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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([288,715]))
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pari:[g,chi] = znchar(Mod(97,2057))
| 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: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
|
\(\chi_{2057}(5,\cdot)\)
\(\chi_{2057}(14,\cdot)\)
\(\chi_{2057}(20,\cdot)\)
\(\chi_{2057}(31,\cdot)\)
\(\chi_{2057}(37,\cdot)\)
\(\chi_{2057}(48,\cdot)\)
\(\chi_{2057}(58,\cdot)\)
\(\chi_{2057}(71,\cdot)\)
\(\chi_{2057}(75,\cdot)\)
\(\chi_{2057}(80,\cdot)\)
\(\chi_{2057}(82,\cdot)\)
\(\chi_{2057}(91,\cdot)\)
\(\chi_{2057}(92,\cdot)\)
\(\chi_{2057}(97,\cdot)\)
\(\chi_{2057}(108,\cdot)\)
\(\chi_{2057}(113,\cdot)\)
\(\chi_{2057}(114,\cdot)\)
\(\chi_{2057}(125,\cdot)\)
\(\chi_{2057}(126,\cdot)\)
\(\chi_{2057}(141,\cdot)\)
\(\chi_{2057}(146,\cdot)\)
\(\chi_{2057}(147,\cdot)\)
\(\chi_{2057}(158,\cdot)\)
\(\chi_{2057}(159,\cdot)\)
\(\chi_{2057}(163,\cdot)\)
\(\chi_{2057}(180,\cdot)\)
\(\chi_{2057}(181,\cdot)\)
\(\chi_{2057}(190,\cdot)\)
\(\chi_{2057}(192,\cdot)\)
\(\chi_{2057}(201,\cdot)\)
...
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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 ]
\((970,122)\) → \((e\left(\frac{18}{55}\right),e\left(\frac{13}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \( \chi_{ 2057 }(97, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{309}{440}\right)\) | \(e\left(\frac{49}{80}\right)\) | \(e\left(\frac{89}{220}\right)\) | \(e\left(\frac{247}{880}\right)\) | \(e\left(\frac{277}{880}\right)\) | \(e\left(\frac{201}{880}\right)\) | \(e\left(\frac{47}{440}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{173}{176}\right)\) | \(e\left(\frac{3}{176}\right)\) |
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sage:chi.jacobi_sum(n)
