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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4332, base_ring=CyclotomicField(38))
M = H._module
chi = DirichletCharacter(H, M([19,19,12]))
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pari:[g,chi] = znchar(Mod(1787,4332))
| Modulus: | \(4332\) | |
| Conductor: | \(4332\) |
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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
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| Minimal: | yes |
| Parity: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{4332}(191,\cdot)\)
\(\chi_{4332}(419,\cdot)\)
\(\chi_{4332}(647,\cdot)\)
\(\chi_{4332}(875,\cdot)\)
\(\chi_{4332}(1103,\cdot)\)
\(\chi_{4332}(1331,\cdot)\)
\(\chi_{4332}(1559,\cdot)\)
\(\chi_{4332}(1787,\cdot)\)
\(\chi_{4332}(2015,\cdot)\)
\(\chi_{4332}(2243,\cdot)\)
\(\chi_{4332}(2471,\cdot)\)
\(\chi_{4332}(2699,\cdot)\)
\(\chi_{4332}(2927,\cdot)\)
\(\chi_{4332}(3155,\cdot)\)
\(\chi_{4332}(3383,\cdot)\)
\(\chi_{4332}(3839,\cdot)\)
\(\chi_{4332}(4067,\cdot)\)
\(\chi_{4332}(4295,\cdot)\)
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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 ]
\((2167,1445,3973)\) → \((-1,-1,e\left(\frac{6}{19}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 4332 }(1787, a) \) |
\(1\) | \(1\) | \(e\left(\frac{29}{38}\right)\) | \(e\left(\frac{33}{38}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{33}{38}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{33}{38}\right)\) | \(e\left(\frac{7}{38}\right)\) | \(e\left(\frac{12}{19}\right)\) |
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sage:chi.jacobi_sum(n)
