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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(85600, base_ring=CyclotomicField(2120))
M = H._module
chi = DirichletCharacter(H, M([1060,1855,1908,1560]))
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pari:[g,chi] = znchar(Mod(19,85600))
| Modulus: | \(85600\) | |
| Conductor: | \(85600\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(2120\) |
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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: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{85600}(19,\cdot)\)
\(\chi_{85600}(539,\cdot)\)
\(\chi_{85600}(579,\cdot)\)
\(\chi_{85600}(779,\cdot)\)
\(\chi_{85600}(859,\cdot)\)
\(\chi_{85600}(939,\cdot)\)
\(\chi_{85600}(979,\cdot)\)
\(\chi_{85600}(1019,\cdot)\)
\(\chi_{85600}(1139,\cdot)\)
\(\chi_{85600}(1219,\cdot)\)
\(\chi_{85600}(1539,\cdot)\)
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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 ]
\((26751,32101,82177,16801)\) → \((-1,e\left(\frac{7}{8}\right),e\left(\frac{9}{10}\right),e\left(\frac{39}{53}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 85600 }(19, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{1981}{2120}\right)\) | \(e\left(\frac{83}{212}\right)\) | \(e\left(\frac{921}{1060}\right)\) | \(e\left(\frac{983}{2120}\right)\) | \(e\left(\frac{1117}{2120}\right)\) | \(e\left(\frac{143}{265}\right)\) | \(e\left(\frac{469}{2120}\right)\) | \(e\left(\frac{691}{2120}\right)\) | \(e\left(\frac{289}{1060}\right)\) | \(e\left(\frac{1703}{2120}\right)\) |
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sage:chi.jacobi_sum(n)
