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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1960, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([0,14,7,26]))
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pari:[g,chi] = znchar(Mod(1637,1960))
| Modulus: | \(1960\) | |
| Conductor: | \(1960\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(28\) |
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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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\(\chi_{1960}(13,\cdot)\)
\(\chi_{1960}(237,\cdot)\)
\(\chi_{1960}(517,\cdot)\)
\(\chi_{1960}(573,\cdot)\)
\(\chi_{1960}(797,\cdot)\)
\(\chi_{1960}(853,\cdot)\)
\(\chi_{1960}(1133,\cdot)\)
\(\chi_{1960}(1357,\cdot)\)
\(\chi_{1960}(1413,\cdot)\)
\(\chi_{1960}(1637,\cdot)\)
\(\chi_{1960}(1693,\cdot)\)
\(\chi_{1960}(1917,\cdot)\)
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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 ]
\((1471,981,1177,1081)\) → \((1,-1,i,e\left(\frac{13}{14}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) |
| \( \chi_{ 1960 }(1637, a) \) |
\(1\) | \(1\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(-1\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(-1\) |
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sage:chi.jacobi_sum(n)
