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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(605, base_ring=CyclotomicField(220))
M = H._module
chi = DirichletCharacter(H, M([55,178]))
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pari:[g,chi] = znchar(Mod(127,605))
| Modulus: | \(605\) | |
| Conductor: | \(605\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(220\) |
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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)
|
\(\chi_{605}(2,\cdot)\)
\(\chi_{605}(7,\cdot)\)
\(\chi_{605}(8,\cdot)\)
\(\chi_{605}(13,\cdot)\)
\(\chi_{605}(17,\cdot)\)
\(\chi_{605}(18,\cdot)\)
\(\chi_{605}(28,\cdot)\)
\(\chi_{605}(52,\cdot)\)
\(\chi_{605}(57,\cdot)\)
\(\chi_{605}(62,\cdot)\)
\(\chi_{605}(63,\cdot)\)
\(\chi_{605}(68,\cdot)\)
\(\chi_{605}(72,\cdot)\)
\(\chi_{605}(73,\cdot)\)
\(\chi_{605}(83,\cdot)\)
\(\chi_{605}(107,\cdot)\)
\(\chi_{605}(117,\cdot)\)
\(\chi_{605}(123,\cdot)\)
\(\chi_{605}(127,\cdot)\)
\(\chi_{605}(128,\cdot)\)
\(\chi_{605}(138,\cdot)\)
\(\chi_{605}(162,\cdot)\)
\(\chi_{605}(167,\cdot)\)
\(\chi_{605}(172,\cdot)\)
\(\chi_{605}(173,\cdot)\)
\(\chi_{605}(178,\cdot)\)
\(\chi_{605}(182,\cdot)\)
\(\chi_{605}(183,\cdot)\)
\(\chi_{605}(193,\cdot)\)
\(\chi_{605}(217,\cdot)\)
...
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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 ]
\((122,486)\) → \((i,e\left(\frac{89}{110}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(13\) | \(14\) |
| \( \chi_{ 605 }(127, a) \) |
\(1\) | \(1\) | \(e\left(\frac{13}{220}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{13}{110}\right)\) | \(e\left(\frac{1}{110}\right)\) | \(e\left(\frac{201}{220}\right)\) | \(e\left(\frac{39}{220}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{103}{220}\right)\) | \(e\left(\frac{107}{110}\right)\) |
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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)
