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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3400, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([0,10,19,5]))
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pari:[g,chi] = znchar(Mod(13,3400))
| Modulus: | \(3400\) | |
| Conductor: | \(3400\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
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| Order: | \(20\) |
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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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\(\chi_{3400}(13,\cdot)\)
\(\chi_{3400}(837,\cdot)\)
\(\chi_{3400}(1373,\cdot)\)
\(\chi_{3400}(1517,\cdot)\)
\(\chi_{3400}(2053,\cdot)\)
\(\chi_{3400}(2197,\cdot)\)
\(\chi_{3400}(2733,\cdot)\)
\(\chi_{3400}(2877,\cdot)\)
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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 ]
\((2551,1701,2177,1601)\) → \((1,-1,e\left(\frac{19}{20}\right),i)\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 3400 }(13, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{2}{5}\right)\) | \(-1\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{13}{20}\right)\) |
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sage:chi.jacobi_sum(n)
