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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2541, base_ring=CyclotomicField(330))
M = H._module
chi = DirichletCharacter(H, M([165,220,51]))
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pari:[g,chi] = znchar(Mod(1481,2541))
| Modulus: | \(2541\) | |
| Conductor: | \(2541\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(330\) |
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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)
|
\(\chi_{2541}(2,\cdot)\)
\(\chi_{2541}(74,\cdot)\)
\(\chi_{2541}(95,\cdot)\)
\(\chi_{2541}(107,\cdot)\)
\(\chi_{2541}(116,\cdot)\)
\(\chi_{2541}(128,\cdot)\)
\(\chi_{2541}(149,\cdot)\)
\(\chi_{2541}(200,\cdot)\)
\(\chi_{2541}(305,\cdot)\)
\(\chi_{2541}(326,\cdot)\)
\(\chi_{2541}(338,\cdot)\)
\(\chi_{2541}(347,\cdot)\)
\(\chi_{2541}(359,\cdot)\)
\(\chi_{2541}(380,\cdot)\)
\(\chi_{2541}(431,\cdot)\)
\(\chi_{2541}(464,\cdot)\)
\(\chi_{2541}(536,\cdot)\)
\(\chi_{2541}(557,\cdot)\)
\(\chi_{2541}(569,\cdot)\)
\(\chi_{2541}(590,\cdot)\)
\(\chi_{2541}(611,\cdot)\)
\(\chi_{2541}(662,\cdot)\)
\(\chi_{2541}(695,\cdot)\)
\(\chi_{2541}(767,\cdot)\)
\(\chi_{2541}(788,\cdot)\)
\(\chi_{2541}(800,\cdot)\)
\(\chi_{2541}(809,\cdot)\)
\(\chi_{2541}(821,\cdot)\)
\(\chi_{2541}(842,\cdot)\)
\(\chi_{2541}(893,\cdot)\)
...
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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 ]
\((848,1816,2059)\) → \((-1,e\left(\frac{2}{3}\right),e\left(\frac{17}{110}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(13\) | \(16\) | \(17\) | \(19\) | \(20\) |
| \( \chi_{ 2541 }(1481, a) \) |
\(1\) | \(1\) | \(e\left(\frac{163}{165}\right)\) | \(e\left(\frac{161}{165}\right)\) | \(e\left(\frac{89}{330}\right)\) | \(e\left(\frac{53}{55}\right)\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{67}{110}\right)\) | \(e\left(\frac{157}{165}\right)\) | \(e\left(\frac{122}{165}\right)\) | \(e\left(\frac{53}{330}\right)\) | \(e\left(\frac{27}{110}\right)\) |
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sage:chi.jacobi_sum(n)
