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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(441, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([28,31]))
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pari:[g,chi] = znchar(Mod(241,441))
| Modulus: | \(441\) | |
| Conductor: | \(441\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
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| Order: | \(42\) |
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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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\(\chi_{441}(40,\cdot)\)
\(\chi_{441}(52,\cdot)\)
\(\chi_{441}(103,\cdot)\)
\(\chi_{441}(115,\cdot)\)
\(\chi_{441}(229,\cdot)\)
\(\chi_{441}(241,\cdot)\)
\(\chi_{441}(292,\cdot)\)
\(\chi_{441}(304,\cdot)\)
\(\chi_{441}(355,\cdot)\)
\(\chi_{441}(367,\cdot)\)
\(\chi_{441}(418,\cdot)\)
\(\chi_{441}(430,\cdot)\)
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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 ]
\((344,199)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{31}{42}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 441 }(241, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{5}{6}\right)\) |
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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)
