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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(225, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([50,21]))
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pari:[g,chi] = znchar(Mod(203,225))
| Modulus: | \(225\) | |
| Conductor: | \(225\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
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| Order: | \(60\) |
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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: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{225}(2,\cdot)\)
\(\chi_{225}(23,\cdot)\)
\(\chi_{225}(38,\cdot)\)
\(\chi_{225}(47,\cdot)\)
\(\chi_{225}(77,\cdot)\)
\(\chi_{225}(83,\cdot)\)
\(\chi_{225}(92,\cdot)\)
\(\chi_{225}(113,\cdot)\)
\(\chi_{225}(122,\cdot)\)
\(\chi_{225}(128,\cdot)\)
\(\chi_{225}(137,\cdot)\)
\(\chi_{225}(158,\cdot)\)
\(\chi_{225}(167,\cdot)\)
\(\chi_{225}(173,\cdot)\)
\(\chi_{225}(203,\cdot)\)
\(\chi_{225}(212,\cdot)\)
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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 ]
\((101,127)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{7}{20}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 225 }(203, a) \) |
\(1\) | \(1\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) |
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sage:chi.jacobi_sum(n)
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sage:chi.gauss_sum(a)
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pari:znchargauss(g,chi,a)
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sage:chi.jacobi_sum(n)
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sage:chi.kloosterman_sum(a,b)
