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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1156, base_ring=CyclotomicField(272))
M = H._module
chi = DirichletCharacter(H, M([136,239]))
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pari:[g,chi] = znchar(Mod(1043,1156))
| Modulus: | \(1156\) | |
| Conductor: | \(1156\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(272\) |
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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)
|
\(\chi_{1156}(3,\cdot)\)
\(\chi_{1156}(7,\cdot)\)
\(\chi_{1156}(11,\cdot)\)
\(\chi_{1156}(23,\cdot)\)
\(\chi_{1156}(27,\cdot)\)
\(\chi_{1156}(31,\cdot)\)
\(\chi_{1156}(39,\cdot)\)
\(\chi_{1156}(63,\cdot)\)
\(\chi_{1156}(71,\cdot)\)
\(\chi_{1156}(79,\cdot)\)
\(\chi_{1156}(91,\cdot)\)
\(\chi_{1156}(95,\cdot)\)
\(\chi_{1156}(99,\cdot)\)
\(\chi_{1156}(107,\cdot)\)
\(\chi_{1156}(139,\cdot)\)
\(\chi_{1156}(143,\cdot)\)
\(\chi_{1156}(147,\cdot)\)
\(\chi_{1156}(159,\cdot)\)
\(\chi_{1156}(163,\cdot)\)
\(\chi_{1156}(167,\cdot)\)
\(\chi_{1156}(175,\cdot)\)
\(\chi_{1156}(199,\cdot)\)
\(\chi_{1156}(207,\cdot)\)
\(\chi_{1156}(211,\cdot)\)
\(\chi_{1156}(215,\cdot)\)
\(\chi_{1156}(227,\cdot)\)
\(\chi_{1156}(231,\cdot)\)
\(\chi_{1156}(235,\cdot)\)
\(\chi_{1156}(243,\cdot)\)
\(\chi_{1156}(267,\cdot)\)
...
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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 ]
\((579,581)\) → \((-1,e\left(\frac{239}{272}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(19\) | \(21\) | \(23\) |
| \( \chi_{ 1156 }(1043, a) \) |
\(1\) | \(1\) | \(e\left(\frac{103}{272}\right)\) | \(e\left(\frac{59}{272}\right)\) | \(e\left(\frac{189}{272}\right)\) | \(e\left(\frac{103}{136}\right)\) | \(e\left(\frac{193}{272}\right)\) | \(e\left(\frac{15}{68}\right)\) | \(e\left(\frac{81}{136}\right)\) | \(e\left(\frac{109}{136}\right)\) | \(e\left(\frac{5}{68}\right)\) | \(e\left(\frac{121}{272}\right)\) |
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sage:chi.jacobi_sum(n)
