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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2475, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([5,3,21]))
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pari:[g,chi] = znchar(Mod(29,2475))
| Modulus: | \(2475\) | |
| Conductor: | \(2475\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
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| 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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\(\chi_{2475}(29,\cdot)\)
\(\chi_{2475}(464,\cdot)\)
\(\chi_{2475}(569,\cdot)\)
\(\chi_{2475}(1289,\cdot)\)
\(\chi_{2475}(1634,\cdot)\)
\(\chi_{2475}(1679,\cdot)\)
\(\chi_{2475}(2219,\cdot)\)
\(\chi_{2475}(2459,\cdot)\)
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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 ]
\((551,2377,2026)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{1}{10}\right),e\left(\frac{7}{10}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) | \(23\) |
| \( \chi_{ 2475 }(29, a) \) |
\(1\) | \(1\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{14}{15}\right)\) |
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sage:chi.jacobi_sum(n)
