
M = H._module
chi = DirichletCharacter(H, M([35,0,14,5]))
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pari:[g,chi] = znchar(Mod(1791,9800))
\(\chi_{9800}(111,\cdot)\)
\(\chi_{9800}(671,\cdot)\)
\(\chi_{9800}(1231,\cdot)\)
\(\chi_{9800}(1511,\cdot)\)
\(\chi_{9800}(1791,\cdot)\)
\(\chi_{9800}(2071,\cdot)\)
\(\chi_{9800}(2631,\cdot)\)
\(\chi_{9800}(2911,\cdot)\)
\(\chi_{9800}(3191,\cdot)\)
\(\chi_{9800}(3471,\cdot)\)
\(\chi_{9800}(4031,\cdot)\)
\(\chi_{9800}(4591,\cdot)\)
\(\chi_{9800}(4871,\cdot)\)
\(\chi_{9800}(5431,\cdot)\)
\(\chi_{9800}(5711,\cdot)\)
\(\chi_{9800}(5991,\cdot)\)
\(\chi_{9800}(6831,\cdot)\)
\(\chi_{9800}(7111,\cdot)\)
\(\chi_{9800}(7391,\cdot)\)
\(\chi_{9800}(7671,\cdot)\)
\(\chi_{9800}(8511,\cdot)\)
\(\chi_{9800}(8791,\cdot)\)
\(\chi_{9800}(9071,\cdot)\)
\(\chi_{9800}(9631,\cdot)\)
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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 ]
\((7351,4901,1177,5001)\) → \((-1,1,e\left(\frac{1}{5}\right),e\left(\frac{1}{14}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) |
| \( \chi_{ 9800 }(1791, a) \) |
\(1\) | \(1\) | \(e\left(\frac{34}{35}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{39}{70}\right)\) | \(e\left(\frac{11}{70}\right)\) | \(e\left(\frac{27}{70}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{29}{70}\right)\) | \(e\left(\frac{32}{35}\right)\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{3}{5}\right)\) |
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sage:chi.jacobi_sum(n)
