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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1904, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([0,36,32,21]))
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pari:[g,chi] = znchar(Mod(1677,1904))
| Modulus: | \(1904\) | |
| Conductor: | \(1904\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(48\) |
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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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\(\chi_{1904}(37,\cdot)\)
\(\chi_{1904}(109,\cdot)\)
\(\chi_{1904}(165,\cdot)\)
\(\chi_{1904}(277,\cdot)\)
\(\chi_{1904}(317,\cdot)\)
\(\chi_{1904}(333,\cdot)\)
\(\chi_{1904}(653,\cdot)\)
\(\chi_{1904}(669,\cdot)\)
\(\chi_{1904}(709,\cdot)\)
\(\chi_{1904}(821,\cdot)\)
\(\chi_{1904}(877,\cdot)\)
\(\chi_{1904}(949,\cdot)\)
\(\chi_{1904}(1213,\cdot)\)
\(\chi_{1904}(1397,\cdot)\)
\(\chi_{1904}(1493,\cdot)\)
\(\chi_{1904}(1677,\cdot)\)
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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 ]
\((239,1429,1361,785)\) → \((1,-i,e\left(\frac{2}{3}\right),e\left(\frac{7}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(19\) | \(23\) | \(25\) | \(27\) |
| \( \chi_{ 1904 }(1677, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(1\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{1}{16}\right)\) |
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sage:chi.jacobi_sum(n)
