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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([162,175]))
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pari:[g,chi] = znchar(Mod(19,925))
| 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)
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\(\chi_{925}(19,\cdot)\)
\(\chi_{925}(39,\cdot)\)
\(\chi_{925}(54,\cdot)\)
\(\chi_{925}(59,\cdot)\)
\(\chi_{925}(69,\cdot)\)
\(\chi_{925}(79,\cdot)\)
\(\chi_{925}(89,\cdot)\)
\(\chi_{925}(94,\cdot)\)
\(\chi_{925}(109,\cdot)\)
\(\chi_{925}(129,\cdot)\)
\(\chi_{925}(204,\cdot)\)
\(\chi_{925}(209,\cdot)\)
\(\chi_{925}(239,\cdot)\)
\(\chi_{925}(244,\cdot)\)
\(\chi_{925}(254,\cdot)\)
\(\chi_{925}(264,\cdot)\)
\(\chi_{925}(279,\cdot)\)
\(\chi_{925}(294,\cdot)\)
\(\chi_{925}(309,\cdot)\)
\(\chi_{925}(314,\cdot)\)
\(\chi_{925}(389,\cdot)\)
\(\chi_{925}(394,\cdot)\)
\(\chi_{925}(409,\cdot)\)
\(\chi_{925}(429,\cdot)\)
\(\chi_{925}(439,\cdot)\)
\(\chi_{925}(459,\cdot)\)
\(\chi_{925}(464,\cdot)\)
\(\chi_{925}(479,\cdot)\)
\(\chi_{925}(494,\cdot)\)
\(\chi_{925}(579,\cdot)\)
...
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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 ]
\((852,76)\) → \((e\left(\frac{9}{10}\right),e\left(\frac{35}{36}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 925 }(19, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{157}{180}\right)\) | \(e\left(\frac{26}{45}\right)\) | \(e\left(\frac{67}{90}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{7}{45}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{29}{90}\right)\) | \(e\left(\frac{143}{180}\right)\) |
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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)
