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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([63,4]))
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pari:[g,chi] = znchar(Mod(58,245))
| 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: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{245}(2,\cdot)\)
\(\chi_{245}(23,\cdot)\)
\(\chi_{245}(32,\cdot)\)
\(\chi_{245}(37,\cdot)\)
\(\chi_{245}(53,\cdot)\)
\(\chi_{245}(58,\cdot)\)
\(\chi_{245}(72,\cdot)\)
\(\chi_{245}(88,\cdot)\)
\(\chi_{245}(93,\cdot)\)
\(\chi_{245}(102,\cdot)\)
\(\chi_{245}(107,\cdot)\)
\(\chi_{245}(123,\cdot)\)
\(\chi_{245}(137,\cdot)\)
\(\chi_{245}(142,\cdot)\)
\(\chi_{245}(158,\cdot)\)
\(\chi_{245}(163,\cdot)\)
\(\chi_{245}(172,\cdot)\)
\(\chi_{245}(193,\cdot)\)
\(\chi_{245}(198,\cdot)\)
\(\chi_{245}(207,\cdot)\)
\(\chi_{245}(212,\cdot)\)
\(\chi_{245}(228,\cdot)\)
\(\chi_{245}(233,\cdot)\)
\(\chi_{245}(242,\cdot)\)
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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 ]
\((197,101)\) → \((-i,e\left(\frac{1}{21}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
| \( \chi_{ 245 }(58, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{83}{84}\right)\) | \(e\left(\frac{25}{84}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{23}{84}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{20}{21}\right)\) |
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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)
