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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2349, base_ring=CyclotomicField(378))
M = H._module
chi = DirichletCharacter(H, M([161,297]))
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pari:[g,chi] = znchar(Mod(5,2349))
| Modulus: | \(2349\) | |
| Conductor: | \(2349\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(378\) |
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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)
|
\(\chi_{2349}(5,\cdot)\)
\(\chi_{2349}(38,\cdot)\)
\(\chi_{2349}(92,\cdot)\)
\(\chi_{2349}(122,\cdot)\)
\(\chi_{2349}(149,\cdot)\)
\(\chi_{2349}(158,\cdot)\)
\(\chi_{2349}(167,\cdot)\)
\(\chi_{2349}(209,\cdot)\)
\(\chi_{2349}(212,\cdot)\)
\(\chi_{2349}(236,\cdot)\)
\(\chi_{2349}(245,\cdot)\)
\(\chi_{2349}(254,\cdot)\)
\(\chi_{2349}(266,\cdot)\)
\(\chi_{2349}(299,\cdot)\)
\(\chi_{2349}(353,\cdot)\)
\(\chi_{2349}(383,\cdot)\)
\(\chi_{2349}(410,\cdot)\)
\(\chi_{2349}(419,\cdot)\)
\(\chi_{2349}(428,\cdot)\)
\(\chi_{2349}(470,\cdot)\)
\(\chi_{2349}(473,\cdot)\)
\(\chi_{2349}(497,\cdot)\)
\(\chi_{2349}(506,\cdot)\)
\(\chi_{2349}(515,\cdot)\)
\(\chi_{2349}(527,\cdot)\)
\(\chi_{2349}(560,\cdot)\)
\(\chi_{2349}(614,\cdot)\)
\(\chi_{2349}(644,\cdot)\)
\(\chi_{2349}(671,\cdot)\)
\(\chi_{2349}(680,\cdot)\)
...
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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 ]
\((407,1945)\) → \((e\left(\frac{23}{54}\right),e\left(\frac{11}{14}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 2349 }(5, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{40}{189}\right)\) | \(e\left(\frac{80}{189}\right)\) | \(e\left(\frac{31}{378}\right)\) | \(e\left(\frac{46}{189}\right)\) | \(e\left(\frac{40}{63}\right)\) | \(e\left(\frac{37}{126}\right)\) | \(e\left(\frac{34}{189}\right)\) | \(e\left(\frac{104}{189}\right)\) | \(e\left(\frac{86}{189}\right)\) | \(e\left(\frac{160}{189}\right)\) |
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sage:chi.jacobi_sum(n)
