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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2600, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,30,24,55]))
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pari:[g,chi] = znchar(Mod(1931,2600))
| Modulus: | \(2600\) | |
| Conductor: | \(2600\) |
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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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\(\chi_{2600}(11,\cdot)\)
\(\chi_{2600}(171,\cdot)\)
\(\chi_{2600}(331,\cdot)\)
\(\chi_{2600}(371,\cdot)\)
\(\chi_{2600}(531,\cdot)\)
\(\chi_{2600}(691,\cdot)\)
\(\chi_{2600}(891,\cdot)\)
\(\chi_{2600}(1211,\cdot)\)
\(\chi_{2600}(1371,\cdot)\)
\(\chi_{2600}(1411,\cdot)\)
\(\chi_{2600}(1571,\cdot)\)
\(\chi_{2600}(1731,\cdot)\)
\(\chi_{2600}(1891,\cdot)\)
\(\chi_{2600}(1931,\cdot)\)
\(\chi_{2600}(2091,\cdot)\)
\(\chi_{2600}(2411,\cdot)\)
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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 ]
\((1951,1301,1977,1601)\) → \((-1,-1,e\left(\frac{2}{5}\right),e\left(\frac{11}{12}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 2600 }(1931, a) \) |
\(1\) | \(1\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{29}{30}\right)\) |
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sage:chi.jacobi_sum(n)
