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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1800, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,30,50,57]))
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pari:[g,chi] = znchar(Mod(1013,1800))
| Modulus: | \(1800\) | |
| Conductor: | \(1800\) |
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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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\(\chi_{1800}(77,\cdot)\)
\(\chi_{1800}(173,\cdot)\)
\(\chi_{1800}(317,\cdot)\)
\(\chi_{1800}(437,\cdot)\)
\(\chi_{1800}(533,\cdot)\)
\(\chi_{1800}(653,\cdot)\)
\(\chi_{1800}(677,\cdot)\)
\(\chi_{1800}(797,\cdot)\)
\(\chi_{1800}(1013,\cdot)\)
\(\chi_{1800}(1037,\cdot)\)
\(\chi_{1800}(1253,\cdot)\)
\(\chi_{1800}(1373,\cdot)\)
\(\chi_{1800}(1397,\cdot)\)
\(\chi_{1800}(1517,\cdot)\)
\(\chi_{1800}(1613,\cdot)\)
\(\chi_{1800}(1733,\cdot)\)
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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 ]
\((1351,901,1001,577)\) → \((1,-1,e\left(\frac{5}{6}\right),e\left(\frac{19}{20}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 1800 }(1013, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{29}{30}\right)\) |
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sage:chi.jacobi_sum(n)
