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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(768, base_ring=CyclotomicField(64))
M = H._module
chi = DirichletCharacter(H, M([32,33,32]))
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pari:[g,chi] = znchar(Mod(635,768))
| Modulus: | \(768\) | |
| Conductor: | \(768\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(64\) |
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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: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{768}(11,\cdot)\)
\(\chi_{768}(35,\cdot)\)
\(\chi_{768}(59,\cdot)\)
\(\chi_{768}(83,\cdot)\)
\(\chi_{768}(107,\cdot)\)
\(\chi_{768}(131,\cdot)\)
\(\chi_{768}(155,\cdot)\)
\(\chi_{768}(179,\cdot)\)
\(\chi_{768}(203,\cdot)\)
\(\chi_{768}(227,\cdot)\)
\(\chi_{768}(251,\cdot)\)
\(\chi_{768}(275,\cdot)\)
\(\chi_{768}(299,\cdot)\)
\(\chi_{768}(323,\cdot)\)
\(\chi_{768}(347,\cdot)\)
\(\chi_{768}(371,\cdot)\)
\(\chi_{768}(395,\cdot)\)
\(\chi_{768}(419,\cdot)\)
\(\chi_{768}(443,\cdot)\)
\(\chi_{768}(467,\cdot)\)
\(\chi_{768}(491,\cdot)\)
\(\chi_{768}(515,\cdot)\)
\(\chi_{768}(539,\cdot)\)
\(\chi_{768}(563,\cdot)\)
\(\chi_{768}(587,\cdot)\)
\(\chi_{768}(611,\cdot)\)
\(\chi_{768}(635,\cdot)\)
\(\chi_{768}(659,\cdot)\)
\(\chi_{768}(683,\cdot)\)
\(\chi_{768}(707,\cdot)\)
...
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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 ]
\((511,517,257)\) → \((-1,e\left(\frac{33}{64}\right),-1)\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 768 }(635, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{64}\right)\) | \(e\left(\frac{21}{32}\right)\) | \(e\left(\frac{53}{64}\right)\) | \(e\left(\frac{15}{64}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{23}{64}\right)\) | \(e\left(\frac{7}{32}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{59}{64}\right)\) | \(e\left(\frac{5}{8}\right)\) |
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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)
