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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([329,324]))
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pari:[g,chi] = znchar(Mod(455,2349))
| 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)
|
\(\chi_{2349}(20,\cdot)\)
\(\chi_{2349}(23,\cdot)\)
\(\chi_{2349}(65,\cdot)\)
\(\chi_{2349}(74,\cdot)\)
\(\chi_{2349}(83,\cdot)\)
\(\chi_{2349}(110,\cdot)\)
\(\chi_{2349}(140,\cdot)\)
\(\chi_{2349}(194,\cdot)\)
\(\chi_{2349}(227,\cdot)\)
\(\chi_{2349}(239,\cdot)\)
\(\chi_{2349}(248,\cdot)\)
\(\chi_{2349}(257,\cdot)\)
\(\chi_{2349}(281,\cdot)\)
\(\chi_{2349}(284,\cdot)\)
\(\chi_{2349}(326,\cdot)\)
\(\chi_{2349}(335,\cdot)\)
\(\chi_{2349}(344,\cdot)\)
\(\chi_{2349}(371,\cdot)\)
\(\chi_{2349}(401,\cdot)\)
\(\chi_{2349}(455,\cdot)\)
\(\chi_{2349}(488,\cdot)\)
\(\chi_{2349}(500,\cdot)\)
\(\chi_{2349}(509,\cdot)\)
\(\chi_{2349}(518,\cdot)\)
\(\chi_{2349}(542,\cdot)\)
\(\chi_{2349}(545,\cdot)\)
\(\chi_{2349}(587,\cdot)\)
\(\chi_{2349}(596,\cdot)\)
\(\chi_{2349}(605,\cdot)\)
\(\chi_{2349}(632,\cdot)\)
...
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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 ]
\((407,1945)\) → \((e\left(\frac{47}{54}\right),e\left(\frac{6}{7}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 2349 }(455, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{275}{378}\right)\) | \(e\left(\frac{86}{189}\right)\) | \(e\left(\frac{331}{378}\right)\) | \(e\left(\frac{40}{189}\right)\) | \(e\left(\frac{23}{126}\right)\) | \(e\left(\frac{38}{63}\right)\) | \(e\left(\frac{281}{378}\right)\) | \(e\left(\frac{74}{189}\right)\) | \(e\left(\frac{355}{378}\right)\) | \(e\left(\frac{172}{189}\right)\) |
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sage:chi.jacobi_sum(n)
