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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(525, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,51,20]))
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pari:[g,chi] = znchar(Mod(422,525))
| Modulus: | \(525\) | |
| Conductor: | \(525\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
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| 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: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{525}(2,\cdot)\)
\(\chi_{525}(23,\cdot)\)
\(\chi_{525}(53,\cdot)\)
\(\chi_{525}(128,\cdot)\)
\(\chi_{525}(137,\cdot)\)
\(\chi_{525}(158,\cdot)\)
\(\chi_{525}(212,\cdot)\)
\(\chi_{525}(233,\cdot)\)
\(\chi_{525}(242,\cdot)\)
\(\chi_{525}(263,\cdot)\)
\(\chi_{525}(317,\cdot)\)
\(\chi_{525}(338,\cdot)\)
\(\chi_{525}(347,\cdot)\)
\(\chi_{525}(422,\cdot)\)
\(\chi_{525}(452,\cdot)\)
\(\chi_{525}(473,\cdot)\)
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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 ]
\((176,127,451)\) → \((-1,e\left(\frac{17}{20}\right),e\left(\frac{1}{3}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) | \(22\) | \(23\) |
| \( \chi_{ 525 }(422, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{31}{60}\right)\) |
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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)
