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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2057, base_ring=CyclotomicField(880))
M = H._module
chi = DirichletCharacter(H, M([152,495]))
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pari:[g,chi] = znchar(Mod(116,2057))
| Modulus: | \(2057\) | |
| Conductor: | \(2057\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(880\) |
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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
|
| Minimal: | yes |
| Parity: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
|
\(\chi_{2057}(6,\cdot)\)
\(\chi_{2057}(7,\cdot)\)
\(\chi_{2057}(24,\cdot)\)
\(\chi_{2057}(28,\cdot)\)
\(\chi_{2057}(29,\cdot)\)
\(\chi_{2057}(39,\cdot)\)
\(\chi_{2057}(41,\cdot)\)
\(\chi_{2057}(46,\cdot)\)
\(\chi_{2057}(57,\cdot)\)
\(\chi_{2057}(61,\cdot)\)
\(\chi_{2057}(62,\cdot)\)
\(\chi_{2057}(63,\cdot)\)
\(\chi_{2057}(73,\cdot)\)
\(\chi_{2057}(74,\cdot)\)
\(\chi_{2057}(79,\cdot)\)
\(\chi_{2057}(90,\cdot)\)
\(\chi_{2057}(95,\cdot)\)
\(\chi_{2057}(96,\cdot)\)
\(\chi_{2057}(105,\cdot)\)
\(\chi_{2057}(107,\cdot)\)
\(\chi_{2057}(116,\cdot)\)
\(\chi_{2057}(129,\cdot)\)
\(\chi_{2057}(139,\cdot)\)
\(\chi_{2057}(150,\cdot)\)
\(\chi_{2057}(156,\cdot)\)
\(\chi_{2057}(160,\cdot)\)
\(\chi_{2057}(167,\cdot)\)
\(\chi_{2057}(173,\cdot)\)
\(\chi_{2057}(182,\cdot)\)
\(\chi_{2057}(184,\cdot)\)
...
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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 ]
\((970,122)\) → \((e\left(\frac{19}{110}\right),e\left(\frac{9}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \( \chi_{ 2057 }(116, a) \) |
\(1\) | \(1\) | \(e\left(\frac{21}{440}\right)\) | \(e\left(\frac{61}{80}\right)\) | \(e\left(\frac{21}{220}\right)\) | \(e\left(\frac{523}{880}\right)\) | \(e\left(\frac{713}{880}\right)\) | \(e\left(\frac{349}{880}\right)\) | \(e\left(\frac{63}{440}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{113}{176}\right)\) | \(e\left(\frac{151}{176}\right)\) |
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sage:chi.jacobi_sum(n)
