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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(925, base_ring=CyclotomicField(90))
M = H._module
chi = DirichletCharacter(H, M([18,10]))
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pari:[g,chi] = znchar(Mod(16,925))
| Modulus: | \(925\) | |
| Conductor: | \(925\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(45\) |
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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
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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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\(\chi_{925}(16,\cdot)\)
\(\chi_{925}(46,\cdot)\)
\(\chi_{925}(71,\cdot)\)
\(\chi_{925}(81,\cdot)\)
\(\chi_{925}(86,\cdot)\)
\(\chi_{925}(181,\cdot)\)
\(\chi_{925}(231,\cdot)\)
\(\chi_{925}(256,\cdot)\)
\(\chi_{925}(266,\cdot)\)
\(\chi_{925}(271,\cdot)\)
\(\chi_{925}(366,\cdot)\)
\(\chi_{925}(386,\cdot)\)
\(\chi_{925}(416,\cdot)\)
\(\chi_{925}(441,\cdot)\)
\(\chi_{925}(456,\cdot)\)
\(\chi_{925}(571,\cdot)\)
\(\chi_{925}(636,\cdot)\)
\(\chi_{925}(641,\cdot)\)
\(\chi_{925}(736,\cdot)\)
\(\chi_{925}(756,\cdot)\)
\(\chi_{925}(786,\cdot)\)
\(\chi_{925}(811,\cdot)\)
\(\chi_{925}(821,\cdot)\)
\(\chi_{925}(921,\cdot)\)
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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 ]
\((852,76)\) → \((e\left(\frac{1}{5}\right),e\left(\frac{1}{9}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 925 }(16, a) \) |
\(1\) | \(1\) | \(e\left(\frac{14}{45}\right)\) | \(e\left(\frac{13}{45}\right)\) | \(e\left(\frac{28}{45}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{26}{45}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{41}{45}\right)\) | \(e\left(\frac{1}{45}\right)\) |
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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)
