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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1309, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([40,12,45]))
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pari:[g,chi] = znchar(Mod(4,1309))
| Modulus: | \(1309\) | |
| Conductor: | \(1309\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(60\) |
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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
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| Minimal: | yes |
| Parity: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{1309}(4,\cdot)\)
\(\chi_{1309}(81,\cdot)\)
\(\chi_{1309}(191,\cdot)\)
\(\chi_{1309}(268,\cdot)\)
\(\chi_{1309}(361,\cdot)\)
\(\chi_{1309}(438,\cdot)\)
\(\chi_{1309}(548,\cdot)\)
\(\chi_{1309}(599,\cdot)\)
\(\chi_{1309}(625,\cdot)\)
\(\chi_{1309}(676,\cdot)\)
\(\chi_{1309}(718,\cdot)\)
\(\chi_{1309}(786,\cdot)\)
\(\chi_{1309}(795,\cdot)\)
\(\chi_{1309}(863,\cdot)\)
\(\chi_{1309}(905,\cdot)\)
\(\chi_{1309}(982,\cdot)\)
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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 ]
\((1123,596,309)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{1}{5}\right),-i)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(12\) | \(13\) |
| \( \chi_{ 1309 }(4, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{5}\right)\) |
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sage:chi.jacobi_sum(n)
