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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2312, base_ring=CyclotomicField(34))
M = H._module
chi = DirichletCharacter(H, M([17,17,23]))
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pari:[g,chi] = znchar(Mod(67,2312))
| Modulus: | \(2312\) | |
| Conductor: | \(2312\) |
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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
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| Minimal: | yes |
| Parity: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{2312}(67,\cdot)\)
\(\chi_{2312}(203,\cdot)\)
\(\chi_{2312}(339,\cdot)\)
\(\chi_{2312}(475,\cdot)\)
\(\chi_{2312}(611,\cdot)\)
\(\chi_{2312}(747,\cdot)\)
\(\chi_{2312}(883,\cdot)\)
\(\chi_{2312}(1019,\cdot)\)
\(\chi_{2312}(1291,\cdot)\)
\(\chi_{2312}(1427,\cdot)\)
\(\chi_{2312}(1563,\cdot)\)
\(\chi_{2312}(1699,\cdot)\)
\(\chi_{2312}(1835,\cdot)\)
\(\chi_{2312}(1971,\cdot)\)
\(\chi_{2312}(2107,\cdot)\)
\(\chi_{2312}(2243,\cdot)\)
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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 ]
\((1735,1157,1737)\) → \((-1,-1,e\left(\frac{23}{34}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(19\) | \(21\) | \(23\) |
| \( \chi_{ 2312 }(67, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{23}{34}\right)\) | \(e\left(\frac{7}{17}\right)\) | \(e\left(\frac{6}{17}\right)\) | \(e\left(\frac{6}{17}\right)\) | \(e\left(\frac{19}{34}\right)\) | \(e\left(\frac{3}{34}\right)\) | \(e\left(\frac{3}{34}\right)\) | \(e\left(\frac{8}{17}\right)\) | \(e\left(\frac{1}{34}\right)\) | \(e\left(\frac{6}{17}\right)\) |
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sage:chi.jacobi_sum(n)
