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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3025, base_ring=CyclotomicField(220))
M = H._module
chi = DirichletCharacter(H, M([55,178]))
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pari:[g,chi] = znchar(Mod(732,3025))
\(\chi_{3025}(7,\cdot)\)
\(\chi_{3025}(18,\cdot)\)
\(\chi_{3025}(57,\cdot)\)
\(\chi_{3025}(68,\cdot)\)
\(\chi_{3025}(107,\cdot)\)
\(\chi_{3025}(182,\cdot)\)
\(\chi_{3025}(193,\cdot)\)
\(\chi_{3025}(293,\cdot)\)
\(\chi_{3025}(332,\cdot)\)
\(\chi_{3025}(343,\cdot)\)
\(\chi_{3025}(382,\cdot)\)
\(\chi_{3025}(393,\cdot)\)
\(\chi_{3025}(468,\cdot)\)
\(\chi_{3025}(557,\cdot)\)
\(\chi_{3025}(568,\cdot)\)
\(\chi_{3025}(607,\cdot)\)
\(\chi_{3025}(618,\cdot)\)
\(\chi_{3025}(657,\cdot)\)
\(\chi_{3025}(668,\cdot)\)
\(\chi_{3025}(732,\cdot)\)
\(\chi_{3025}(743,\cdot)\)
\(\chi_{3025}(832,\cdot)\)
\(\chi_{3025}(843,\cdot)\)
\(\chi_{3025}(882,\cdot)\)
\(\chi_{3025}(893,\cdot)\)
\(\chi_{3025}(932,\cdot)\)
\(\chi_{3025}(943,\cdot)\)
\(\chi_{3025}(1007,\cdot)\)
\(\chi_{3025}(1018,\cdot)\)
\(\chi_{3025}(1107,\cdot)\)
...
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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 ]
\((727,2301)\) → \((i,e\left(\frac{89}{110}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(13\) | \(14\) |
| \( \chi_{ 3025 }(732, a) \) |
\(1\) | \(1\) | \(e\left(\frac{13}{220}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{13}{110}\right)\) | \(e\left(\frac{1}{110}\right)\) | \(e\left(\frac{201}{220}\right)\) | \(e\left(\frac{39}{220}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{103}{220}\right)\) | \(e\left(\frac{107}{110}\right)\) |
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sage:chi.jacobi_sum(n)
