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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2001, base_ring=CyclotomicField(154))
M = H._module
chi = DirichletCharacter(H, M([77,49,88]))
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pari:[g,chi] = znchar(Mod(431,2001))
| Modulus: | \(2001\) | |
| Conductor: | \(2001\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(154\) |
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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)
|
\(\chi_{2001}(20,\cdot)\)
\(\chi_{2001}(53,\cdot)\)
\(\chi_{2001}(65,\cdot)\)
\(\chi_{2001}(74,\cdot)\)
\(\chi_{2001}(83,\cdot)\)
\(\chi_{2001}(107,\cdot)\)
\(\chi_{2001}(152,\cdot)\)
\(\chi_{2001}(194,\cdot)\)
\(\chi_{2001}(227,\cdot)\)
\(\chi_{2001}(281,\cdot)\)
\(\chi_{2001}(314,\cdot)\)
\(\chi_{2001}(401,\cdot)\)
\(\chi_{2001}(431,\cdot)\)
\(\chi_{2001}(458,\cdot)\)
\(\chi_{2001}(488,\cdot)\)
\(\chi_{2001}(500,\cdot)\)
\(\chi_{2001}(596,\cdot)\)
\(\chi_{2001}(605,\cdot)\)
\(\chi_{2001}(632,\cdot)\)
\(\chi_{2001}(674,\cdot)\)
\(\chi_{2001}(770,\cdot)\)
\(\chi_{2001}(779,\cdot)\)
\(\chi_{2001}(803,\cdot)\)
\(\chi_{2001}(848,\cdot)\)
\(\chi_{2001}(866,\cdot)\)
\(\chi_{2001}(893,\cdot)\)
\(\chi_{2001}(935,\cdot)\)
\(\chi_{2001}(953,\cdot)\)
\(\chi_{2001}(977,\cdot)\)
\(\chi_{2001}(980,\cdot)\)
...
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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 ]
\((668,1132,553)\) → \((-1,e\left(\frac{7}{22}\right),e\left(\frac{4}{7}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 2001 }(431, a) \) |
\(1\) | \(1\) | \(e\left(\frac{109}{154}\right)\) | \(e\left(\frac{32}{77}\right)\) | \(e\left(\frac{30}{77}\right)\) | \(e\left(\frac{139}{154}\right)\) | \(e\left(\frac{19}{154}\right)\) | \(e\left(\frac{15}{154}\right)\) | \(e\left(\frac{50}{77}\right)\) | \(e\left(\frac{57}{77}\right)\) | \(e\left(\frac{47}{77}\right)\) | \(e\left(\frac{64}{77}\right)\) |
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sage:chi.jacobi_sum(n)
