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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(608, base_ring=CyclotomicField(72))
M = H._module
chi = DirichletCharacter(H, M([0,63,20]))
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pari:[g,chi] = znchar(Mod(13,608))
| Modulus: | \(608\) | |
| Conductor: | \(608\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(72\) |
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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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\(\chi_{608}(13,\cdot)\)
\(\chi_{608}(21,\cdot)\)
\(\chi_{608}(29,\cdot)\)
\(\chi_{608}(53,\cdot)\)
\(\chi_{608}(109,\cdot)\)
\(\chi_{608}(117,\cdot)\)
\(\chi_{608}(165,\cdot)\)
\(\chi_{608}(173,\cdot)\)
\(\chi_{608}(181,\cdot)\)
\(\chi_{608}(205,\cdot)\)
\(\chi_{608}(261,\cdot)\)
\(\chi_{608}(269,\cdot)\)
\(\chi_{608}(317,\cdot)\)
\(\chi_{608}(325,\cdot)\)
\(\chi_{608}(333,\cdot)\)
\(\chi_{608}(357,\cdot)\)
\(\chi_{608}(413,\cdot)\)
\(\chi_{608}(421,\cdot)\)
\(\chi_{608}(469,\cdot)\)
\(\chi_{608}(477,\cdot)\)
\(\chi_{608}(485,\cdot)\)
\(\chi_{608}(509,\cdot)\)
\(\chi_{608}(565,\cdot)\)
\(\chi_{608}(573,\cdot)\)
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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 ]
\((191,229,97)\) → \((1,e\left(\frac{7}{8}\right),e\left(\frac{5}{18}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(21\) | \(23\) |
| \( \chi_{ 608 }(13, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{17}{72}\right)\) | \(e\left(\frac{23}{72}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{37}{72}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{47}{72}\right)\) | \(e\left(\frac{29}{36}\right)\) |
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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)
