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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1000, base_ring=CyclotomicField(50))
M = H._module
chi = DirichletCharacter(H, M([0,25,1]))
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pari:[g,chi] = znchar(Mod(629,1000))
| Modulus: | \(1000\) | |
| Conductor: | \(1000\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(50\) |
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sage:chi.multiplicative_order()
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pari:charorder(g,chi)
|
| Real: | no |
| Primitive: | yes |
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sage:chi.is_primitive()
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pari:#znconreyconductor(g,chi)==1
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| Minimal: | yes |
| Parity: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{1000}(29,\cdot)\)
\(\chi_{1000}(69,\cdot)\)
\(\chi_{1000}(109,\cdot)\)
\(\chi_{1000}(189,\cdot)\)
\(\chi_{1000}(229,\cdot)\)
\(\chi_{1000}(269,\cdot)\)
\(\chi_{1000}(309,\cdot)\)
\(\chi_{1000}(389,\cdot)\)
\(\chi_{1000}(429,\cdot)\)
\(\chi_{1000}(469,\cdot)\)
\(\chi_{1000}(509,\cdot)\)
\(\chi_{1000}(589,\cdot)\)
\(\chi_{1000}(629,\cdot)\)
\(\chi_{1000}(669,\cdot)\)
\(\chi_{1000}(709,\cdot)\)
\(\chi_{1000}(789,\cdot)\)
\(\chi_{1000}(829,\cdot)\)
\(\chi_{1000}(869,\cdot)\)
\(\chi_{1000}(909,\cdot)\)
\(\chi_{1000}(989,\cdot)\)
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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 ]
\((751,501,377)\) → \((1,-1,e\left(\frac{1}{50}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 1000 }(629, a) \) |
\(1\) | \(1\) | \(e\left(\frac{16}{25}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{7}{25}\right)\) | \(e\left(\frac{1}{50}\right)\) | \(e\left(\frac{7}{25}\right)\) | \(e\left(\frac{23}{50}\right)\) | \(e\left(\frac{43}{50}\right)\) | \(e\left(\frac{17}{50}\right)\) | \(e\left(\frac{31}{50}\right)\) | \(e\left(\frac{23}{25}\right)\) |
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sage:chi.jacobi_sum(n)
