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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(127072, base_ring=CyclotomicField(2280))
M = H._module
chi = DirichletCharacter(H, M([0,285,1368,2240]))
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pari:[g,chi] = znchar(Mod(581,127072))
| Modulus: | \(127072\) | |
| Conductor: | \(127072\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(2280\) |
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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)
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\(\chi_{127072}(125,\cdot)\)
\(\chi_{127072}(581,\cdot)\)
\(\chi_{127072}(885,\cdot)\)
\(\chi_{127072}(1037,\cdot)\)
\(\chi_{127072}(1109,\cdot)\)
\(\chi_{127072}(1413,\cdot)\)
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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 ]
\((39711,47653,69313,14081)\) → \((1,e\left(\frac{1}{8}\right),e\left(\frac{3}{5}\right),e\left(\frac{56}{57}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(13\) | \(15\) | \(17\) | \(21\) | \(23\) | \(25\) |
| \( \chi_{ 127072 }(581, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1679}{2280}\right)\) | \(e\left(\frac{1037}{2280}\right)\) | \(e\left(\frac{311}{380}\right)\) | \(e\left(\frac{539}{1140}\right)\) | \(e\left(\frac{1603}{2280}\right)\) | \(e\left(\frac{109}{570}\right)\) | \(e\left(\frac{533}{570}\right)\) | \(e\left(\frac{253}{456}\right)\) | \(e\left(\frac{43}{228}\right)\) | \(e\left(\frac{1037}{1140}\right)\) |
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sage:chi.jacobi_sum(n)
