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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2205, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([35,21,33]))
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pari:[g,chi] = znchar(Mod(1679,2205))
| Modulus: | \(2205\) | |
| Conductor: | \(2205\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(42\) |
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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)
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\(\chi_{2205}(104,\cdot)\)
\(\chi_{2205}(209,\cdot)\)
\(\chi_{2205}(419,\cdot)\)
\(\chi_{2205}(524,\cdot)\)
\(\chi_{2205}(839,\cdot)\)
\(\chi_{2205}(1049,\cdot)\)
\(\chi_{2205}(1154,\cdot)\)
\(\chi_{2205}(1364,\cdot)\)
\(\chi_{2205}(1679,\cdot)\)
\(\chi_{2205}(1784,\cdot)\)
\(\chi_{2205}(1994,\cdot)\)
\(\chi_{2205}(2099,\cdot)\)
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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 ]
\((1226,442,1081)\) → \((e\left(\frac{5}{6}\right),-1,e\left(\frac{11}{14}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) | \(22\) | \(23\) |
| \( \chi_{ 2205 }(1679, a) \) |
\(1\) | \(1\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(-1\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{11}{21}\right)\) |
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sage:chi.jacobi_sum(n)
