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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3675, base_ring=CyclotomicField(420))
M = H._module
chi = DirichletCharacter(H, M([210,273,20]))
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pari:[g,chi] = znchar(Mod(842,3675))
| Modulus: | \(3675\) | |
| Conductor: | \(3675\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(420\) |
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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
|
| Minimal: | yes |
| Parity: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
|
\(\chi_{3675}(2,\cdot)\)
\(\chi_{3675}(23,\cdot)\)
\(\chi_{3675}(53,\cdot)\)
\(\chi_{3675}(137,\cdot)\)
\(\chi_{3675}(158,\cdot)\)
\(\chi_{3675}(212,\cdot)\)
\(\chi_{3675}(233,\cdot)\)
\(\chi_{3675}(242,\cdot)\)
\(\chi_{3675}(317,\cdot)\)
\(\chi_{3675}(338,\cdot)\)
\(\chi_{3675}(347,\cdot)\)
\(\chi_{3675}(452,\cdot)\)
\(\chi_{3675}(473,\cdot)\)
\(\chi_{3675}(527,\cdot)\)
\(\chi_{3675}(548,\cdot)\)
\(\chi_{3675}(578,\cdot)\)
\(\chi_{3675}(653,\cdot)\)
\(\chi_{3675}(662,\cdot)\)
\(\chi_{3675}(683,\cdot)\)
\(\chi_{3675}(737,\cdot)\)
\(\chi_{3675}(758,\cdot)\)
\(\chi_{3675}(767,\cdot)\)
\(\chi_{3675}(788,\cdot)\)
\(\chi_{3675}(842,\cdot)\)
\(\chi_{3675}(872,\cdot)\)
\(\chi_{3675}(947,\cdot)\)
\(\chi_{3675}(977,\cdot)\)
\(\chi_{3675}(1052,\cdot)\)
\(\chi_{3675}(1073,\cdot)\)
\(\chi_{3675}(1103,\cdot)\)
...
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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 ]
\((1226,1177,2551)\) → \((-1,e\left(\frac{13}{20}\right),e\left(\frac{1}{21}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) | \(22\) | \(23\) |
| \( \chi_{ 3675 }(842, a) \) |
\(1\) | \(1\) | \(e\left(\frac{163}{420}\right)\) | \(e\left(\frac{163}{210}\right)\) | \(e\left(\frac{23}{140}\right)\) | \(e\left(\frac{169}{210}\right)\) | \(e\left(\frac{129}{140}\right)\) | \(e\left(\frac{58}{105}\right)\) | \(e\left(\frac{59}{420}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{27}{140}\right)\) | \(e\left(\frac{193}{420}\right)\) |
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sage:chi.jacobi_sum(n)
