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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(875, base_ring=CyclotomicField(50))
M = H._module
chi = DirichletCharacter(H, M([33,25]))
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pari:[g,chi] = znchar(Mod(839,875))
| Modulus: | \(875\) | |
| Conductor: | \(875\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
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| Order: | \(50\) |
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sage:chi.multiplicative_order()
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pari:charorder(g,chi)
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| 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: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{875}(34,\cdot)\)
\(\chi_{875}(69,\cdot)\)
\(\chi_{875}(104,\cdot)\)
\(\chi_{875}(139,\cdot)\)
\(\chi_{875}(209,\cdot)\)
\(\chi_{875}(244,\cdot)\)
\(\chi_{875}(279,\cdot)\)
\(\chi_{875}(314,\cdot)\)
\(\chi_{875}(384,\cdot)\)
\(\chi_{875}(419,\cdot)\)
\(\chi_{875}(454,\cdot)\)
\(\chi_{875}(489,\cdot)\)
\(\chi_{875}(559,\cdot)\)
\(\chi_{875}(594,\cdot)\)
\(\chi_{875}(629,\cdot)\)
\(\chi_{875}(664,\cdot)\)
\(\chi_{875}(734,\cdot)\)
\(\chi_{875}(769,\cdot)\)
\(\chi_{875}(804,\cdot)\)
\(\chi_{875}(839,\cdot)\)
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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 ]
\((127,626)\) → \((e\left(\frac{33}{50}\right),-1)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
| \( \chi_{ 875 }(839, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{33}{50}\right)\) | \(e\left(\frac{3}{25}\right)\) | \(e\left(\frac{8}{25}\right)\) | \(e\left(\frac{39}{50}\right)\) | \(e\left(\frac{49}{50}\right)\) | \(e\left(\frac{6}{25}\right)\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{6}{25}\right)\) | \(e\left(\frac{16}{25}\right)\) |
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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)
