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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3400, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([20,20,36,35]))
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pari:[g,chi] = znchar(Mod(19,3400))
| Modulus: | \(3400\) | |
| Conductor: | \(3400\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
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| 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: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{3400}(19,\cdot)\)
\(\chi_{3400}(59,\cdot)\)
\(\chi_{3400}(179,\cdot)\)
\(\chi_{3400}(219,\cdot)\)
\(\chi_{3400}(739,\cdot)\)
\(\chi_{3400}(859,\cdot)\)
\(\chi_{3400}(1379,\cdot)\)
\(\chi_{3400}(1419,\cdot)\)
\(\chi_{3400}(1539,\cdot)\)
\(\chi_{3400}(1579,\cdot)\)
\(\chi_{3400}(2059,\cdot)\)
\(\chi_{3400}(2219,\cdot)\)
\(\chi_{3400}(2259,\cdot)\)
\(\chi_{3400}(2739,\cdot)\)
\(\chi_{3400}(2779,\cdot)\)
\(\chi_{3400}(2939,\cdot)\)
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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 ]
\((2551,1701,2177,1601)\) → \((-1,-1,e\left(\frac{9}{10}\right),e\left(\frac{7}{8}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 3400 }(19, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{27}{40}\right)\) |
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sage:chi.jacobi_sum(n)
