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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1700, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([20,18,5]))
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pari:[g,chi] = znchar(Mod(587,1700))
| Modulus: | \(1700\) | |
| Conductor: | \(1700\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(40\) |
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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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\(\chi_{1700}(83,\cdot)\)
\(\chi_{1700}(127,\cdot)\)
\(\chi_{1700}(223,\cdot)\)
\(\chi_{1700}(247,\cdot)\)
\(\chi_{1700}(423,\cdot)\)
\(\chi_{1700}(467,\cdot)\)
\(\chi_{1700}(563,\cdot)\)
\(\chi_{1700}(587,\cdot)\)
\(\chi_{1700}(763,\cdot)\)
\(\chi_{1700}(903,\cdot)\)
\(\chi_{1700}(927,\cdot)\)
\(\chi_{1700}(1103,\cdot)\)
\(\chi_{1700}(1147,\cdot)\)
\(\chi_{1700}(1267,\cdot)\)
\(\chi_{1700}(1487,\cdot)\)
\(\chi_{1700}(1583,\cdot)\)
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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 ]
\((851,477,1601)\) → \((-1,e\left(\frac{9}{20}\right),e\left(\frac{1}{8}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 1700 }(587, a) \) |
\(1\) | \(1\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{21}{40}\right)\) |
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sage:chi.jacobi_sum(n)
