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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(33957, base_ring=CyclotomicField(1470))
M = H._module
chi = DirichletCharacter(H, M([980,1415,441]))
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pari:[g,chi] = znchar(Mod(13912,33957))
| Modulus: | \(33957\) | |
| Conductor: | \(33957\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(1470\) |
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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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\(\chi_{33957}(40,\cdot)\)
\(\chi_{33957}(52,\cdot)\)
\(\chi_{33957}(292,\cdot)\)
\(\chi_{33957}(304,\cdot)\)
\(\chi_{33957}(481,\cdot)\)
\(\chi_{33957}(556,\cdot)\)
\(\chi_{33957}(733,\cdot)\)
\(\chi_{33957}(745,\cdot)\)
\(\chi_{33957}(871,\cdot)\)
\(\chi_{33957}(985,\cdot)\)
\(\chi_{33957}(997,\cdot)\)
\(\chi_{33957}(1174,\cdot)\)
\(\chi_{33957}(1249,\cdot)\)
\(\chi_{33957}(1300,\cdot)\)
\(\chi_{33957}(1426,\cdot)\)
\(\chi_{33957}(1438,\cdot)\)
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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 ]
\((18866,14752,24697)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{283}{294}\right),e\left(\frac{3}{10}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(13\) | \(16\) | \(17\) | \(19\) | \(20\) |
| \( \chi_{ 33957 }(13912, a) \) |
\(1\) | \(1\) | \(e\left(\frac{347}{490}\right)\) | \(e\left(\frac{102}{245}\right)\) | \(e\left(\frac{659}{1470}\right)\) | \(e\left(\frac{61}{490}\right)\) | \(e\left(\frac{23}{147}\right)\) | \(e\left(\frac{713}{735}\right)\) | \(e\left(\frac{204}{245}\right)\) | \(e\left(\frac{562}{735}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{1271}{1470}\right)\) |
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sage:chi.jacobi_sum(n)
