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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2349, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([46,27]))
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pari:[g,chi] = znchar(Mod(1159,2349))
| Modulus: | \(2349\) | |
| Conductor: | \(2349\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(54\) |
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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
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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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\(\chi_{2349}(115,\cdot)\)
\(\chi_{2349}(202,\cdot)\)
\(\chi_{2349}(376,\cdot)\)
\(\chi_{2349}(463,\cdot)\)
\(\chi_{2349}(637,\cdot)\)
\(\chi_{2349}(724,\cdot)\)
\(\chi_{2349}(898,\cdot)\)
\(\chi_{2349}(985,\cdot)\)
\(\chi_{2349}(1159,\cdot)\)
\(\chi_{2349}(1246,\cdot)\)
\(\chi_{2349}(1420,\cdot)\)
\(\chi_{2349}(1507,\cdot)\)
\(\chi_{2349}(1681,\cdot)\)
\(\chi_{2349}(1768,\cdot)\)
\(\chi_{2349}(1942,\cdot)\)
\(\chi_{2349}(2029,\cdot)\)
\(\chi_{2349}(2203,\cdot)\)
\(\chi_{2349}(2290,\cdot)\)
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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 ]
\((407,1945)\) → \((e\left(\frac{23}{27}\right),-1)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 2349 }(1159, a) \) |
\(1\) | \(1\) | \(e\left(\frac{19}{54}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{31}{54}\right)\) | \(e\left(\frac{22}{27}\right)\) | \(e\left(\frac{53}{54}\right)\) | \(e\left(\frac{11}{27}\right)\) |
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sage:chi.jacobi_sum(n)
