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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1001, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([10,48,55]))
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pari:[g,chi] = znchar(Mod(696,1001))
| Modulus: | \(1001\) | |
| Conductor: | \(1001\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
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| Order: | \(60\) |
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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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\(\chi_{1001}(59,\cdot)\)
\(\chi_{1001}(136,\cdot)\)
\(\chi_{1001}(180,\cdot)\)
\(\chi_{1001}(236,\cdot)\)
\(\chi_{1001}(423,\cdot)\)
\(\chi_{1001}(500,\cdot)\)
\(\chi_{1001}(509,\cdot)\)
\(\chi_{1001}(544,\cdot)\)
\(\chi_{1001}(691,\cdot)\)
\(\chi_{1001}(696,\cdot)\)
\(\chi_{1001}(773,\cdot)\)
\(\chi_{1001}(817,\cdot)\)
\(\chi_{1001}(873,\cdot)\)
\(\chi_{1001}(878,\cdot)\)
\(\chi_{1001}(955,\cdot)\)
\(\chi_{1001}(999,\cdot)\)
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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 ]
\((430,365,925)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{4}{5}\right),e\left(\frac{11}{12}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(12\) | \(15\) |
| \( \chi_{ 1001 }(696, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{31}{60}\right)\) |
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sage:chi.jacobi_sum(n)
