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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(19404, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,7,13,21]))
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pari:[g,chi] = znchar(Mod(18875,19404))
| Modulus: | \(19404\) | |
| Conductor: | \(19404\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(42\) |
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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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\(\chi_{19404}(131,\cdot)\)
\(\chi_{19404}(2243,\cdot)\)
\(\chi_{19404}(2903,\cdot)\)
\(\chi_{19404}(5015,\cdot)\)
\(\chi_{19404}(5675,\cdot)\)
\(\chi_{19404}(7787,\cdot)\)
\(\chi_{19404}(10559,\cdot)\)
\(\chi_{19404}(11219,\cdot)\)
\(\chi_{19404}(13331,\cdot)\)
\(\chi_{19404}(13991,\cdot)\)
\(\chi_{19404}(16763,\cdot)\)
\(\chi_{19404}(18875,\cdot)\)
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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 ]
\((9703,4313,9901,5293)\) → \((-1,e\left(\frac{1}{6}\right),e\left(\frac{13}{42}\right),-1)\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 19404 }(18875, a) \) |
\(1\) | \(1\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(1\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{41}{42}\right)\) |
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sage:chi.jacobi_sum(n)
