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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2057, base_ring=CyclotomicField(176))
M = H._module
chi = DirichletCharacter(H, M([144,143]))
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pari:[g,chi] = znchar(Mod(12,2057))
| Modulus: | \(2057\) | |
| Conductor: | \(2057\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(176\) |
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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)
|
\(\chi_{2057}(12,\cdot)\)
\(\chi_{2057}(23,\cdot)\)
\(\chi_{2057}(45,\cdot)\)
\(\chi_{2057}(56,\cdot)\)
\(\chi_{2057}(78,\cdot)\)
\(\chi_{2057}(133,\cdot)\)
\(\chi_{2057}(177,\cdot)\)
\(\chi_{2057}(199,\cdot)\)
\(\chi_{2057}(210,\cdot)\)
\(\chi_{2057}(232,\cdot)\)
\(\chi_{2057}(265,\cdot)\)
\(\chi_{2057}(309,\cdot)\)
\(\chi_{2057}(320,\cdot)\)
\(\chi_{2057}(386,\cdot)\)
\(\chi_{2057}(397,\cdot)\)
\(\chi_{2057}(419,\cdot)\)
\(\chi_{2057}(430,\cdot)\)
\(\chi_{2057}(452,\cdot)\)
\(\chi_{2057}(496,\cdot)\)
\(\chi_{2057}(507,\cdot)\)
\(\chi_{2057}(551,\cdot)\)
\(\chi_{2057}(573,\cdot)\)
\(\chi_{2057}(584,\cdot)\)
\(\chi_{2057}(617,\cdot)\)
\(\chi_{2057}(639,\cdot)\)
\(\chi_{2057}(683,\cdot)\)
\(\chi_{2057}(694,\cdot)\)
\(\chi_{2057}(738,\cdot)\)
\(\chi_{2057}(760,\cdot)\)
\(\chi_{2057}(771,\cdot)\)
...
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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 ]
\((970,122)\) → \((e\left(\frac{9}{11}\right),e\left(\frac{13}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \( \chi_{ 2057 }(12, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{17}{88}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{107}{176}\right)\) | \(e\left(\frac{1}{176}\right)\) | \(e\left(\frac{117}{176}\right)\) | \(e\left(\frac{51}{88}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{141}{176}\right)\) | \(e\left(\frac{35}{176}\right)\) |
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sage:chi.jacobi_sum(n)
