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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1925, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([18,20,3]))
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pari:[g,chi] = znchar(Mod(46,1925))
| Modulus: | \(1925\) | |
| Conductor: | \(1925\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
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| Order: | \(30\) |
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sage:chi.multiplicative_order()
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pari:charorder(g,chi)
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| 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: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{1925}(46,\cdot)\)
\(\chi_{1925}(261,\cdot)\)
\(\chi_{1925}(536,\cdot)\)
\(\chi_{1925}(1381,\cdot)\)
\(\chi_{1925}(1591,\cdot)\)
\(\chi_{1925}(1656,\cdot)\)
\(\chi_{1925}(1696,\cdot)\)
\(\chi_{1925}(1866,\cdot)\)
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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 ]
\((1002,276,1751)\) → \((e\left(\frac{3}{5}\right),e\left(\frac{2}{3}\right),e\left(\frac{1}{10}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(12\) | \(13\) | \(16\) | \(17\) |
| \( \chi_{ 1925 }(46, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(-1\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{11}{30}\right)\) |
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sage:chi.jacobi_sum(n)
