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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(925, base_ring=CyclotomicField(180))
M = H._module
chi = DirichletCharacter(H, M([81,5]))
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pari:[g,chi] = znchar(Mod(187,925))
| Modulus: | \(925\) | |
| Conductor: | \(925\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(180\) |
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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
|
| Minimal: | yes |
| Parity: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{925}(2,\cdot)\)
\(\chi_{925}(13,\cdot)\)
\(\chi_{925}(52,\cdot)\)
\(\chi_{925}(92,\cdot)\)
\(\chi_{925}(128,\cdot)\)
\(\chi_{925}(133,\cdot)\)
\(\chi_{925}(153,\cdot)\)
\(\chi_{925}(172,\cdot)\)
\(\chi_{925}(183,\cdot)\)
\(\chi_{925}(187,\cdot)\)
\(\chi_{925}(198,\cdot)\)
\(\chi_{925}(217,\cdot)\)
\(\chi_{925}(237,\cdot)\)
\(\chi_{925}(242,\cdot)\)
\(\chi_{925}(277,\cdot)\)
\(\chi_{925}(278,\cdot)\)
\(\chi_{925}(313,\cdot)\)
\(\chi_{925}(338,\cdot)\)
\(\chi_{925}(372,\cdot)\)
\(\chi_{925}(383,\cdot)\)
\(\chi_{925}(402,\cdot)\)
\(\chi_{925}(422,\cdot)\)
\(\chi_{925}(427,\cdot)\)
\(\chi_{925}(462,\cdot)\)
\(\chi_{925}(463,\cdot)\)
\(\chi_{925}(498,\cdot)\)
\(\chi_{925}(503,\cdot)\)
\(\chi_{925}(523,\cdot)\)
\(\chi_{925}(542,\cdot)\)
\(\chi_{925}(553,\cdot)\)
...
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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 ]
\((852,76)\) → \((e\left(\frac{9}{20}\right),e\left(\frac{1}{36}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 925 }(187, a) \) |
\(1\) | \(1\) | \(e\left(\frac{43}{90}\right)\) | \(e\left(\frac{157}{180}\right)\) | \(e\left(\frac{43}{45}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{67}{90}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{149}{180}\right)\) | \(e\left(\frac{77}{90}\right)\) |
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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)
