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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3800, base_ring=CyclotomicField(90))
M = H._module
chi = DirichletCharacter(H, M([45,0,36,35]))
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pari:[g,chi] = znchar(Mod(831,3800))
\(\chi_{3800}(71,\cdot)\)
\(\chi_{3800}(231,\cdot)\)
\(\chi_{3800}(431,\cdot)\)
\(\chi_{3800}(471,\cdot)\)
\(\chi_{3800}(591,\cdot)\)
\(\chi_{3800}(831,\cdot)\)
\(\chi_{3800}(991,\cdot)\)
\(\chi_{3800}(1191,\cdot)\)
\(\chi_{3800}(1231,\cdot)\)
\(\chi_{3800}(1511,\cdot)\)
\(\chi_{3800}(1591,\cdot)\)
\(\chi_{3800}(1991,\cdot)\)
\(\chi_{3800}(2111,\cdot)\)
\(\chi_{3800}(2271,\cdot)\)
\(\chi_{3800}(2511,\cdot)\)
\(\chi_{3800}(2711,\cdot)\)
\(\chi_{3800}(2871,\cdot)\)
\(\chi_{3800}(3031,\cdot)\)
\(\chi_{3800}(3111,\cdot)\)
\(\chi_{3800}(3271,\cdot)\)
\(\chi_{3800}(3471,\cdot)\)
\(\chi_{3800}(3511,\cdot)\)
\(\chi_{3800}(3631,\cdot)\)
\(\chi_{3800}(3791,\cdot)\)
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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 ]
\((951,1901,1977,401)\) → \((-1,1,e\left(\frac{2}{5}\right),e\left(\frac{7}{18}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 3800 }(831, a) \) |
\(1\) | \(1\) | \(e\left(\frac{16}{45}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{32}{45}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{49}{90}\right)\) | \(e\left(\frac{4}{45}\right)\) | \(e\left(\frac{17}{90}\right)\) | \(e\left(\frac{61}{90}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{37}{90}\right)\) |
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sage:chi.jacobi_sum(n)
