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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2001, base_ring=CyclotomicField(308))
M = H._module
chi = DirichletCharacter(H, M([154,224,55]))
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pari:[g,chi] = znchar(Mod(1337,2001))
| Modulus: | \(2001\) | |
| Conductor: | \(2001\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(308\) |
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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)
|
\(\chi_{2001}(2,\cdot)\)
\(\chi_{2001}(8,\cdot)\)
\(\chi_{2001}(26,\cdot)\)
\(\chi_{2001}(32,\cdot)\)
\(\chi_{2001}(50,\cdot)\)
\(\chi_{2001}(77,\cdot)\)
\(\chi_{2001}(95,\cdot)\)
\(\chi_{2001}(98,\cdot)\)
\(\chi_{2001}(101,\cdot)\)
\(\chi_{2001}(119,\cdot)\)
\(\chi_{2001}(131,\cdot)\)
\(\chi_{2001}(164,\cdot)\)
\(\chi_{2001}(188,\cdot)\)
\(\chi_{2001}(200,\cdot)\)
\(\chi_{2001}(242,\cdot)\)
\(\chi_{2001}(269,\cdot)\)
\(\chi_{2001}(305,\cdot)\)
\(\chi_{2001}(308,\cdot)\)
\(\chi_{2001}(311,\cdot)\)
\(\chi_{2001}(317,\cdot)\)
\(\chi_{2001}(338,\cdot)\)
\(\chi_{2001}(374,\cdot)\)
\(\chi_{2001}(380,\cdot)\)
\(\chi_{2001}(395,\cdot)\)
\(\chi_{2001}(404,\cdot)\)
\(\chi_{2001}(416,\cdot)\)
\(\chi_{2001}(443,\cdot)\)
\(\chi_{2001}(446,\cdot)\)
\(\chi_{2001}(449,\cdot)\)
\(\chi_{2001}(485,\cdot)\)
...
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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 ]
\((668,1132,553)\) → \((-1,e\left(\frac{8}{11}\right),e\left(\frac{5}{28}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 2001 }(1337, a) \) |
\(1\) | \(1\) | \(e\left(\frac{41}{308}\right)\) | \(e\left(\frac{41}{154}\right)\) | \(e\left(\frac{12}{77}\right)\) | \(e\left(\frac{74}{77}\right)\) | \(e\left(\frac{123}{308}\right)\) | \(e\left(\frac{89}{308}\right)\) | \(e\left(\frac{157}{308}\right)\) | \(e\left(\frac{61}{154}\right)\) | \(e\left(\frac{29}{308}\right)\) | \(e\left(\frac{41}{77}\right)\) |
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sage:chi.jacobi_sum(n)
