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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2668, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([22,12,33]))
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pari:[g,chi] = znchar(Mod(307,2668))
| Modulus: | \(2668\) | |
| Conductor: | \(2668\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(44\) |
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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: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{2668}(75,\cdot)\)
\(\chi_{2668}(215,\cdot)\)
\(\chi_{2668}(307,\cdot)\)
\(\chi_{2668}(331,\cdot)\)
\(\chi_{2668}(423,\cdot)\)
\(\chi_{2668}(679,\cdot)\)
\(\chi_{2668}(771,\cdot)\)
\(\chi_{2668}(795,\cdot)\)
\(\chi_{2668}(887,\cdot)\)
\(\chi_{2668}(1143,\cdot)\)
\(\chi_{2668}(1235,\cdot)\)
\(\chi_{2668}(1375,\cdot)\)
\(\chi_{2668}(1467,\cdot)\)
\(\chi_{2668}(2187,\cdot)\)
\(\chi_{2668}(2279,\cdot)\)
\(\chi_{2668}(2303,\cdot)\)
\(\chi_{2668}(2395,\cdot)\)
\(\chi_{2668}(2419,\cdot)\)
\(\chi_{2668}(2511,\cdot)\)
\(\chi_{2668}(2651,\cdot)\)
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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 ]
\((1335,465,553)\) → \((-1,e\left(\frac{3}{11}\right),-i)\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 2668 }(307, a) \) |
\(1\) | \(1\) | \(e\left(\frac{27}{44}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{13}{44}\right)\) |
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sage:chi.jacobi_sum(n)
