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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([16,55]))
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pari:[g,chi] = znchar(Mod(41,425))
| 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: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
|
\(\chi_{425}(6,\cdot)\)
\(\chi_{425}(11,\cdot)\)
\(\chi_{425}(31,\cdot)\)
\(\chi_{425}(41,\cdot)\)
\(\chi_{425}(46,\cdot)\)
\(\chi_{425}(56,\cdot)\)
\(\chi_{425}(61,\cdot)\)
\(\chi_{425}(71,\cdot)\)
\(\chi_{425}(91,\cdot)\)
\(\chi_{425}(96,\cdot)\)
\(\chi_{425}(116,\cdot)\)
\(\chi_{425}(131,\cdot)\)
\(\chi_{425}(141,\cdot)\)
\(\chi_{425}(146,\cdot)\)
\(\chi_{425}(156,\cdot)\)
\(\chi_{425}(181,\cdot)\)
\(\chi_{425}(211,\cdot)\)
\(\chi_{425}(216,\cdot)\)
\(\chi_{425}(231,\cdot)\)
\(\chi_{425}(241,\cdot)\)
\(\chi_{425}(261,\cdot)\)
\(\chi_{425}(266,\cdot)\)
\(\chi_{425}(286,\cdot)\)
\(\chi_{425}(296,\cdot)\)
\(\chi_{425}(311,\cdot)\)
\(\chi_{425}(316,\cdot)\)
\(\chi_{425}(346,\cdot)\)
\(\chi_{425}(371,\cdot)\)
\(\chi_{425}(381,\cdot)\)
\(\chi_{425}(386,\cdot)\)
...
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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 ]
\((52,326)\) → \((e\left(\frac{1}{5}\right),e\left(\frac{11}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 425 }(41, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{7}{80}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{73}{80}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{1}{80}\right)\) | \(e\left(\frac{59}{80}\right)\) | \(e\left(\frac{11}{20}\right)\) |
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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)
