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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5175, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([20,51,30]))
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pari:[g,chi] = znchar(Mod(22,5175))
| Modulus: | \(5175\) | |
| Conductor: | \(5175\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(60\) |
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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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\(\chi_{5175}(22,\cdot)\)
\(\chi_{5175}(367,\cdot)\)
\(\chi_{5175}(988,\cdot)\)
\(\chi_{5175}(1402,\cdot)\)
\(\chi_{5175}(1678,\cdot)\)
\(\chi_{5175}(2023,\cdot)\)
\(\chi_{5175}(2092,\cdot)\)
\(\chi_{5175}(2437,\cdot)\)
\(\chi_{5175}(2713,\cdot)\)
\(\chi_{5175}(3058,\cdot)\)
\(\chi_{5175}(3127,\cdot)\)
\(\chi_{5175}(3472,\cdot)\)
\(\chi_{5175}(3748,\cdot)\)
\(\chi_{5175}(4162,\cdot)\)
\(\chi_{5175}(4783,\cdot)\)
\(\chi_{5175}(5128,\cdot)\)
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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 ]
\((4601,3727,2926)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{17}{20}\right),-1)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 5175 }(22, a) \) |
\(1\) | \(1\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{4}{5}\right)\) |
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sage:chi.jacobi_sum(n)
