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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1016, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,21,26]))
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pari:[g,chi] = znchar(Mod(61,1016))
| Modulus: | \(1016\) | |
| Conductor: | \(1016\) |
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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)
|
\(\chi_{1016}(61,\cdot)\)
\(\chi_{1016}(117,\cdot)\)
\(\chi_{1016}(165,\cdot)\)
\(\chi_{1016}(221,\cdot)\)
\(\chi_{1016}(301,\cdot)\)
\(\chi_{1016}(341,\cdot)\)
\(\chi_{1016}(533,\cdot)\)
\(\chi_{1016}(581,\cdot)\)
\(\chi_{1016}(685,\cdot)\)
\(\chi_{1016}(757,\cdot)\)
\(\chi_{1016}(965,\cdot)\)
\(\chi_{1016}(989,\cdot)\)
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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 ]
\((255,509,257)\) → \((1,-1,e\left(\frac{13}{21}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 1016 }(61, a) \) |
\(1\) | \(1\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(-1\) | \(e\left(\frac{13}{42}\right)\) |
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sage:chi.jacobi_sum(n)
