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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(373527, base_ring=CyclotomicField(16170))
M = H._module
chi = DirichletCharacter(H, M([5390,7865,4557]))
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pari:[g,chi] = znchar(Mod(1300,373527))
| Modulus: | \(373527\) | |
| Conductor: | \(373527\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(16170\) |
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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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\(\chi_{373527}(52,\cdot)\)
\(\chi_{373527}(292,\cdot)\)
\(\chi_{373527}(304,\cdot)\)
\(\chi_{373527}(556,\cdot)\)
\(\chi_{373527}(733,\cdot)\)
\(\chi_{373527}(745,\cdot)\)
\(\chi_{373527}(871,\cdot)\)
\(\chi_{373527}(985,\cdot)\)
\(\chi_{373527}(997,\cdot)\)
\(\chi_{373527}(1174,\cdot)\)
\(\chi_{373527}(1249,\cdot)\)
\(\chi_{373527}(1300,\cdot)\)
\(\chi_{373527}(1426,\cdot)\)
\(\chi_{373527}(1438,\cdot)\)
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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 ]
\((290522,286408,126568)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{143}{294}\right),e\left(\frac{31}{110}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(13\) | \(16\) | \(17\) | \(19\) | \(20\) |
| \( \chi_{ 373527 }(1300, a) \) |
\(1\) | \(1\) | \(e\left(\frac{5259}{5390}\right)\) | \(e\left(\frac{2564}{2695}\right)\) | \(e\left(\frac{10133}{16170}\right)\) | \(e\left(\frac{4997}{5390}\right)\) | \(e\left(\frac{974}{1617}\right)\) | \(e\left(\frac{6086}{8085}\right)\) | \(e\left(\frac{2433}{2695}\right)\) | \(e\left(\frac{7834}{8085}\right)\) | \(e\left(\frac{92}{165}\right)\) | \(e\left(\frac{9347}{16170}\right)\) |
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sage:chi.jacobi_sum(n)
