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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3185, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,16,7]))
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pari:[g,chi] = znchar(Mod(2279,3185))
| Modulus: | \(3185\) | |
| Conductor: | \(3185\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
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| Order: | \(42\) |
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sage:chi.multiplicative_order()
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pari:charorder(g,chi)
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| 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: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{3185}(4,\cdot)\)
\(\chi_{3185}(114,\cdot)\)
\(\chi_{3185}(914,\cdot)\)
\(\chi_{3185}(1024,\cdot)\)
\(\chi_{3185}(1369,\cdot)\)
\(\chi_{3185}(1479,\cdot)\)
\(\chi_{3185}(1824,\cdot)\)
\(\chi_{3185}(1934,\cdot)\)
\(\chi_{3185}(2279,\cdot)\)
\(\chi_{3185}(2389,\cdot)\)
\(\chi_{3185}(2734,\cdot)\)
\(\chi_{3185}(2844,\cdot)\)
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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 ]
\((1912,2796,1471)\) → \((-1,e\left(\frac{8}{21}\right),e\left(\frac{1}{6}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(16\) | \(17\) |
| \( \chi_{ 3185 }(2279, a) \) |
\(1\) | \(1\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{5}{14}\right)\) |
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sage:chi.jacobi_sum(n)
