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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1984, base_ring=CyclotomicField(80))
M = H._module
chi = DirichletCharacter(H, M([0,55,16]))
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pari:[g,chi] = znchar(Mod(605,1984))
| Modulus: | \(1984\) | |
| Conductor: | \(1984\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(80\) |
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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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\(\chi_{1984}(101,\cdot)\)
\(\chi_{1984}(109,\cdot)\)
\(\chi_{1984}(157,\cdot)\)
\(\chi_{1984}(221,\cdot)\)
\(\chi_{1984}(349,\cdot)\)
\(\chi_{1984}(357,\cdot)\)
\(\chi_{1984}(405,\cdot)\)
\(\chi_{1984}(469,\cdot)\)
\(\chi_{1984}(597,\cdot)\)
\(\chi_{1984}(605,\cdot)\)
\(\chi_{1984}(653,\cdot)\)
\(\chi_{1984}(717,\cdot)\)
\(\chi_{1984}(845,\cdot)\)
\(\chi_{1984}(853,\cdot)\)
\(\chi_{1984}(901,\cdot)\)
\(\chi_{1984}(965,\cdot)\)
\(\chi_{1984}(1093,\cdot)\)
\(\chi_{1984}(1101,\cdot)\)
\(\chi_{1984}(1149,\cdot)\)
\(\chi_{1984}(1213,\cdot)\)
\(\chi_{1984}(1341,\cdot)\)
\(\chi_{1984}(1349,\cdot)\)
\(\chi_{1984}(1397,\cdot)\)
\(\chi_{1984}(1461,\cdot)\)
\(\chi_{1984}(1589,\cdot)\)
\(\chi_{1984}(1597,\cdot)\)
\(\chi_{1984}(1645,\cdot)\)
\(\chi_{1984}(1709,\cdot)\)
\(\chi_{1984}(1837,\cdot)\)
\(\chi_{1984}(1845,\cdot)\)
...
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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 ]
\((63,1861,65)\) → \((1,e\left(\frac{11}{16}\right),e\left(\frac{1}{5}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 1984 }(605, a) \) |
\(1\) | \(1\) | \(e\left(\frac{21}{80}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{3}{80}\right)\) | \(e\left(\frac{41}{80}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{49}{80}\right)\) | \(e\left(\frac{59}{80}\right)\) |
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sage:chi.jacobi_sum(n)
