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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1275, base_ring=CyclotomicField(80))
M = H._module
chi = DirichletCharacter(H, M([40,16,75]))
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pari:[g,chi] = znchar(Mod(941,1275))
| Modulus: | \(1275\) | |
| Conductor: | \(1275\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(80\) |
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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)
|
\(\chi_{1275}(11,\cdot)\)
\(\chi_{1275}(41,\cdot)\)
\(\chi_{1275}(56,\cdot)\)
\(\chi_{1275}(71,\cdot)\)
\(\chi_{1275}(116,\cdot)\)
\(\chi_{1275}(131,\cdot)\)
\(\chi_{1275}(146,\cdot)\)
\(\chi_{1275}(266,\cdot)\)
\(\chi_{1275}(296,\cdot)\)
\(\chi_{1275}(311,\cdot)\)
\(\chi_{1275}(371,\cdot)\)
\(\chi_{1275}(386,\cdot)\)
\(\chi_{1275}(431,\cdot)\)
\(\chi_{1275}(521,\cdot)\)
\(\chi_{1275}(566,\cdot)\)
\(\chi_{1275}(581,\cdot)\)
\(\chi_{1275}(641,\cdot)\)
\(\chi_{1275}(656,\cdot)\)
\(\chi_{1275}(686,\cdot)\)
\(\chi_{1275}(806,\cdot)\)
\(\chi_{1275}(821,\cdot)\)
\(\chi_{1275}(836,\cdot)\)
\(\chi_{1275}(881,\cdot)\)
\(\chi_{1275}(896,\cdot)\)
\(\chi_{1275}(911,\cdot)\)
\(\chi_{1275}(941,\cdot)\)
\(\chi_{1275}(1031,\cdot)\)
\(\chi_{1275}(1061,\cdot)\)
\(\chi_{1275}(1091,\cdot)\)
\(\chi_{1275}(1136,\cdot)\)
...
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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 ]
\((851,52,751)\) → \((-1,e\left(\frac{1}{5}\right),e\left(\frac{15}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(19\) | \(22\) |
| \( \chi_{ 1275 }(941, a) \) |
\(1\) | \(1\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{21}{80}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{11}{80}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{7}{80}\right)\) |
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sage:chi.jacobi_sum(n)
