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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(14700, base_ring=CyclotomicField(420))
M = H._module
chi = DirichletCharacter(H, M([0,210,399,170]))
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pari:[g,chi] = znchar(Mod(3113,14700))
\(\chi_{14700}(17,\cdot)\)
\(\chi_{14700}(173,\cdot)\)
\(\chi_{14700}(353,\cdot)\)
\(\chi_{14700}(437,\cdot)\)
\(\chi_{14700}(677,\cdot)\)
\(\chi_{14700}(773,\cdot)\)
\(\chi_{14700}(1013,\cdot)\)
\(\chi_{14700}(1277,\cdot)\)
\(\chi_{14700}(1433,\cdot)\)
\(\chi_{14700}(1517,\cdot)\)
\(\chi_{14700}(1613,\cdot)\)
\(\chi_{14700}(1853,\cdot)\)
\(\chi_{14700}(1937,\cdot)\)
\(\chi_{14700}(2033,\cdot)\)
\(\chi_{14700}(2117,\cdot)\)
\(\chi_{14700}(2453,\cdot)\)
\(\chi_{14700}(2537,\cdot)\)
\(\chi_{14700}(2777,\cdot)\)
\(\chi_{14700}(3113,\cdot)\)
\(\chi_{14700}(3197,\cdot)\)
\(\chi_{14700}(3377,\cdot)\)
\(\chi_{14700}(3533,\cdot)\)
\(\chi_{14700}(3617,\cdot)\)
\(\chi_{14700}(3713,\cdot)\)
\(\chi_{14700}(3797,\cdot)\)
\(\chi_{14700}(3953,\cdot)\)
\(\chi_{14700}(4133,\cdot)\)
\(\chi_{14700}(4217,\cdot)\)
\(\chi_{14700}(4373,\cdot)\)
\(\chi_{14700}(4553,\cdot)\)
...
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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 ]
\((7351,4901,1177,9901)\) → \((1,-1,e\left(\frac{19}{20}\right),e\left(\frac{17}{42}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 14700 }(3113, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{187}{210}\right)\) | \(e\left(\frac{57}{140}\right)\) | \(e\left(\frac{407}{420}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{139}{420}\right)\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{211}{420}\right)\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{19}{28}\right)\) |
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sage:chi.jacobi_sum(n)
