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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2480, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,15,45,52]))
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pari:[g,chi] = znchar(Mod(1723,2480))
| Modulus: | \(2480\) | |
| Conductor: | \(2480\) |
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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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\(\chi_{2480}(227,\cdot)\)
\(\chi_{2480}(307,\cdot)\)
\(\chi_{2480}(443,\cdot)\)
\(\chi_{2480}(547,\cdot)\)
\(\chi_{2480}(603,\cdot)\)
\(\chi_{2480}(627,\cdot)\)
\(\chi_{2480}(763,\cdot)\)
\(\chi_{2480}(1187,\cdot)\)
\(\chi_{2480}(1347,\cdot)\)
\(\chi_{2480}(1507,\cdot)\)
\(\chi_{2480}(1723,\cdot)\)
\(\chi_{2480}(1963,\cdot)\)
\(\chi_{2480}(2043,\cdot)\)
\(\chi_{2480}(2283,\cdot)\)
\(\chi_{2480}(2363,\cdot)\)
\(\chi_{2480}(2467,\cdot)\)
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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 ]
\((1551,1861,497,561)\) → \((-1,i,-i,e\left(\frac{13}{15}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 2480 }(1723, a) \) |
\(1\) | \(1\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) |
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sage:chi.jacobi_sum(n)
