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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1352, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([0,13,22]))
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pari:[g,chi] = znchar(Mod(261,1352))
| Modulus: | \(1352\) | |
| Conductor: | \(1352\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(26\) |
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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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\(\chi_{1352}(53,\cdot)\)
\(\chi_{1352}(157,\cdot)\)
\(\chi_{1352}(261,\cdot)\)
\(\chi_{1352}(365,\cdot)\)
\(\chi_{1352}(469,\cdot)\)
\(\chi_{1352}(573,\cdot)\)
\(\chi_{1352}(781,\cdot)\)
\(\chi_{1352}(885,\cdot)\)
\(\chi_{1352}(989,\cdot)\)
\(\chi_{1352}(1093,\cdot)\)
\(\chi_{1352}(1197,\cdot)\)
\(\chi_{1352}(1301,\cdot)\)
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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 ]
\((1015,677,1185)\) → \((1,-1,e\left(\frac{11}{13}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(15\) | \(17\) | \(19\) | \(21\) | \(23\) |
| \( \chi_{ 1352 }(261, a) \) |
\(1\) | \(1\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(-1\) | \(e\left(\frac{25}{26}\right)\) | \(1\) |
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sage:chi.jacobi_sum(n)
