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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([25,25,9]))
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pari:[g,chi] = znchar(Mod(19,1000))
| 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: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{1000}(19,\cdot)\)
\(\chi_{1000}(59,\cdot)\)
\(\chi_{1000}(139,\cdot)\)
\(\chi_{1000}(179,\cdot)\)
\(\chi_{1000}(219,\cdot)\)
\(\chi_{1000}(259,\cdot)\)
\(\chi_{1000}(339,\cdot)\)
\(\chi_{1000}(379,\cdot)\)
\(\chi_{1000}(419,\cdot)\)
\(\chi_{1000}(459,\cdot)\)
\(\chi_{1000}(539,\cdot)\)
\(\chi_{1000}(579,\cdot)\)
\(\chi_{1000}(619,\cdot)\)
\(\chi_{1000}(659,\cdot)\)
\(\chi_{1000}(739,\cdot)\)
\(\chi_{1000}(779,\cdot)\)
\(\chi_{1000}(819,\cdot)\)
\(\chi_{1000}(859,\cdot)\)
\(\chi_{1000}(939,\cdot)\)
\(\chi_{1000}(979,\cdot)\)
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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 ]
\((751,501,377)\) → \((-1,-1,e\left(\frac{9}{50}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 1000 }(19, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{13}{50}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{13}{25}\right)\) | \(e\left(\frac{17}{25}\right)\) | \(e\left(\frac{13}{25}\right)\) | \(e\left(\frac{7}{50}\right)\) | \(e\left(\frac{6}{25}\right)\) | \(e\left(\frac{3}{50}\right)\) | \(e\left(\frac{2}{25}\right)\) | \(e\left(\frac{39}{50}\right)\) |
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sage:chi.jacobi_sum(n)
