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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(508288, base_ring=CyclotomicField(5472))
M = H._module
chi = DirichletCharacter(H, M([0,2223,2736,4624]))
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pari:[g,chi] = znchar(Mod(21,508288))
| Modulus: | \(508288\) | |
| Conductor: | \(508288\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(5472\) |
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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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\(\chi_{508288}(21,\cdot)\)
\(\chi_{508288}(109,\cdot)\)
\(\chi_{508288}(637,\cdot)\)
\(\chi_{508288}(725,\cdot)\)
\(\chi_{508288}(813,\cdot)\)
\(\chi_{508288}(1077,\cdot)\)
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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 ]
\((166783,174725,323457,14081)\) → \((1,e\left(\frac{13}{32}\right),-1,e\left(\frac{289}{342}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(13\) | \(15\) | \(17\) | \(21\) | \(23\) | \(25\) |
| \( \chi_{ 508288 }(21, a) \) |
\(1\) | \(1\) | \(e\left(\frac{3709}{5472}\right)\) | \(e\left(\frac{2479}{5472}\right)\) | \(e\left(\frac{289}{912}\right)\) | \(e\left(\frac{973}{2736}\right)\) | \(e\left(\frac{3329}{5472}\right)\) | \(e\left(\frac{179}{1368}\right)\) | \(e\left(\frac{1165}{1368}\right)\) | \(e\left(\frac{5443}{5472}\right)\) | \(e\left(\frac{169}{2736}\right)\) | \(e\left(\frac{2479}{2736}\right)\) |
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sage:chi.jacobi_sum(n)
