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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(508288, base_ring=CyclotomicField(27360))
M = H._module
chi = DirichletCharacter(H, M([0,2565,21888,3760]))
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pari:[g,chi] = znchar(Mod(509,508288))
| Modulus: | \(508288\) | |
| Conductor: | \(508288\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(27360\) |
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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)
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\(\chi_{508288}(53,\cdot)\)
\(\chi_{508288}(181,\cdot)\)
\(\chi_{508288}(269,\cdot)\)
\(\chi_{508288}(317,\cdot)\)
\(\chi_{508288}(357,\cdot)\)
\(\chi_{508288}(421,\cdot)\)
\(\chi_{508288}(509,\cdot)\)
\(\chi_{508288}(565,\cdot)\)
\(\chi_{508288}(621,\cdot)\)
\(\chi_{508288}(773,\cdot)\)
\(\chi_{508288}(933,\cdot)\)
\(\chi_{508288}(1093,\cdot)\)
\(\chi_{508288}(1181,\cdot)\)
\(\chi_{508288}(1237,\cdot)\)
\(\chi_{508288}(1269,\cdot)\)
\(\chi_{508288}(1325,\cdot)\)
\(\chi_{508288}(1389,\cdot)\)
\(\chi_{508288}(1477,\cdot)\)
\(\chi_{508288}(1533,\cdot)\)
\(\chi_{508288}(1549,\cdot)\)
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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 ]
\((166783,174725,323457,14081)\) → \((1,e\left(\frac{3}{32}\right),e\left(\frac{4}{5}\right),e\left(\frac{47}{342}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(13\) | \(15\) | \(17\) | \(21\) | \(23\) | \(25\) |
| \( \chi_{ 508288 }(509, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{21439}{27360}\right)\) | \(e\left(\frac{4837}{27360}\right)\) | \(e\left(\frac{691}{4560}\right)\) | \(e\left(\frac{7759}{13680}\right)\) | \(e\left(\frac{11483}{27360}\right)\) | \(e\left(\frac{6569}{6840}\right)\) | \(e\left(\frac{6043}{6840}\right)\) | \(e\left(\frac{5117}{5472}\right)\) | \(e\left(\frac{1031}{2736}\right)\) | \(e\left(\frac{4837}{13680}\right)\) |
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sage:chi.jacobi_sum(n)
