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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([2695,6765,11172]))
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pari:[g,chi] = znchar(Mod(20,373527))
| 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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\(\chi_{373527}(20,\cdot)\)
\(\chi_{373527}(104,\cdot)\)
\(\chi_{373527}(335,\cdot)\)
\(\chi_{373527}(356,\cdot)\)
\(\chi_{373527}(482,\cdot)\)
\(\chi_{373527}(713,\cdot)\)
\(\chi_{373527}(797,\cdot)\)
\(\chi_{373527}(839,\cdot)\)
\(\chi_{373527}(1280,\cdot)\)
\(\chi_{373527}(1301,\cdot)\)
\(\chi_{373527}(1406,\cdot)\)
\(\chi_{373527}(1490,\cdot)\)
\(\chi_{373527}(1532,\cdot)\)
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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 ]
\((290522,286408,126568)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{41}{98}\right),e\left(\frac{38}{55}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(13\) | \(16\) | \(17\) | \(19\) | \(20\) |
| \( \chi_{ 373527 }(20, a) \) |
\(1\) | \(1\) | \(e\left(\frac{337}{16170}\right)\) | \(e\left(\frac{337}{8085}\right)\) | \(e\left(\frac{754}{8085}\right)\) | \(e\left(\frac{337}{5390}\right)\) | \(e\left(\frac{123}{1078}\right)\) | \(e\left(\frac{5657}{16170}\right)\) | \(e\left(\frac{674}{8085}\right)\) | \(e\left(\frac{2193}{2695}\right)\) | \(e\left(\frac{93}{110}\right)\) | \(e\left(\frac{1091}{8085}\right)\) |
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sage:chi.jacobi_sum(n)
