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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(603, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([22,28]))
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pari:[g,chi] = znchar(Mod(157,603))
| Modulus: | \(603\) | |
| Conductor: | \(603\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(33\) |
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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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\(\chi_{603}(49,\cdot)\)
\(\chi_{603}(88,\cdot)\)
\(\chi_{603}(121,\cdot)\)
\(\chi_{603}(157,\cdot)\)
\(\chi_{603}(160,\cdot)\)
\(\chi_{603}(169,\cdot)\)
\(\chi_{603}(205,\cdot)\)
\(\chi_{603}(211,\cdot)\)
\(\chi_{603}(220,\cdot)\)
\(\chi_{603}(274,\cdot)\)
\(\chi_{603}(304,\cdot)\)
\(\chi_{603}(400,\cdot)\)
\(\chi_{603}(418,\cdot)\)
\(\chi_{603}(457,\cdot)\)
\(\chi_{603}(502,\cdot)\)
\(\chi_{603}(508,\cdot)\)
\(\chi_{603}(529,\cdot)\)
\(\chi_{603}(553,\cdot)\)
\(\chi_{603}(583,\cdot)\)
\(\chi_{603}(592,\cdot)\)
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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 ]
\((470,136)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{14}{33}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 603 }(157, a) \) |
\(1\) | \(1\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{1}{33}\right)\) |
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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)
