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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2415, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([22,33,22,26]))
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pari:[g,chi] = znchar(Mod(2183,2415))
| Modulus: | \(2415\) | |
| Conductor: | \(2415\) |
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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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\(\chi_{2415}(83,\cdot)\)
\(\chi_{2415}(272,\cdot)\)
\(\chi_{2415}(293,\cdot)\)
\(\chi_{2415}(398,\cdot)\)
\(\chi_{2415}(503,\cdot)\)
\(\chi_{2415}(608,\cdot)\)
\(\chi_{2415}(797,\cdot)\)
\(\chi_{2415}(902,\cdot)\)
\(\chi_{2415}(1217,\cdot)\)
\(\chi_{2415}(1238,\cdot)\)
\(\chi_{2415}(1322,\cdot)\)
\(\chi_{2415}(1532,\cdot)\)
\(\chi_{2415}(1742,\cdot)\)
\(\chi_{2415}(1763,\cdot)\)
\(\chi_{2415}(1847,\cdot)\)
\(\chi_{2415}(1868,\cdot)\)
\(\chi_{2415}(1952,\cdot)\)
\(\chi_{2415}(2057,\cdot)\)
\(\chi_{2415}(2183,\cdot)\)
\(\chi_{2415}(2288,\cdot)\)
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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 ]
\((806,967,346,1891)\) → \((-1,-i,-1,e\left(\frac{13}{22}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) | \(22\) | \(26\) |
| \( \chi_{ 2415 }(2183, a) \) |
\(1\) | \(1\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(i\) | \(e\left(\frac{5}{11}\right)\) |
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sage:chi.jacobi_sum(n)
