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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(605, base_ring=CyclotomicField(110))
M = H._module
chi = DirichletCharacter(H, M([55,71]))
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pari:[g,chi] = znchar(Mod(409,605))
| Modulus: | \(605\) | |
| Conductor: | \(605\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(110\) |
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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)
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\(\chi_{605}(19,\cdot)\)
\(\chi_{605}(24,\cdot)\)
\(\chi_{605}(29,\cdot)\)
\(\chi_{605}(39,\cdot)\)
\(\chi_{605}(74,\cdot)\)
\(\chi_{605}(79,\cdot)\)
\(\chi_{605}(84,\cdot)\)
\(\chi_{605}(129,\cdot)\)
\(\chi_{605}(134,\cdot)\)
\(\chi_{605}(139,\cdot)\)
\(\chi_{605}(149,\cdot)\)
\(\chi_{605}(184,\cdot)\)
\(\chi_{605}(189,\cdot)\)
\(\chi_{605}(194,\cdot)\)
\(\chi_{605}(204,\cdot)\)
\(\chi_{605}(244,\cdot)\)
\(\chi_{605}(249,\cdot)\)
\(\chi_{605}(259,\cdot)\)
\(\chi_{605}(294,\cdot)\)
\(\chi_{605}(299,\cdot)\)
\(\chi_{605}(304,\cdot)\)
\(\chi_{605}(314,\cdot)\)
\(\chi_{605}(349,\cdot)\)
\(\chi_{605}(359,\cdot)\)
\(\chi_{605}(369,\cdot)\)
\(\chi_{605}(404,\cdot)\)
\(\chi_{605}(409,\cdot)\)
\(\chi_{605}(414,\cdot)\)
\(\chi_{605}(424,\cdot)\)
\(\chi_{605}(459,\cdot)\)
...
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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 ]
\((122,486)\) → \((-1,e\left(\frac{71}{110}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(13\) | \(14\) |
| \( \chi_{ 605 }(409, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{8}{55}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{16}{55}\right)\) | \(e\left(\frac{49}{110}\right)\) | \(e\left(\frac{1}{55}\right)\) | \(e\left(\frac{24}{55}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{38}{55}\right)\) | \(e\left(\frac{9}{55}\right)\) |
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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)
