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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3152, base_ring=CyclotomicField(196))
M = H._module
chi = DirichletCharacter(H, M([0,49,178]))
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pari:[g,chi] = znchar(Mod(2389,3152))
| Modulus: | \(3152\) | |
| Conductor: | \(3152\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(196\) |
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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)
|
\(\chi_{3152}(109,\cdot)\)
\(\chi_{3152}(157,\cdot)\)
\(\chi_{3152}(173,\cdot)\)
\(\chi_{3152}(181,\cdot)\)
\(\chi_{3152}(261,\cdot)\)
\(\chi_{3152}(293,\cdot)\)
\(\chi_{3152}(309,\cdot)\)
\(\chi_{3152}(333,\cdot)\)
\(\chi_{3152}(341,\cdot)\)
\(\chi_{3152}(357,\cdot)\)
\(\chi_{3152}(365,\cdot)\)
\(\chi_{3152}(437,\cdot)\)
\(\chi_{3152}(501,\cdot)\)
\(\chi_{3152}(549,\cdot)\)
\(\chi_{3152}(557,\cdot)\)
\(\chi_{3152}(613,\cdot)\)
\(\chi_{3152}(653,\cdot)\)
\(\chi_{3152}(725,\cdot)\)
\(\chi_{3152}(765,\cdot)\)
\(\chi_{3152}(797,\cdot)\)
\(\chi_{3152}(813,\cdot)\)
\(\chi_{3152}(829,\cdot)\)
\(\chi_{3152}(853,\cdot)\)
\(\chi_{3152}(885,\cdot)\)
\(\chi_{3152}(909,\cdot)\)
\(\chi_{3152}(925,\cdot)\)
\(\chi_{3152}(957,\cdot)\)
\(\chi_{3152}(989,\cdot)\)
\(\chi_{3152}(1077,\cdot)\)
\(\chi_{3152}(1101,\cdot)\)
...
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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 ]
\((1183,789,593)\) → \((1,i,e\left(\frac{89}{98}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 3152 }(2389, a) \) |
\(1\) | \(1\) | \(e\left(\frac{25}{196}\right)\) | \(e\left(\frac{15}{196}\right)\) | \(e\left(\frac{9}{98}\right)\) | \(e\left(\frac{25}{98}\right)\) | \(e\left(\frac{115}{196}\right)\) | \(e\left(\frac{89}{196}\right)\) | \(e\left(\frac{10}{49}\right)\) | \(e\left(\frac{39}{98}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{43}{196}\right)\) |
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sage:chi.jacobi_sum(n)
