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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(961, base_ring=CyclotomicField(62))
M = H._module
chi = DirichletCharacter(H, M([26]))
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pari:[g,chi] = znchar(Mod(125,961))
| Modulus: | \(961\) | |
| Conductor: | \(961\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(31\) |
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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)
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\(\chi_{961}(32,\cdot)\)
\(\chi_{961}(63,\cdot)\)
\(\chi_{961}(94,\cdot)\)
\(\chi_{961}(125,\cdot)\)
\(\chi_{961}(156,\cdot)\)
\(\chi_{961}(187,\cdot)\)
\(\chi_{961}(218,\cdot)\)
\(\chi_{961}(249,\cdot)\)
\(\chi_{961}(280,\cdot)\)
\(\chi_{961}(311,\cdot)\)
\(\chi_{961}(342,\cdot)\)
\(\chi_{961}(373,\cdot)\)
\(\chi_{961}(404,\cdot)\)
\(\chi_{961}(435,\cdot)\)
\(\chi_{961}(466,\cdot)\)
\(\chi_{961}(497,\cdot)\)
\(\chi_{961}(528,\cdot)\)
\(\chi_{961}(559,\cdot)\)
\(\chi_{961}(590,\cdot)\)
\(\chi_{961}(621,\cdot)\)
\(\chi_{961}(652,\cdot)\)
\(\chi_{961}(683,\cdot)\)
\(\chi_{961}(714,\cdot)\)
\(\chi_{961}(745,\cdot)\)
\(\chi_{961}(776,\cdot)\)
\(\chi_{961}(807,\cdot)\)
\(\chi_{961}(838,\cdot)\)
\(\chi_{961}(869,\cdot)\)
\(\chi_{961}(900,\cdot)\)
\(\chi_{961}(931,\cdot)\)
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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 ]
\(3\) → \(e\left(\frac{13}{31}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 961 }(125, a) \) |
\(1\) | \(1\) | \(e\left(\frac{21}{31}\right)\) | \(e\left(\frac{13}{31}\right)\) | \(e\left(\frac{11}{31}\right)\) | \(e\left(\frac{16}{31}\right)\) | \(e\left(\frac{3}{31}\right)\) | \(e\left(\frac{29}{31}\right)\) | \(e\left(\frac{1}{31}\right)\) | \(e\left(\frac{26}{31}\right)\) | \(e\left(\frac{6}{31}\right)\) | \(e\left(\frac{14}{31}\right)\) |
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sage:chi.jacobi_sum(n)
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sage:chi.gauss_sum(a)
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pari:znchargauss(g,chi,a)
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sage:chi.jacobi_sum(n)
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sage:chi.kloosterman_sum(a,b)
