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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(735, base_ring=CyclotomicField(84))
M = H._module
chi = DirichletCharacter(H, M([42,63,16]))
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pari:[g,chi] = znchar(Mod(338,735))
| Modulus: | \(735\) | |
| Conductor: | \(735\) |
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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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\(\chi_{735}(2,\cdot)\)
\(\chi_{735}(23,\cdot)\)
\(\chi_{735}(32,\cdot)\)
\(\chi_{735}(53,\cdot)\)
\(\chi_{735}(107,\cdot)\)
\(\chi_{735}(137,\cdot)\)
\(\chi_{735}(158,\cdot)\)
\(\chi_{735}(212,\cdot)\)
\(\chi_{735}(233,\cdot)\)
\(\chi_{735}(242,\cdot)\)
\(\chi_{735}(317,\cdot)\)
\(\chi_{735}(338,\cdot)\)
\(\chi_{735}(347,\cdot)\)
\(\chi_{735}(368,\cdot)\)
\(\chi_{735}(443,\cdot)\)
\(\chi_{735}(452,\cdot)\)
\(\chi_{735}(473,\cdot)\)
\(\chi_{735}(527,\cdot)\)
\(\chi_{735}(548,\cdot)\)
\(\chi_{735}(578,\cdot)\)
\(\chi_{735}(632,\cdot)\)
\(\chi_{735}(653,\cdot)\)
\(\chi_{735}(662,\cdot)\)
\(\chi_{735}(683,\cdot)\)
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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 ]
\((491,442,346)\) → \((-1,-i,e\left(\frac{4}{21}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) | \(22\) | \(23\) |
| \( \chi_{ 735 }(338, a) \) |
\(1\) | \(1\) | \(e\left(\frac{17}{84}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{1}{84}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{83}{84}\right)\) |
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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)
