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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2580, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,21,21,8]))
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pari:[g,chi] = znchar(Mod(2519,2580))
| Modulus: | \(2580\) | |
| Conductor: | \(2580\) |
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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
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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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\(\chi_{2580}(239,\cdot)\)
\(\chi_{2580}(359,\cdot)\)
\(\chi_{2580}(539,\cdot)\)
\(\chi_{2580}(599,\cdot)\)
\(\chi_{2580}(659,\cdot)\)
\(\chi_{2580}(719,\cdot)\)
\(\chi_{2580}(959,\cdot)\)
\(\chi_{2580}(1199,\cdot)\)
\(\chi_{2580}(1859,\cdot)\)
\(\chi_{2580}(2159,\cdot)\)
\(\chi_{2580}(2339,\cdot)\)
\(\chi_{2580}(2519,\cdot)\)
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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 ]
\((1291,1721,517,1981)\) → \((-1,-1,-1,e\left(\frac{4}{21}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 2580 }(2519, a) \) |
\(1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{9}{14}\right)\) |
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sage:chi.jacobi_sum(n)
