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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4851, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([7,18,21]))
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pari:[g,chi] = znchar(Mod(1352,4851))
| Modulus: | \(4851\) | |
| Conductor: | \(4851\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(42\) |
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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: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{4851}(428,\cdot)\)
\(\chi_{4851}(659,\cdot)\)
\(\chi_{4851}(1121,\cdot)\)
\(\chi_{4851}(1352,\cdot)\)
\(\chi_{4851}(2045,\cdot)\)
\(\chi_{4851}(2507,\cdot)\)
\(\chi_{4851}(2738,\cdot)\)
\(\chi_{4851}(3200,\cdot)\)
\(\chi_{4851}(3893,\cdot)\)
\(\chi_{4851}(4124,\cdot)\)
\(\chi_{4851}(4586,\cdot)\)
\(\chi_{4851}(4817,\cdot)\)
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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 ]
\((4313,199,442)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{3}{7}\right),-1)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(13\) | \(16\) | \(17\) | \(19\) | \(20\) |
| \( \chi_{ 4851 }(1352, a) \) |
\(1\) | \(1\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(-1\) | \(e\left(\frac{37}{42}\right)\) |
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sage:chi.jacobi_sum(n)
