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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1300, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,39,20]))
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pari:[g,chi] = znchar(Mod(367,1300))
| Modulus: | \(1300\) | |
| Conductor: | \(1300\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(60\) |
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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: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{1300}(3,\cdot)\)
\(\chi_{1300}(87,\cdot)\)
\(\chi_{1300}(263,\cdot)\)
\(\chi_{1300}(347,\cdot)\)
\(\chi_{1300}(367,\cdot)\)
\(\chi_{1300}(503,\cdot)\)
\(\chi_{1300}(523,\cdot)\)
\(\chi_{1300}(627,\cdot)\)
\(\chi_{1300}(763,\cdot)\)
\(\chi_{1300}(783,\cdot)\)
\(\chi_{1300}(867,\cdot)\)
\(\chi_{1300}(887,\cdot)\)
\(\chi_{1300}(1023,\cdot)\)
\(\chi_{1300}(1127,\cdot)\)
\(\chi_{1300}(1147,\cdot)\)
\(\chi_{1300}(1283,\cdot)\)
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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 ]
\((651,677,301)\) → \((-1,e\left(\frac{13}{20}\right),e\left(\frac{1}{3}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 1300 }(367, a) \) |
\(1\) | \(1\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{19}{30}\right)\) |
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sage:chi.jacobi_sum(n)
