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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6069, base_ring=CyclotomicField(68))
M = H._module
chi = DirichletCharacter(H, M([34,34,37]))
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pari:[g,chi] = znchar(Mod(608,6069))
| Modulus: | \(6069\) | |
| Conductor: | \(6069\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(68\) |
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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)
|
\(\chi_{6069}(293,\cdot)\)
\(\chi_{6069}(608,\cdot)\)
\(\chi_{6069}(650,\cdot)\)
\(\chi_{6069}(965,\cdot)\)
\(\chi_{6069}(1007,\cdot)\)
\(\chi_{6069}(1322,\cdot)\)
\(\chi_{6069}(1364,\cdot)\)
\(\chi_{6069}(1679,\cdot)\)
\(\chi_{6069}(1721,\cdot)\)
\(\chi_{6069}(2036,\cdot)\)
\(\chi_{6069}(2078,\cdot)\)
\(\chi_{6069}(2393,\cdot)\)
\(\chi_{6069}(2435,\cdot)\)
\(\chi_{6069}(2750,\cdot)\)
\(\chi_{6069}(2792,\cdot)\)
\(\chi_{6069}(3107,\cdot)\)
\(\chi_{6069}(3149,\cdot)\)
\(\chi_{6069}(3464,\cdot)\)
\(\chi_{6069}(3821,\cdot)\)
\(\chi_{6069}(3863,\cdot)\)
\(\chi_{6069}(4178,\cdot)\)
\(\chi_{6069}(4220,\cdot)\)
\(\chi_{6069}(4535,\cdot)\)
\(\chi_{6069}(4577,\cdot)\)
\(\chi_{6069}(4892,\cdot)\)
\(\chi_{6069}(4934,\cdot)\)
\(\chi_{6069}(5249,\cdot)\)
\(\chi_{6069}(5291,\cdot)\)
\(\chi_{6069}(5606,\cdot)\)
\(\chi_{6069}(5648,\cdot)\)
...
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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 ]
\((2024,4336,3760)\) → \((-1,-1,e\left(\frac{37}{68}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(19\) | \(20\) |
| \( \chi_{ 6069 }(608, a) \) |
\(1\) | \(1\) | \(e\left(\frac{15}{17}\right)\) | \(e\left(\frac{13}{17}\right)\) | \(e\left(\frac{41}{68}\right)\) | \(e\left(\frac{11}{17}\right)\) | \(e\left(\frac{33}{68}\right)\) | \(e\left(\frac{1}{68}\right)\) | \(e\left(\frac{5}{34}\right)\) | \(e\left(\frac{9}{17}\right)\) | \(e\left(\frac{2}{17}\right)\) | \(e\left(\frac{25}{68}\right)\) |
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sage:chi.jacobi_sum(n)
