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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(175, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([3,20]))
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pari:[g,chi] = znchar(Mod(2,175))
| Modulus: | \(175\) | |
| Conductor: | \(175\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(60\) |
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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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\(\chi_{175}(2,\cdot)\)
\(\chi_{175}(23,\cdot)\)
\(\chi_{175}(37,\cdot)\)
\(\chi_{175}(53,\cdot)\)
\(\chi_{175}(58,\cdot)\)
\(\chi_{175}(67,\cdot)\)
\(\chi_{175}(72,\cdot)\)
\(\chi_{175}(88,\cdot)\)
\(\chi_{175}(102,\cdot)\)
\(\chi_{175}(123,\cdot)\)
\(\chi_{175}(128,\cdot)\)
\(\chi_{175}(137,\cdot)\)
\(\chi_{175}(142,\cdot)\)
\(\chi_{175}(158,\cdot)\)
\(\chi_{175}(163,\cdot)\)
\(\chi_{175}(172,\cdot)\)
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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 ]
\((127,101)\) → \((e\left(\frac{1}{20}\right),e\left(\frac{1}{3}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
| \( \chi_{ 175 }(2, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{13}{15}\right)\) |
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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)
