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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1521, base_ring=CyclotomicField(156))
M = H._module
chi = DirichletCharacter(H, M([104,89]))
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pari:[g,chi] = znchar(Mod(1501,1521))
| Modulus: | \(1521\) | |
| Conductor: | \(1521\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(156\) |
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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: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
|
\(\chi_{1521}(7,\cdot)\)
\(\chi_{1521}(67,\cdot)\)
\(\chi_{1521}(76,\cdot)\)
\(\chi_{1521}(97,\cdot)\)
\(\chi_{1521}(124,\cdot)\)
\(\chi_{1521}(184,\cdot)\)
\(\chi_{1521}(193,\cdot)\)
\(\chi_{1521}(214,\cdot)\)
\(\chi_{1521}(241,\cdot)\)
\(\chi_{1521}(301,\cdot)\)
\(\chi_{1521}(310,\cdot)\)
\(\chi_{1521}(331,\cdot)\)
\(\chi_{1521}(358,\cdot)\)
\(\chi_{1521}(448,\cdot)\)
\(\chi_{1521}(475,\cdot)\)
\(\chi_{1521}(535,\cdot)\)
\(\chi_{1521}(544,\cdot)\)
\(\chi_{1521}(565,\cdot)\)
\(\chi_{1521}(592,\cdot)\)
\(\chi_{1521}(652,\cdot)\)
\(\chi_{1521}(661,\cdot)\)
\(\chi_{1521}(682,\cdot)\)
\(\chi_{1521}(709,\cdot)\)
\(\chi_{1521}(769,\cdot)\)
\(\chi_{1521}(778,\cdot)\)
\(\chi_{1521}(799,\cdot)\)
\(\chi_{1521}(886,\cdot)\)
\(\chi_{1521}(895,\cdot)\)
\(\chi_{1521}(916,\cdot)\)
\(\chi_{1521}(943,\cdot)\)
...
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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 ]
\((677,847)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{89}{156}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(14\) | \(16\) | \(17\) |
| \( \chi_{ 1521 }(1501, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{37}{156}\right)\) | \(e\left(\frac{37}{78}\right)\) | \(e\left(\frac{73}{156}\right)\) | \(e\left(\frac{37}{52}\right)\) | \(e\left(\frac{37}{52}\right)\) | \(e\left(\frac{55}{78}\right)\) | \(e\left(\frac{67}{156}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{23}{78}\right)\) |
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sage:chi.jacobi_sum(n)
