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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([4,45]))
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pari:[g,chi] = znchar(Mod(252,425))
| 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)
|
\(\chi_{425}(12,\cdot)\)
\(\chi_{425}(22,\cdot)\)
\(\chi_{425}(23,\cdot)\)
\(\chi_{425}(28,\cdot)\)
\(\chi_{425}(37,\cdot)\)
\(\chi_{425}(58,\cdot)\)
\(\chi_{425}(78,\cdot)\)
\(\chi_{425}(97,\cdot)\)
\(\chi_{425}(108,\cdot)\)
\(\chi_{425}(113,\cdot)\)
\(\chi_{425}(122,\cdot)\)
\(\chi_{425}(163,\cdot)\)
\(\chi_{425}(167,\cdot)\)
\(\chi_{425}(192,\cdot)\)
\(\chi_{425}(198,\cdot)\)
\(\chi_{425}(228,\cdot)\)
\(\chi_{425}(248,\cdot)\)
\(\chi_{425}(252,\cdot)\)
\(\chi_{425}(267,\cdot)\)
\(\chi_{425}(277,\cdot)\)
\(\chi_{425}(278,\cdot)\)
\(\chi_{425}(283,\cdot)\)
\(\chi_{425}(292,\cdot)\)
\(\chi_{425}(313,\cdot)\)
\(\chi_{425}(333,\cdot)\)
\(\chi_{425}(337,\cdot)\)
\(\chi_{425}(352,\cdot)\)
\(\chi_{425}(362,\cdot)\)
\(\chi_{425}(363,\cdot)\)
\(\chi_{425}(377,\cdot)\)
...
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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 ]
\((52,326)\) → \((e\left(\frac{1}{20}\right),e\left(\frac{9}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 425 }(252, a) \) |
\(1\) | \(1\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{73}{80}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{67}{80}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{59}{80}\right)\) | \(e\left(\frac{61}{80}\right)\) | \(e\left(\frac{1}{5}\right)\) |
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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)
