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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6069, base_ring=CyclotomicField(34))
M = H._module
chi = DirichletCharacter(H, M([17,17,20]))
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pari:[g,chi] = znchar(Mod(545,6069))
| Modulus: | \(6069\) | |
| Conductor: | \(6069\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(34\) |
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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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\(\chi_{6069}(188,\cdot)\)
\(\chi_{6069}(545,\cdot)\)
\(\chi_{6069}(902,\cdot)\)
\(\chi_{6069}(1259,\cdot)\)
\(\chi_{6069}(1616,\cdot)\)
\(\chi_{6069}(1973,\cdot)\)
\(\chi_{6069}(2330,\cdot)\)
\(\chi_{6069}(2687,\cdot)\)
\(\chi_{6069}(3044,\cdot)\)
\(\chi_{6069}(3401,\cdot)\)
\(\chi_{6069}(4115,\cdot)\)
\(\chi_{6069}(4472,\cdot)\)
\(\chi_{6069}(4829,\cdot)\)
\(\chi_{6069}(5186,\cdot)\)
\(\chi_{6069}(5543,\cdot)\)
\(\chi_{6069}(5900,\cdot)\)
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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 ]
\((2024,4336,3760)\) → \((-1,-1,e\left(\frac{10}{17}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(19\) | \(20\) |
| \( \chi_{ 6069 }(545, a) \) |
\(1\) | \(1\) | \(e\left(\frac{9}{34}\right)\) | \(e\left(\frac{9}{17}\right)\) | \(e\left(\frac{12}{17}\right)\) | \(e\left(\frac{27}{34}\right)\) | \(e\left(\frac{33}{34}\right)\) | \(e\left(\frac{1}{34}\right)\) | \(e\left(\frac{27}{34}\right)\) | \(e\left(\frac{1}{17}\right)\) | \(e\left(\frac{25}{34}\right)\) | \(e\left(\frac{4}{17}\right)\) |
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sage:chi.jacobi_sum(n)
