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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3751, base_ring=CyclotomicField(330))
M = H._module
chi = DirichletCharacter(H, M([303,253]))
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pari:[g,chi] = znchar(Mod(1344,3751))
| Modulus: | \(3751\) | |
| Conductor: | \(3751\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(330\) |
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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)
|
\(\chi_{3751}(17,\cdot)\)
\(\chi_{3751}(24,\cdot)\)
\(\chi_{3751}(52,\cdot)\)
\(\chi_{3751}(74,\cdot)\)
\(\chi_{3751}(106,\cdot)\)
\(\chi_{3751}(270,\cdot)\)
\(\chi_{3751}(321,\cdot)\)
\(\chi_{3751}(358,\cdot)\)
\(\chi_{3751}(365,\cdot)\)
\(\chi_{3751}(393,\cdot)\)
\(\chi_{3751}(415,\cdot)\)
\(\chi_{3751}(447,\cdot)\)
\(\chi_{3751}(611,\cdot)\)
\(\chi_{3751}(623,\cdot)\)
\(\chi_{3751}(662,\cdot)\)
\(\chi_{3751}(706,\cdot)\)
\(\chi_{3751}(734,\cdot)\)
\(\chi_{3751}(756,\cdot)\)
\(\chi_{3751}(788,\cdot)\)
\(\chi_{3751}(952,\cdot)\)
\(\chi_{3751}(964,\cdot)\)
\(\chi_{3751}(1003,\cdot)\)
\(\chi_{3751}(1040,\cdot)\)
\(\chi_{3751}(1047,\cdot)\)
\(\chi_{3751}(1075,\cdot)\)
\(\chi_{3751}(1097,\cdot)\)
\(\chi_{3751}(1293,\cdot)\)
\(\chi_{3751}(1305,\cdot)\)
\(\chi_{3751}(1344,\cdot)\)
\(\chi_{3751}(1381,\cdot)\)
...
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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 ]
\((2543,2421)\) → \((e\left(\frac{101}{110}\right),e\left(\frac{23}{30}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \( \chi_{ 3751 }(1344, a) \) |
\(1\) | \(1\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{46}{165}\right)\) | \(e\left(\frac{146}{165}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{197}{330}\right)\) | \(e\left(\frac{67}{330}\right)\) |
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sage:chi.jacobi_sum(n)
