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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4563, base_ring=CyclotomicField(468))
M = H._module
chi = DirichletCharacter(H, M([130,285]))
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pari:[g,chi] = znchar(Mod(410,4563))
| Modulus: | \(4563\) | |
| Conductor: | \(4563\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(468\) |
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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: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
|
\(\chi_{4563}(2,\cdot)\)
\(\chi_{4563}(11,\cdot)\)
\(\chi_{4563}(32,\cdot)\)
\(\chi_{4563}(59,\cdot)\)
\(\chi_{4563}(119,\cdot)\)
\(\chi_{4563}(128,\cdot)\)
\(\chi_{4563}(149,\cdot)\)
\(\chi_{4563}(176,\cdot)\)
\(\chi_{4563}(236,\cdot)\)
\(\chi_{4563}(245,\cdot)\)
\(\chi_{4563}(266,\cdot)\)
\(\chi_{4563}(293,\cdot)\)
\(\chi_{4563}(353,\cdot)\)
\(\chi_{4563}(362,\cdot)\)
\(\chi_{4563}(383,\cdot)\)
\(\chi_{4563}(410,\cdot)\)
\(\chi_{4563}(470,\cdot)\)
\(\chi_{4563}(479,\cdot)\)
\(\chi_{4563}(500,\cdot)\)
\(\chi_{4563}(527,\cdot)\)
\(\chi_{4563}(617,\cdot)\)
\(\chi_{4563}(644,\cdot)\)
\(\chi_{4563}(704,\cdot)\)
\(\chi_{4563}(713,\cdot)\)
\(\chi_{4563}(734,\cdot)\)
\(\chi_{4563}(761,\cdot)\)
\(\chi_{4563}(821,\cdot)\)
\(\chi_{4563}(830,\cdot)\)
\(\chi_{4563}(851,\cdot)\)
\(\chi_{4563}(878,\cdot)\)
...
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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 ]
\((677,3889)\) → \((e\left(\frac{5}{18}\right),e\left(\frac{95}{156}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(14\) | \(16\) | \(17\) |
| \( \chi_{ 4563 }(410, a) \) |
\(1\) | \(1\) | \(e\left(\frac{415}{468}\right)\) | \(e\left(\frac{181}{234}\right)\) | \(e\left(\frac{407}{468}\right)\) | \(e\left(\frac{283}{468}\right)\) | \(e\left(\frac{103}{156}\right)\) | \(e\left(\frac{59}{78}\right)\) | \(e\left(\frac{157}{468}\right)\) | \(e\left(\frac{115}{234}\right)\) | \(e\left(\frac{64}{117}\right)\) | \(e\left(\frac{1}{13}\right)\) |
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sage:chi.jacobi_sum(n)
