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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(788, base_ring=CyclotomicField(98))
M = H._module
chi = DirichletCharacter(H, M([49,54]))
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pari:[g,chi] = znchar(Mod(355,788))
| Modulus: | \(788\) | |
| Conductor: | \(788\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(98\) |
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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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\(\chi_{788}(23,\cdot)\)
\(\chi_{788}(51,\cdot)\)
\(\chi_{788}(59,\cdot)\)
\(\chi_{788}(63,\cdot)\)
\(\chi_{788}(135,\cdot)\)
\(\chi_{788}(171,\cdot)\)
\(\chi_{788}(175,\cdot)\)
\(\chi_{788}(187,\cdot)\)
\(\chi_{788}(231,\cdot)\)
\(\chi_{788}(239,\cdot)\)
\(\chi_{788}(251,\cdot)\)
\(\chi_{788}(267,\cdot)\)
\(\chi_{788}(287,\cdot)\)
\(\chi_{788}(339,\cdot)\)
\(\chi_{788}(347,\cdot)\)
\(\chi_{788}(351,\cdot)\)
\(\chi_{788}(355,\cdot)\)
\(\chi_{788}(379,\cdot)\)
\(\chi_{788}(387,\cdot)\)
\(\chi_{788}(423,\cdot)\)
\(\chi_{788}(431,\cdot)\)
\(\chi_{788}(443,\cdot)\)
\(\chi_{788}(447,\cdot)\)
\(\chi_{788}(455,\cdot)\)
\(\chi_{788}(475,\cdot)\)
\(\chi_{788}(479,\cdot)\)
\(\chi_{788}(495,\cdot)\)
\(\chi_{788}(499,\cdot)\)
\(\chi_{788}(527,\cdot)\)
\(\chi_{788}(587,\cdot)\)
...
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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 ]
\((395,593)\) → \((-1,e\left(\frac{27}{49}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 788 }(355, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{23}{98}\right)\) | \(e\left(\frac{2}{49}\right)\) | \(e\left(\frac{93}{98}\right)\) | \(e\left(\frac{23}{49}\right)\) | \(e\left(\frac{47}{98}\right)\) | \(e\left(\frac{38}{49}\right)\) | \(e\left(\frac{27}{98}\right)\) | \(e\left(\frac{30}{49}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{9}{49}\right)\) |
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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)
