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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1920, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([0,31,16,16]))
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pari:[g,chi] = znchar(Mod(1229,1920))
| Modulus: | \(1920\) | |
| Conductor: | \(1920\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(32\) |
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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: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{1920}(29,\cdot)\)
\(\chi_{1920}(149,\cdot)\)
\(\chi_{1920}(269,\cdot)\)
\(\chi_{1920}(389,\cdot)\)
\(\chi_{1920}(509,\cdot)\)
\(\chi_{1920}(629,\cdot)\)
\(\chi_{1920}(749,\cdot)\)
\(\chi_{1920}(869,\cdot)\)
\(\chi_{1920}(989,\cdot)\)
\(\chi_{1920}(1109,\cdot)\)
\(\chi_{1920}(1229,\cdot)\)
\(\chi_{1920}(1349,\cdot)\)
\(\chi_{1920}(1469,\cdot)\)
\(\chi_{1920}(1589,\cdot)\)
\(\chi_{1920}(1709,\cdot)\)
\(\chi_{1920}(1829,\cdot)\)
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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 ]
\((511,901,641,1537)\) → \((1,e\left(\frac{31}{32}\right),-1,-1)\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 1920 }(1229, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{9}{32}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{21}{32}\right)\) | \(-i\) | \(e\left(\frac{23}{32}\right)\) | \(e\left(\frac{9}{16}\right)\) |
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sage:chi.jacobi_sum(n)
