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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(245, base_ring=CyclotomicField(84))
M = H._module
chi = DirichletCharacter(H, M([21,26]))
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pari:[g,chi] = znchar(Mod(157,245))
| Modulus: | \(245\) | |
| Conductor: | \(245\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(84\) |
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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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\(\chi_{245}(3,\cdot)\)
\(\chi_{245}(12,\cdot)\)
\(\chi_{245}(17,\cdot)\)
\(\chi_{245}(33,\cdot)\)
\(\chi_{245}(38,\cdot)\)
\(\chi_{245}(47,\cdot)\)
\(\chi_{245}(52,\cdot)\)
\(\chi_{245}(73,\cdot)\)
\(\chi_{245}(82,\cdot)\)
\(\chi_{245}(87,\cdot)\)
\(\chi_{245}(103,\cdot)\)
\(\chi_{245}(108,\cdot)\)
\(\chi_{245}(122,\cdot)\)
\(\chi_{245}(138,\cdot)\)
\(\chi_{245}(143,\cdot)\)
\(\chi_{245}(152,\cdot)\)
\(\chi_{245}(157,\cdot)\)
\(\chi_{245}(173,\cdot)\)
\(\chi_{245}(187,\cdot)\)
\(\chi_{245}(192,\cdot)\)
\(\chi_{245}(208,\cdot)\)
\(\chi_{245}(213,\cdot)\)
\(\chi_{245}(222,\cdot)\)
\(\chi_{245}(243,\cdot)\)
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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 ]
\((197,101)\) → \((i,e\left(\frac{13}{42}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
| \( \chi_{ 245 }(157, a) \) |
\(1\) | \(1\) | \(e\left(\frac{25}{84}\right)\) | \(e\left(\frac{5}{84}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{55}{84}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{4}{21}\right)\) |
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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)
