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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8664, base_ring=CyclotomicField(38))
M = H._module
chi = DirichletCharacter(H, M([19,19,19,34]))
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pari:[g,chi] = znchar(Mod(1331,8664))
| Modulus: | \(8664\) | |
| Conductor: | \(8664\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(38\) |
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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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\(\chi_{8664}(419,\cdot)\)
\(\chi_{8664}(875,\cdot)\)
\(\chi_{8664}(1331,\cdot)\)
\(\chi_{8664}(1787,\cdot)\)
\(\chi_{8664}(2243,\cdot)\)
\(\chi_{8664}(2699,\cdot)\)
\(\chi_{8664}(3155,\cdot)\)
\(\chi_{8664}(4067,\cdot)\)
\(\chi_{8664}(4523,\cdot)\)
\(\chi_{8664}(4979,\cdot)\)
\(\chi_{8664}(5435,\cdot)\)
\(\chi_{8664}(5891,\cdot)\)
\(\chi_{8664}(6347,\cdot)\)
\(\chi_{8664}(6803,\cdot)\)
\(\chi_{8664}(7259,\cdot)\)
\(\chi_{8664}(7715,\cdot)\)
\(\chi_{8664}(8171,\cdot)\)
\(\chi_{8664}(8627,\cdot)\)
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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 ]
\((2167,4333,5777,8305)\) → \((-1,-1,-1,e\left(\frac{17}{19}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 8664 }(1331, a) \) |
\(1\) | \(1\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{27}{38}\right)\) | \(e\left(\frac{29}{38}\right)\) | \(e\left(\frac{33}{38}\right)\) | \(e\left(\frac{27}{38}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{23}{38}\right)\) | \(e\left(\frac{11}{38}\right)\) |
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sage:chi.jacobi_sum(n)
