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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1573, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([14,33]))
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pari:[g,chi] = znchar(Mod(109,1573))
| Modulus: | \(1573\) | |
| Conductor: | \(1573\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(44\) |
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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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\(\chi_{1573}(21,\cdot)\)
\(\chi_{1573}(109,\cdot)\)
\(\chi_{1573}(164,\cdot)\)
\(\chi_{1573}(252,\cdot)\)
\(\chi_{1573}(307,\cdot)\)
\(\chi_{1573}(395,\cdot)\)
\(\chi_{1573}(450,\cdot)\)
\(\chi_{1573}(538,\cdot)\)
\(\chi_{1573}(593,\cdot)\)
\(\chi_{1573}(681,\cdot)\)
\(\chi_{1573}(736,\cdot)\)
\(\chi_{1573}(824,\cdot)\)
\(\chi_{1573}(879,\cdot)\)
\(\chi_{1573}(1022,\cdot)\)
\(\chi_{1573}(1110,\cdot)\)
\(\chi_{1573}(1165,\cdot)\)
\(\chi_{1573}(1253,\cdot)\)
\(\chi_{1573}(1308,\cdot)\)
\(\chi_{1573}(1396,\cdot)\)
\(\chi_{1573}(1539,\cdot)\)
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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 ]
\((365,1211)\) → \((e\left(\frac{7}{22}\right),-i)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \( \chi_{ 1573 }(109, a) \) |
\(1\) | \(1\) | \(e\left(\frac{3}{44}\right)\) | \(1\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(1\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{3}{22}\right)\) |
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sage:chi.jacobi_sum(n)
