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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1575, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([50,39,50]))
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pari:[g,chi] = znchar(Mod(1517,1575))
| Modulus: | \(1575\) | |
| Conductor: | \(1575\) |
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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: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{1575}(38,\cdot)\)
\(\chi_{1575}(227,\cdot)\)
\(\chi_{1575}(353,\cdot)\)
\(\chi_{1575}(383,\cdot)\)
\(\chi_{1575}(542,\cdot)\)
\(\chi_{1575}(572,\cdot)\)
\(\chi_{1575}(698,\cdot)\)
\(\chi_{1575}(887,\cdot)\)
\(\chi_{1575}(983,\cdot)\)
\(\chi_{1575}(1013,\cdot)\)
\(\chi_{1575}(1172,\cdot)\)
\(\chi_{1575}(1202,\cdot)\)
\(\chi_{1575}(1298,\cdot)\)
\(\chi_{1575}(1328,\cdot)\)
\(\chi_{1575}(1487,\cdot)\)
\(\chi_{1575}(1517,\cdot)\)
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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 ]
\((1226,127,451)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{13}{20}\right),e\left(\frac{5}{6}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) | \(22\) | \(23\) |
| \( \chi_{ 1575 }(1517, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{59}{60}\right)\) |
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sage:chi.jacobi_sum(n)
