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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1216, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([0,21,16]))
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pari:[g,chi] = znchar(Mod(45,1216))
| Modulus: | \(1216\) | |
| Conductor: | \(1216\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(48\) |
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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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\(\chi_{1216}(45,\cdot)\)
\(\chi_{1216}(125,\cdot)\)
\(\chi_{1216}(197,\cdot)\)
\(\chi_{1216}(277,\cdot)\)
\(\chi_{1216}(349,\cdot)\)
\(\chi_{1216}(429,\cdot)\)
\(\chi_{1216}(501,\cdot)\)
\(\chi_{1216}(581,\cdot)\)
\(\chi_{1216}(653,\cdot)\)
\(\chi_{1216}(733,\cdot)\)
\(\chi_{1216}(805,\cdot)\)
\(\chi_{1216}(885,\cdot)\)
\(\chi_{1216}(957,\cdot)\)
\(\chi_{1216}(1037,\cdot)\)
\(\chi_{1216}(1109,\cdot)\)
\(\chi_{1216}(1189,\cdot)\)
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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 ]
\((191,837,705)\) → \((1,e\left(\frac{7}{16}\right),e\left(\frac{1}{3}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(21\) | \(23\) |
| \( \chi_{ 1216 }(45, a) \) |
\(1\) | \(1\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{19}{24}\right)\) |
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sage:chi.jacobi_sum(n)
