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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6300, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([15,20,9,5]))
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pari:[g,chi] = znchar(Mod(3139,6300))
| Modulus: | \(6300\) | |
| Conductor: | \(6300\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(30\) |
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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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\(\chi_{6300}(619,\cdot)\)
\(\chi_{6300}(859,\cdot)\)
\(\chi_{6300}(1879,\cdot)\)
\(\chi_{6300}(2119,\cdot)\)
\(\chi_{6300}(3139,\cdot)\)
\(\chi_{6300}(3379,\cdot)\)
\(\chi_{6300}(4639,\cdot)\)
\(\chi_{6300}(5659,\cdot)\)
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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 ]
\((3151,2801,3277,3601)\) → \((-1,e\left(\frac{2}{3}\right),e\left(\frac{3}{10}\right),e\left(\frac{1}{6}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 6300 }(3139, a) \) |
\(1\) | \(1\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{2}{3}\right)\) |
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sage:chi.jacobi_sum(n)
