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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2475, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([20,3,54]))
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pari:[g,chi] = znchar(Mod(1777,2475))
| Modulus: | \(2475\) | |
| Conductor: | \(2475\) |
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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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\(\chi_{2475}(13,\cdot)\)
\(\chi_{2475}(112,\cdot)\)
\(\chi_{2475}(547,\cdot)\)
\(\chi_{2475}(733,\cdot)\)
\(\chi_{2475}(778,\cdot)\)
\(\chi_{2475}(952,\cdot)\)
\(\chi_{2475}(1348,\cdot)\)
\(\chi_{2475}(1372,\cdot)\)
\(\chi_{2475}(1492,\cdot)\)
\(\chi_{2475}(1663,\cdot)\)
\(\chi_{2475}(1762,\cdot)\)
\(\chi_{2475}(1777,\cdot)\)
\(\chi_{2475}(2173,\cdot)\)
\(\chi_{2475}(2317,\cdot)\)
\(\chi_{2475}(2383,\cdot)\)
\(\chi_{2475}(2428,\cdot)\)
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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 ]
\((551,2377,2026)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{1}{20}\right),e\left(\frac{9}{10}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) | \(23\) |
| \( \chi_{ 2475 }(1777, a) \) |
\(1\) | \(1\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(-i\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{13}{60}\right)\) |
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sage:chi.jacobi_sum(n)
