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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6304, base_ring=CyclotomicField(392))
M = H._module
chi = DirichletCharacter(H, M([0,343,260]))
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pari:[g,chi] = znchar(Mod(109,6304))
| Modulus: | \(6304\) | |
| Conductor: | \(6304\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(392\) |
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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)
|
\(\chi_{6304}(109,\cdot)\)
\(\chi_{6304}(157,\cdot)\)
\(\chi_{6304}(173,\cdot)\)
\(\chi_{6304}(181,\cdot)\)
\(\chi_{6304}(261,\cdot)\)
\(\chi_{6304}(293,\cdot)\)
\(\chi_{6304}(309,\cdot)\)
\(\chi_{6304}(333,\cdot)\)
\(\chi_{6304}(341,\cdot)\)
\(\chi_{6304}(357,\cdot)\)
\(\chi_{6304}(365,\cdot)\)
\(\chi_{6304}(437,\cdot)\)
\(\chi_{6304}(501,\cdot)\)
\(\chi_{6304}(549,\cdot)\)
\(\chi_{6304}(557,\cdot)\)
\(\chi_{6304}(613,\cdot)\)
\(\chi_{6304}(653,\cdot)\)
\(\chi_{6304}(725,\cdot)\)
\(\chi_{6304}(765,\cdot)\)
\(\chi_{6304}(797,\cdot)\)
\(\chi_{6304}(813,\cdot)\)
\(\chi_{6304}(829,\cdot)\)
\(\chi_{6304}(853,\cdot)\)
\(\chi_{6304}(885,\cdot)\)
\(\chi_{6304}(909,\cdot)\)
\(\chi_{6304}(925,\cdot)\)
\(\chi_{6304}(957,\cdot)\)
\(\chi_{6304}(989,\cdot)\)
\(\chi_{6304}(1077,\cdot)\)
\(\chi_{6304}(1101,\cdot)\)
...
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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 ]
\((1183,3941,3745)\) → \((1,e\left(\frac{7}{8}\right),e\left(\frac{65}{98}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 6304 }(109, a) \) |
\(1\) | \(1\) | \(e\left(\frac{265}{392}\right)\) | \(e\left(\frac{355}{392}\right)\) | \(e\left(\frac{115}{196}\right)\) | \(e\left(\frac{69}{196}\right)\) | \(e\left(\frac{239}{392}\right)\) | \(e\left(\frac{277}{392}\right)\) | \(e\left(\frac{57}{98}\right)\) | \(e\left(\frac{47}{49}\right)\) | \(e\left(\frac{15}{56}\right)\) | \(e\left(\frac{103}{392}\right)\) |
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sage:chi.jacobi_sum(n)
