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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1764, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,7,40]))
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pari:[g,chi] = znchar(Mod(11,1764))
| Modulus: | \(1764\) | |
| Conductor: | \(1764\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(42\) |
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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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\(\chi_{1764}(11,\cdot)\)
\(\chi_{1764}(23,\cdot)\)
\(\chi_{1764}(515,\cdot)\)
\(\chi_{1764}(527,\cdot)\)
\(\chi_{1764}(767,\cdot)\)
\(\chi_{1764}(779,\cdot)\)
\(\chi_{1764}(1019,\cdot)\)
\(\chi_{1764}(1031,\cdot)\)
\(\chi_{1764}(1271,\cdot)\)
\(\chi_{1764}(1283,\cdot)\)
\(\chi_{1764}(1523,\cdot)\)
\(\chi_{1764}(1535,\cdot)\)
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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 ]
\((883,785,1081)\) → \((-1,e\left(\frac{1}{6}\right),e\left(\frac{20}{21}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 1764 }(11, a) \) |
\(1\) | \(1\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(-1\) | \(e\left(\frac{10}{21}\right)\) |
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sage:chi.jacobi_sum(n)
