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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1815, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([11,11,17]))
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pari:[g,chi] = znchar(Mod(1649,1815))
| Modulus: | \(1815\) | |
| Conductor: | \(1815\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(22\) |
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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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\(\chi_{1815}(164,\cdot)\)
\(\chi_{1815}(329,\cdot)\)
\(\chi_{1815}(494,\cdot)\)
\(\chi_{1815}(659,\cdot)\)
\(\chi_{1815}(824,\cdot)\)
\(\chi_{1815}(989,\cdot)\)
\(\chi_{1815}(1154,\cdot)\)
\(\chi_{1815}(1319,\cdot)\)
\(\chi_{1815}(1484,\cdot)\)
\(\chi_{1815}(1649,\cdot)\)
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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 ]
\((1211,727,1696)\) → \((-1,-1,e\left(\frac{17}{22}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) | \(23\) |
| \( \chi_{ 1815 }(1649, a) \) |
\(1\) | \(1\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{1}{11}\right)\) |
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sage:chi.jacobi_sum(n)
