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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1309, base_ring=CyclotomicField(80))
M = H._module
chi = DirichletCharacter(H, M([40,8,45]))
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pari:[g,chi] = znchar(Mod(167,1309))
| Modulus: | \(1309\) | |
| Conductor: | \(1309\) |
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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: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
|
\(\chi_{1309}(6,\cdot)\)
\(\chi_{1309}(41,\cdot)\)
\(\chi_{1309}(62,\cdot)\)
\(\chi_{1309}(90,\cdot)\)
\(\chi_{1309}(139,\cdot)\)
\(\chi_{1309}(160,\cdot)\)
\(\chi_{1309}(167,\cdot)\)
\(\chi_{1309}(216,\cdot)\)
\(\chi_{1309}(244,\cdot)\)
\(\chi_{1309}(398,\cdot)\)
\(\chi_{1309}(447,\cdot)\)
\(\chi_{1309}(503,\cdot)\)
\(\chi_{1309}(524,\cdot)\)
\(\chi_{1309}(601,\cdot)\)
\(\chi_{1309}(622,\cdot)\)
\(\chi_{1309}(657,\cdot)\)
\(\chi_{1309}(734,\cdot)\)
\(\chi_{1309}(755,\cdot)\)
\(\chi_{1309}(776,\cdot)\)
\(\chi_{1309}(811,\cdot)\)
\(\chi_{1309}(853,\cdot)\)
\(\chi_{1309}(860,\cdot)\)
\(\chi_{1309}(930,\cdot)\)
\(\chi_{1309}(1014,\cdot)\)
\(\chi_{1309}(1042,\cdot)\)
\(\chi_{1309}(1091,\cdot)\)
\(\chi_{1309}(1119,\cdot)\)
\(\chi_{1309}(1161,\cdot)\)
\(\chi_{1309}(1168,\cdot)\)
\(\chi_{1309}(1196,\cdot)\)
...
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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 ]
\((1123,596,309)\) → \((-1,e\left(\frac{1}{10}\right),e\left(\frac{9}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(12\) | \(13\) |
| \( \chi_{ 1309 }(167, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{69}{80}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{57}{80}\right)\) | \(e\left(\frac{67}{80}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{17}{20}\right)\) |
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sage:chi.jacobi_sum(n)
