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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1404, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,10,33]))
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pari:[g,chi] = znchar(Mod(59,1404))
| Modulus: | \(1404\) | |
| Conductor: | \(1404\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
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| Order: | \(36\) |
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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: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{1404}(11,\cdot)\)
\(\chi_{1404}(59,\cdot)\)
\(\chi_{1404}(119,\cdot)\)
\(\chi_{1404}(383,\cdot)\)
\(\chi_{1404}(479,\cdot)\)
\(\chi_{1404}(527,\cdot)\)
\(\chi_{1404}(587,\cdot)\)
\(\chi_{1404}(851,\cdot)\)
\(\chi_{1404}(947,\cdot)\)
\(\chi_{1404}(995,\cdot)\)
\(\chi_{1404}(1055,\cdot)\)
\(\chi_{1404}(1319,\cdot)\)
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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 ]
\((703,677,1081)\) → \((-1,e\left(\frac{5}{18}\right),e\left(\frac{11}{12}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 1404 }(59, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(1\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{2}{3}\right)\) |
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sage:chi.jacobi_sum(n)
