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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(231, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([15,20,24]))
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pari:[g,chi] = znchar(Mod(179,231))
| Modulus: | \(231\) | |
| Conductor: | \(231\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
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| Order: | \(30\) |
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sage:chi.multiplicative_order()
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pari:charorder(g,chi)
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| 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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\(\chi_{231}(53,\cdot)\)
\(\chi_{231}(86,\cdot)\)
\(\chi_{231}(137,\cdot)\)
\(\chi_{231}(158,\cdot)\)
\(\chi_{231}(170,\cdot)\)
\(\chi_{231}(179,\cdot)\)
\(\chi_{231}(191,\cdot)\)
\(\chi_{231}(212,\cdot)\)
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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 ]
\((155,199,211)\) → \((-1,e\left(\frac{2}{3}\right),e\left(\frac{4}{5}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(13\) | \(16\) | \(17\) | \(19\) | \(20\) |
| \( \chi_{ 231 }(179, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{3}{10}\right)\) |
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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)
