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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1309, base_ring=CyclotomicField(240))
M = H._module
chi = DirichletCharacter(H, M([160,216,75]))
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pari:[g,chi] = znchar(Mod(39,1309))
| Modulus: | \(1309\) | |
| Conductor: | \(1309\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(240\) |
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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)
|
\(\chi_{1309}(39,\cdot)\)
\(\chi_{1309}(46,\cdot)\)
\(\chi_{1309}(74,\cdot)\)
\(\chi_{1309}(79,\cdot)\)
\(\chi_{1309}(95,\cdot)\)
\(\chi_{1309}(107,\cdot)\)
\(\chi_{1309}(116,\cdot)\)
\(\chi_{1309}(156,\cdot)\)
\(\chi_{1309}(184,\cdot)\)
\(\chi_{1309}(193,\cdot)\)
\(\chi_{1309}(226,\cdot)\)
\(\chi_{1309}(228,\cdot)\)
\(\chi_{1309}(233,\cdot)\)
\(\chi_{1309}(249,\cdot)\)
\(\chi_{1309}(261,\cdot)\)
\(\chi_{1309}(277,\cdot)\)
\(\chi_{1309}(282,\cdot)\)
\(\chi_{1309}(303,\cdot)\)
\(\chi_{1309}(326,\cdot)\)
\(\chi_{1309}(347,\cdot)\)
\(\chi_{1309}(354,\cdot)\)
\(\chi_{1309}(380,\cdot)\)
\(\chi_{1309}(403,\cdot)\)
\(\chi_{1309}(415,\cdot)\)
\(\chi_{1309}(431,\cdot)\)
\(\chi_{1309}(436,\cdot)\)
\(\chi_{1309}(464,\cdot)\)
\(\chi_{1309}(513,\cdot)\)
\(\chi_{1309}(534,\cdot)\)
\(\chi_{1309}(541,\cdot)\)
...
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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 ]
\((1123,596,309)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{9}{10}\right),e\left(\frac{5}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(12\) | \(13\) |
| \( \chi_{ 1309 }(39, a) \) |
\(1\) | \(1\) | \(e\left(\frac{73}{120}\right)\) | \(e\left(\frac{43}{240}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{119}{240}\right)\) | \(e\left(\frac{63}{80}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{43}{120}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{3}{20}\right)\) |
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sage:chi.jacobi_sum(n)
