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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([20,12,5]))
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pari:[g,chi] = znchar(Mod(1539,1700))
| 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
|
| Minimal: | yes |
| Parity: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{1700}(19,\cdot)\)
\(\chi_{1700}(59,\cdot)\)
\(\chi_{1700}(179,\cdot)\)
\(\chi_{1700}(219,\cdot)\)
\(\chi_{1700}(359,\cdot)\)
\(\chi_{1700}(519,\cdot)\)
\(\chi_{1700}(559,\cdot)\)
\(\chi_{1700}(739,\cdot)\)
\(\chi_{1700}(859,\cdot)\)
\(\chi_{1700}(1039,\cdot)\)
\(\chi_{1700}(1079,\cdot)\)
\(\chi_{1700}(1239,\cdot)\)
\(\chi_{1700}(1379,\cdot)\)
\(\chi_{1700}(1419,\cdot)\)
\(\chi_{1700}(1539,\cdot)\)
\(\chi_{1700}(1579,\cdot)\)
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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 ]
\((851,477,1601)\) → \((-1,e\left(\frac{3}{10}\right),e\left(\frac{1}{8}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 1700 }(1539, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{9}{40}\right)\) |
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sage:chi.jacobi_sum(n)
