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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1323, base_ring=CyclotomicField(126))
M = H._module
chi = DirichletCharacter(H, M([77,114]))
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pari:[g,chi] = znchar(Mod(23,1323))
| Modulus: | \(1323\) | |
| Conductor: | \(1323\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(126\) |
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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: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
|
\(\chi_{1323}(11,\cdot)\)
\(\chi_{1323}(23,\cdot)\)
\(\chi_{1323}(74,\cdot)\)
\(\chi_{1323}(86,\cdot)\)
\(\chi_{1323}(137,\cdot)\)
\(\chi_{1323}(149,\cdot)\)
\(\chi_{1323}(200,\cdot)\)
\(\chi_{1323}(212,\cdot)\)
\(\chi_{1323}(326,\cdot)\)
\(\chi_{1323}(338,\cdot)\)
\(\chi_{1323}(389,\cdot)\)
\(\chi_{1323}(401,\cdot)\)
\(\chi_{1323}(452,\cdot)\)
\(\chi_{1323}(464,\cdot)\)
\(\chi_{1323}(515,\cdot)\)
\(\chi_{1323}(527,\cdot)\)
\(\chi_{1323}(578,\cdot)\)
\(\chi_{1323}(590,\cdot)\)
\(\chi_{1323}(641,\cdot)\)
\(\chi_{1323}(653,\cdot)\)
\(\chi_{1323}(767,\cdot)\)
\(\chi_{1323}(779,\cdot)\)
\(\chi_{1323}(830,\cdot)\)
\(\chi_{1323}(842,\cdot)\)
\(\chi_{1323}(893,\cdot)\)
\(\chi_{1323}(905,\cdot)\)
\(\chi_{1323}(956,\cdot)\)
\(\chi_{1323}(968,\cdot)\)
\(\chi_{1323}(1019,\cdot)\)
\(\chi_{1323}(1031,\cdot)\)
...
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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 ]
\((785,1081)\) → \((e\left(\frac{11}{18}\right),e\left(\frac{19}{21}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 1323 }(23, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{17}{126}\right)\) | \(e\left(\frac{17}{63}\right)\) | \(e\left(\frac{37}{126}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{17}{126}\right)\) | \(e\left(\frac{47}{63}\right)\) | \(e\left(\frac{34}{63}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(1\) |
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sage:chi.jacobi_sum(n)
