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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(11025, base_ring=CyclotomicField(140))
M = H._module
chi = DirichletCharacter(H, M([0,49,10]))
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pari:[g,chi] = znchar(Mod(6103,11025))
\(\chi_{11025}(433,\cdot)\)
\(\chi_{11025}(622,\cdot)\)
\(\chi_{11025}(748,\cdot)\)
\(\chi_{11025}(937,\cdot)\)
\(\chi_{11025}(1063,\cdot)\)
\(\chi_{11025}(1252,\cdot)\)
\(\chi_{11025}(1378,\cdot)\)
\(\chi_{11025}(2197,\cdot)\)
\(\chi_{11025}(2323,\cdot)\)
\(\chi_{11025}(2512,\cdot)\)
\(\chi_{11025}(2638,\cdot)\)
\(\chi_{11025}(2827,\cdot)\)
\(\chi_{11025}(2953,\cdot)\)
\(\chi_{11025}(3142,\cdot)\)
\(\chi_{11025}(3583,\cdot)\)
\(\chi_{11025}(3898,\cdot)\)
\(\chi_{11025}(4087,\cdot)\)
\(\chi_{11025}(4402,\cdot)\)
\(\chi_{11025}(4528,\cdot)\)
\(\chi_{11025}(4717,\cdot)\)
\(\chi_{11025}(5158,\cdot)\)
\(\chi_{11025}(5347,\cdot)\)
\(\chi_{11025}(5473,\cdot)\)
\(\chi_{11025}(5662,\cdot)\)
\(\chi_{11025}(5788,\cdot)\)
\(\chi_{11025}(6103,\cdot)\)
\(\chi_{11025}(6292,\cdot)\)
\(\chi_{11025}(6733,\cdot)\)
\(\chi_{11025}(6922,\cdot)\)
\(\chi_{11025}(7048,\cdot)\)
...
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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 ]
\((1226,4852,9901)\) → \((1,e\left(\frac{7}{20}\right),e\left(\frac{1}{14}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) | \(22\) | \(23\) |
| \( \chi_{ 11025 }(6103, a) \) |
\(1\) | \(1\) | \(e\left(\frac{29}{140}\right)\) | \(e\left(\frac{29}{70}\right)\) | \(e\left(\frac{87}{140}\right)\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{1}{140}\right)\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{47}{140}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{93}{140}\right)\) | \(e\left(\frac{79}{140}\right)\) |
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sage:chi.jacobi_sum(n)
