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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(196, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,29]))
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pari:[g,chi] = znchar(Mod(103,196))
| Modulus: | \(196\) | |
| Conductor: | \(196\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
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| Order: | \(42\) |
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sage:chi.multiplicative_order()
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pari:charorder(g,chi)
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| Real: | no |
| Primitive: | yes |
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sage:chi.is_primitive()
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pari:#znconreyconductor(g,chi)==1
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| Minimal: | yes |
| Parity: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{196}(3,\cdot)\)
\(\chi_{196}(47,\cdot)\)
\(\chi_{196}(59,\cdot)\)
\(\chi_{196}(75,\cdot)\)
\(\chi_{196}(87,\cdot)\)
\(\chi_{196}(103,\cdot)\)
\(\chi_{196}(115,\cdot)\)
\(\chi_{196}(131,\cdot)\)
\(\chi_{196}(143,\cdot)\)
\(\chi_{196}(159,\cdot)\)
\(\chi_{196}(171,\cdot)\)
\(\chi_{196}(187,\cdot)\)
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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 ]
\((99,101)\) → \((-1,e\left(\frac{29}{42}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
| \( \chi_{ 196 }(103, a) \) |
\(1\) | \(1\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{1}{21}\right)\) |
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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)
