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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1547, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([32,44,21]))
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pari:[g,chi] = znchar(Mod(1320,1547))
| Modulus: | \(1547\) | |
| Conductor: | \(1547\) |
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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: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{1547}(184,\cdot)\)
\(\chi_{1547}(228,\cdot)\)
\(\chi_{1547}(275,\cdot)\)
\(\chi_{1547}(396,\cdot)\)
\(\chi_{1547}(401,\cdot)\)
\(\chi_{1547}(674,\cdot)\)
\(\chi_{1547}(760,\cdot)\)
\(\chi_{1547}(821,\cdot)\)
\(\chi_{1547}(856,\cdot)\)
\(\chi_{1547}(1047,\cdot)\)
\(\chi_{1547}(1129,\cdot)\)
\(\chi_{1547}(1185,\cdot)\)
\(\chi_{1547}(1306,\cdot)\)
\(\chi_{1547}(1320,\cdot)\)
\(\chi_{1547}(1397,\cdot)\)
\(\chi_{1547}(1502,\cdot)\)
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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 ]
\((885,834,547)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{11}{12}\right),e\left(\frac{7}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 1547 }(1320, a) \) |
\(1\) | \(1\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(-i\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{25}{48}\right)\) |
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sage:chi.jacobi_sum(n)
