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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1925, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([13,10,14]))
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pari:[g,chi] = znchar(Mod(1217,1925))
| Modulus: | \(1925\) | |
| Conductor: | \(1925\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
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| Order: | \(20\) |
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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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\(\chi_{1925}(13,\cdot)\)
\(\chi_{1925}(272,\cdot)\)
\(\chi_{1925}(503,\cdot)\)
\(\chi_{1925}(937,\cdot)\)
\(\chi_{1925}(1217,\cdot)\)
\(\chi_{1925}(1623,\cdot)\)
\(\chi_{1925}(1777,\cdot)\)
\(\chi_{1925}(1833,\cdot)\)
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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 ]
\((1002,276,1751)\) → \((e\left(\frac{13}{20}\right),-1,e\left(\frac{7}{10}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(12\) | \(13\) | \(16\) | \(17\) |
| \( \chi_{ 1925 }(1217, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(1\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(i\) |
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sage:chi.jacobi_sum(n)
