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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([99,5]))
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pari:[g,chi] = znchar(Mod(298,925))
| 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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\(\chi_{925}(17,\cdot)\)
\(\chi_{925}(22,\cdot)\)
\(\chi_{925}(42,\cdot)\)
\(\chi_{925}(72,\cdot)\)
\(\chi_{925}(87,\cdot)\)
\(\chi_{925}(98,\cdot)\)
\(\chi_{925}(113,\cdot)\)
\(\chi_{925}(163,\cdot)\)
\(\chi_{925}(167,\cdot)\)
\(\chi_{925}(202,\cdot)\)
\(\chi_{925}(203,\cdot)\)
\(\chi_{925}(227,\cdot)\)
\(\chi_{925}(272,\cdot)\)
\(\chi_{925}(283,\cdot)\)
\(\chi_{925}(298,\cdot)\)
\(\chi_{925}(328,\cdot)\)
\(\chi_{925}(348,\cdot)\)
\(\chi_{925}(352,\cdot)\)
\(\chi_{925}(353,\cdot)\)
\(\chi_{925}(387,\cdot)\)
\(\chi_{925}(388,\cdot)\)
\(\chi_{925}(392,\cdot)\)
\(\chi_{925}(412,\cdot)\)
\(\chi_{925}(442,\cdot)\)
\(\chi_{925}(483,\cdot)\)
\(\chi_{925}(513,\cdot)\)
\(\chi_{925}(533,\cdot)\)
\(\chi_{925}(537,\cdot)\)
\(\chi_{925}(538,\cdot)\)
\(\chi_{925}(572,\cdot)\)
...
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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 ]
\((852,76)\) → \((e\left(\frac{11}{20}\right),e\left(\frac{1}{36}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 925 }(298, a) \) |
\(1\) | \(1\) | \(e\left(\frac{26}{45}\right)\) | \(e\left(\frac{103}{180}\right)\) | \(e\left(\frac{7}{45}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{13}{90}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{131}{180}\right)\) | \(e\left(\frac{34}{45}\right)\) |
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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)
