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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1309, base_ring=CyclotomicField(80))
M = H._module
chi = DirichletCharacter(H, M([40,48,5]))
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pari:[g,chi] = znchar(Mod(20,1309))
| Modulus: | \(1309\) | |
| Conductor: | \(1309\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(80\) |
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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)
|
\(\chi_{1309}(20,\cdot)\)
\(\chi_{1309}(27,\cdot)\)
\(\chi_{1309}(48,\cdot)\)
\(\chi_{1309}(97,\cdot)\)
\(\chi_{1309}(125,\cdot)\)
\(\chi_{1309}(146,\cdot)\)
\(\chi_{1309}(181,\cdot)\)
\(\chi_{1309}(258,\cdot)\)
\(\chi_{1309}(279,\cdot)\)
\(\chi_{1309}(300,\cdot)\)
\(\chi_{1309}(328,\cdot)\)
\(\chi_{1309}(335,\cdot)\)
\(\chi_{1309}(377,\cdot)\)
\(\chi_{1309}(405,\cdot)\)
\(\chi_{1309}(454,\cdot)\)
\(\chi_{1309}(482,\cdot)\)
\(\chi_{1309}(566,\cdot)\)
\(\chi_{1309}(636,\cdot)\)
\(\chi_{1309}(643,\cdot)\)
\(\chi_{1309}(685,\cdot)\)
\(\chi_{1309}(720,\cdot)\)
\(\chi_{1309}(741,\cdot)\)
\(\chi_{1309}(762,\cdot)\)
\(\chi_{1309}(839,\cdot)\)
\(\chi_{1309}(874,\cdot)\)
\(\chi_{1309}(895,\cdot)\)
\(\chi_{1309}(972,\cdot)\)
\(\chi_{1309}(993,\cdot)\)
\(\chi_{1309}(1049,\cdot)\)
\(\chi_{1309}(1098,\cdot)\)
...
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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 ]
\((1123,596,309)\) → \((-1,e\left(\frac{3}{5}\right),e\left(\frac{1}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(12\) | \(13\) |
| \( \chi_{ 1309 }(20, a) \) |
\(1\) | \(1\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{29}{80}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{17}{80}\right)\) | \(e\left(\frac{67}{80}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{7}{20}\right)\) |
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sage:chi.jacobi_sum(n)
