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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([10,24,45]))
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pari:[g,chi] = znchar(Mod(38,1309))
| 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: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{1309}(38,\cdot)\)
\(\chi_{1309}(47,\cdot)\)
\(\chi_{1309}(115,\cdot)\)
\(\chi_{1309}(157,\cdot)\)
\(\chi_{1309}(234,\cdot)\)
\(\chi_{1309}(565,\cdot)\)
\(\chi_{1309}(642,\cdot)\)
\(\chi_{1309}(752,\cdot)\)
\(\chi_{1309}(829,\cdot)\)
\(\chi_{1309}(922,\cdot)\)
\(\chi_{1309}(999,\cdot)\)
\(\chi_{1309}(1109,\cdot)\)
\(\chi_{1309}(1160,\cdot)\)
\(\chi_{1309}(1186,\cdot)\)
\(\chi_{1309}(1237,\cdot)\)
\(\chi_{1309}(1279,\cdot)\)
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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 ]
\((1123,596,309)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{2}{5}\right),-i)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(12\) | \(13\) |
| \( \chi_{ 1309 }(38, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{9}{10}\right)\) |
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sage:chi.jacobi_sum(n)
