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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(31360, base_ring=CyclotomicField(672))
M = H._module
chi = DirichletCharacter(H, M([0,315,504,320]))
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pari:[g,chi] = znchar(Mod(653,31360))
| Modulus: | \(31360\) | |
| Conductor: | \(31360\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(672\) |
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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)
|
\(\chi_{31360}(37,\cdot)\)
\(\chi_{31360}(93,\cdot)\)
\(\chi_{31360}(277,\cdot)\)
\(\chi_{31360}(333,\cdot)\)
\(\chi_{31360}(597,\cdot)\)
\(\chi_{31360}(653,\cdot)\)
\(\chi_{31360}(837,\cdot)\)
\(\chi_{31360}(893,\cdot)\)
\(\chi_{31360}(1213,\cdot)\)
\(\chi_{31360}(1397,\cdot)\)
\(\chi_{31360}(1453,\cdot)\)
\(\chi_{31360}(1717,\cdot)\)
\(\chi_{31360}(1773,\cdot)\)
\(\chi_{31360}(1957,\cdot)\)
\(\chi_{31360}(2013,\cdot)\)
\(\chi_{31360}(2277,\cdot)\)
\(\chi_{31360}(2573,\cdot)\)
\(\chi_{31360}(2837,\cdot)\)
\(\chi_{31360}(2893,\cdot)\)
\(\chi_{31360}(3077,\cdot)\)
\(\chi_{31360}(3133,\cdot)\)
\(\chi_{31360}(3397,\cdot)\)
\(\chi_{31360}(3453,\cdot)\)
\(\chi_{31360}(3637,\cdot)\)
\(\chi_{31360}(3957,\cdot)\)
\(\chi_{31360}(4013,\cdot)\)
\(\chi_{31360}(4197,\cdot)\)
\(\chi_{31360}(4253,\cdot)\)
\(\chi_{31360}(4517,\cdot)\)
\(\chi_{31360}(4573,\cdot)\)
...
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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 ]
\((17151,28421,18817,10881)\) → \((1,e\left(\frac{15}{32}\right),-i,e\left(\frac{10}{21}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) |
| \( \chi_{ 31360 }(653, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{89}{672}\right)\) | \(e\left(\frac{89}{336}\right)\) | \(e\left(\frac{599}{672}\right)\) | \(e\left(\frac{223}{224}\right)\) | \(e\left(\frac{131}{168}\right)\) | \(e\left(\frac{91}{96}\right)\) | \(e\left(\frac{305}{336}\right)\) | \(e\left(\frac{89}{224}\right)\) | \(e\left(\frac{163}{224}\right)\) | \(e\left(\frac{1}{12}\right)\) |
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sage:chi.jacobi_sum(n)
