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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(425, base_ring=CyclotomicField(80))
M = H._module
chi = DirichletCharacter(H, M([76,25]))
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pari:[g,chi] = znchar(Mod(413,425))
| Modulus: | \(425\) | |
| Conductor: | \(425\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(80\) |
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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
|
| Minimal: | yes |
| Parity: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
|
\(\chi_{425}(3,\cdot)\)
\(\chi_{425}(27,\cdot)\)
\(\chi_{425}(48,\cdot)\)
\(\chi_{425}(62,\cdot)\)
\(\chi_{425}(63,\cdot)\)
\(\chi_{425}(73,\cdot)\)
\(\chi_{425}(88,\cdot)\)
\(\chi_{425}(92,\cdot)\)
\(\chi_{425}(112,\cdot)\)
\(\chi_{425}(133,\cdot)\)
\(\chi_{425}(142,\cdot)\)
\(\chi_{425}(147,\cdot)\)
\(\chi_{425}(148,\cdot)\)
\(\chi_{425}(158,\cdot)\)
\(\chi_{425}(173,\cdot)\)
\(\chi_{425}(177,\cdot)\)
\(\chi_{425}(197,\cdot)\)
\(\chi_{425}(227,\cdot)\)
\(\chi_{425}(233,\cdot)\)
\(\chi_{425}(258,\cdot)\)
\(\chi_{425}(262,\cdot)\)
\(\chi_{425}(303,\cdot)\)
\(\chi_{425}(312,\cdot)\)
\(\chi_{425}(317,\cdot)\)
\(\chi_{425}(328,\cdot)\)
\(\chi_{425}(347,\cdot)\)
\(\chi_{425}(367,\cdot)\)
\(\chi_{425}(388,\cdot)\)
\(\chi_{425}(397,\cdot)\)
\(\chi_{425}(402,\cdot)\)
...
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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 ]
\((52,326)\) → \((e\left(\frac{19}{20}\right),e\left(\frac{5}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 425 }(413, a) \) |
\(1\) | \(1\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{77}{80}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{23}{80}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{31}{80}\right)\) | \(e\left(\frac{49}{80}\right)\) | \(e\left(\frac{3}{10}\right)\) |
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sage:chi.jacobi_sum(n)
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sage:chi.gauss_sum(a)
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pari:znchargauss(g,chi,a)
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sage:chi.jacobi_sum(n)
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sage:chi.kloosterman_sum(a,b)
