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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([117,100]))
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pari:[g,chi] = znchar(Mod(292,925))
| 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: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
|
\(\chi_{925}(12,\cdot)\)
\(\chi_{925}(33,\cdot)\)
\(\chi_{925}(53,\cdot)\)
\(\chi_{925}(83,\cdot)\)
\(\chi_{925}(108,\cdot)\)
\(\chi_{925}(123,\cdot)\)
\(\chi_{925}(127,\cdot)\)
\(\chi_{925}(192,\cdot)\)
\(\chi_{925}(197,\cdot)\)
\(\chi_{925}(238,\cdot)\)
\(\chi_{925}(292,\cdot)\)
\(\chi_{925}(303,\cdot)\)
\(\chi_{925}(308,\cdot)\)
\(\chi_{925}(312,\cdot)\)
\(\chi_{925}(342,\cdot)\)
\(\chi_{925}(367,\cdot)\)
\(\chi_{925}(377,\cdot)\)
\(\chi_{925}(403,\cdot)\)
\(\chi_{925}(423,\cdot)\)
\(\chi_{925}(453,\cdot)\)
\(\chi_{925}(477,\cdot)\)
\(\chi_{925}(478,\cdot)\)
\(\chi_{925}(488,\cdot)\)
\(\chi_{925}(497,\cdot)\)
\(\chi_{925}(527,\cdot)\)
\(\chi_{925}(552,\cdot)\)
\(\chi_{925}(562,\cdot)\)
\(\chi_{925}(567,\cdot)\)
\(\chi_{925}(588,\cdot)\)
\(\chi_{925}(608,\cdot)\)
...
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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 ]
\((852,76)\) → \((e\left(\frac{13}{20}\right),e\left(\frac{5}{9}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 925 }(292, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{37}{180}\right)\) | \(e\left(\frac{179}{180}\right)\) | \(e\left(\frac{37}{90}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{89}{90}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{73}{180}\right)\) | \(e\left(\frac{83}{180}\right)\) |
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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)
