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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1760, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([0,35,20,24]))
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pari:[g,chi] = znchar(Mod(1549,1760))
| Modulus: | \(1760\) | |
| Conductor: | \(1760\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(40\) |
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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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\(\chi_{1760}(69,\cdot)\)
\(\chi_{1760}(229,\cdot)\)
\(\chi_{1760}(269,\cdot)\)
\(\chi_{1760}(389,\cdot)\)
\(\chi_{1760}(509,\cdot)\)
\(\chi_{1760}(669,\cdot)\)
\(\chi_{1760}(709,\cdot)\)
\(\chi_{1760}(829,\cdot)\)
\(\chi_{1760}(949,\cdot)\)
\(\chi_{1760}(1109,\cdot)\)
\(\chi_{1760}(1149,\cdot)\)
\(\chi_{1760}(1269,\cdot)\)
\(\chi_{1760}(1389,\cdot)\)
\(\chi_{1760}(1549,\cdot)\)
\(\chi_{1760}(1589,\cdot)\)
\(\chi_{1760}(1709,\cdot)\)
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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 ]
\((991,1541,1057,321)\) → \((1,e\left(\frac{7}{8}\right),-1,e\left(\frac{3}{5}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 1760 }(1549, a) \) |
\(1\) | \(1\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(-i\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{33}{40}\right)\) |
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sage:chi.jacobi_sum(n)
