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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1309, base_ring=CyclotomicField(240))
M = H._module
chi = DirichletCharacter(H, M([40,216,45]))
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pari:[g,chi] = znchar(Mod(248,1309))
| Modulus: | \(1309\) | |
| Conductor: | \(1309\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(240\) |
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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)
|
\(\chi_{1309}(24,\cdot)\)
\(\chi_{1309}(40,\cdot)\)
\(\chi_{1309}(61,\cdot)\)
\(\chi_{1309}(73,\cdot)\)
\(\chi_{1309}(96,\cdot)\)
\(\chi_{1309}(129,\cdot)\)
\(\chi_{1309}(150,\cdot)\)
\(\chi_{1309}(173,\cdot)\)
\(\chi_{1309}(194,\cdot)\)
\(\chi_{1309}(215,\cdot)\)
\(\chi_{1309}(227,\cdot)\)
\(\chi_{1309}(248,\cdot)\)
\(\chi_{1309}(250,\cdot)\)
\(\chi_{1309}(283,\cdot)\)
\(\chi_{1309}(292,\cdot)\)
\(\chi_{1309}(299,\cdot)\)
\(\chi_{1309}(360,\cdot)\)
\(\chi_{1309}(369,\cdot)\)
\(\chi_{1309}(381,\cdot)\)
\(\chi_{1309}(402,\cdot)\)
\(\chi_{1309}(437,\cdot)\)
\(\chi_{1309}(453,\cdot)\)
\(\chi_{1309}(479,\cdot)\)
\(\chi_{1309}(481,\cdot)\)
\(\chi_{1309}(486,\cdot)\)
\(\chi_{1309}(530,\cdot)\)
\(\chi_{1309}(556,\cdot)\)
\(\chi_{1309}(558,\cdot)\)
\(\chi_{1309}(600,\cdot)\)
\(\chi_{1309}(607,\cdot)\)
...
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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 ]
\((1123,596,309)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{9}{10}\right),e\left(\frac{3}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(12\) | \(13\) |
| \( \chi_{ 1309 }(248, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{103}{120}\right)\) | \(e\left(\frac{133}{240}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{89}{240}\right)\) | \(e\left(\frac{33}{80}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{13}{120}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{3}{20}\right)\) |
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sage:chi.jacobi_sum(n)
